# Riser Interference and VIV Amplification in Tandem Configuration

Emmanuel Fontaine, Jean Paul Morel, Yves-Marie Scolan, T. Rippol

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Emmanuel Fontaine, Jean Paul Morel, Yves-Marie Scolan, T. Rippol. Riser Interference and VIV Amplification in Tandem Configuration. International Journal of Offshore and Polar Engineering (IJOPE), 2006, 16 (1), pp.33-40; ISOPE-06-16-1-033. hal-00458047

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# Riser Interference and VIV Amplification in Tandem Configuration

E. Fontaine<fnref n="1" />, J.P. Morel<fnref n="2" />, Y.M. Scolan<fnref n="3" />, T. Rippol<fnref n="4" />

## ABSTRACT

Wake Induced Oscillations (WIO) and amplification of Vortex Induced Vibration (VIV) are studied for two risers in tandem configuration. Criticality of riser interference is first investigated in term of non dimensional numbers for different types of risers. Theoretical, numerical and experimental approaches are then used simultaneously to define similitude rules and to investigate scale effects with respect to Reynolds number. Finally, experimental results demonstrating amplification of vortex induced vibration are given.

**KEY WORDS:** Riser interference, clashing, VIV amplification.

## NOMENCLATURE

* $\rho$ water density

* $V(z)$ current profile

* $(x,y,z)$ Coordinate system ($z$ is vertical downward)

* $V_0$ effective current velocity

* $\psi_n(z)$ modal shape associated to structural mode number $n$

* $f$ first natural frequency (and associated pulsation $\omega$)

* $i$ subscript to identify riser number

* $D$ riser diameter ($D_i$ for riser number $i$)

* $m$ riser mass per unit length in air including internal fluid (also defined as $m_i$)

* $\mu$ reduced mass ($\mu = 4m/\pi\rho D^2$)

* $ma$ fluid added mass per unit length ($ma = \pi\rho D_i^2/4$)

* $\zeta_{x, drag}$ hydrodynamic damping due to drag (in $x$ and $y$ directions respectively)

* $\zeta_i$ structural damping of riser $i$

* $\zeta$ total damping (structural and hydrodynamic)

* $Ur, V_R$ reduced velocity ($Ur = U/\omega D, V_R = 2\pi Ur$)

* $C_Y$ Cauchy number ($C_Y = (Ur)^2/\mu$)

* $Sc$ Scruton number (also noted $A_R$)

* $X_i$ position of riser $i$

* $L$ initial spacing between risers

* $Cd$ drag coefficient (also $Cd_i$)

* $\tau$ non dimensional time ($\tau = \omega t$)

* $K$ structural stiffness of first mode

* $K_c$ contact stiffness between risers

## INTRODUCTION

Hydrodynamic riser interference is becoming a crucial issue as offshore technology moves into deep and ultra-deep water depth. The main concern about riser interference is to be able to predict the risk of collision between closely spaced risers, and eventually to design risers able to withstand possible impact events between each other. Besides turbulence buffeting, hydrodynamic interference between risers can be classified in two categories, namely Wake Induced Oscillation (WIO) and Vortex Induced Vibration (VIV).

Wake Induced Oscillation (WIO), also referred to as galloping, is a classical type of instability, see e.g. Blevins (1977, 1990, 2004), Axisa (2001) for a simplified description of the mechanism underlining the phenomena. Basically, spatial variations of the time averaged hydrodynamic coefficients (lift and drag) can create such instability at high reduced velocity. WIO is a low frequency motion which arises at the first natural period of the riser. Large amplitude, which may rise up to several tens of diameters, are predicted (see e.g. Wu, 2001).

Vortex Induced Vibration are known to be self limited motion with a maximum of the order of one diameter for a single riser. However, in cylinders array, measurements have shown VIV amplification, up to two diameters. VIV frequencies may be high as they are closely correlated with current velocity through Strouhal relationship.

Literature survey indicates that hydrodynamic risers interference have been investigated recently, see e.g. Wu (2001), Blevins (2004). A Recommended Practice RP F-203 should be issued during 2005 by DNV, as a result of NDP research, representing a first milestone for industrial design. Clearly, the next stage consists in better understanding the hydrodynamic loads acting on a riser placed in the wake of an upstream one.

## CRITICALITY OF RISER INTERFERENCE IN DEEP AND ULTRA-DEEP WATER

In ultra-deep water depth, the first natural period of riser is generally of the order of 50 to 100 seconds (see table 1) and the modal stiffness associated to the first natural frequency of the riser at mid span is very low due to the riser length. Consequently, large displacements of the WIO type are susceptible to arise and must therefore be studied. When the instantaneous distance between risers get below 3 to 4 diameters,

<footnote n="1">¹IFP, 1&4 av. de Bois Préau, Rueil-Malmaison, France.</footnote>

<footnote n="2">²IFREMER, Centre de Brest, Plouzané, France.</footnote>

<footnote n="3">³Ecole Supérieure d'Ingénieurs de Marseille, Marseille, France.</footnote>

<footnote n="4">⁴Oceanide, BGO-FIRST, La Seyne / Mer, France.</footnote>
the upstream riser may attract the downstream one due to negative drag coefficient leading to possible contact. Deep water developments involve a large number of risers and umbilicals. Spacing in between these components is therefore of critical importance. The amplitude of WIO has to be considered when performing a clearance analysis, not only for vertical risers, but also between adjacent jumpers or catenary risers, for which first mode motion, of the "swinging rope" type, has very low stiffness. Criticality of riser interference is first studied, in term of non dimensional numbers. Reduced velocity $V_R=V_0/(f.D)$ and Scruton number $Sc=A_R=(2\pi m\zeta)/(\rho D^2)$ have been evaluated for various type of risers: Top Tension Riser (TTR), Steel Catenary Risers (SCR), Hybrid Catenary Risers (HCR), Composite (carbon fiber) and Drilling risers, for some production or drilling configurations in water depths ranging from 600 to 2300 m. Results are summarized in table 1. The reduced mass is defined as the ratio between linear structural mass and the displaced mass of water. The first natural period is computed using FEM DeepLines™ riser analysis software package. The reduced velocity is computed based on the effective velocity $V_0$, i.e. accounting for the modal shape function $\psi_n(z)$ depending on the riser type

$$V_0 = \int V(z) \psi_n^2(z) dz / \int \psi_n^2(z) dz \quad (1)$$

For vertical risers, a sinusoidal shape is assumed, while a unit shape function is chosen for catenary risers, in view of the associated pendulum motion. A typical value of 0.4 m/s for the effective velocity is obtained for the studied cases though other data suggest values as high as 1.2 m/s in Northern Norway. The Cauchy number is defined as the reduced velocity squared divided by reduced mass, i.e. $C_y=(Ur)^2/\mu$. As will be seen in the simplified modeling, the Cauchy number indicates the magnification factor to be applied to the spatial variations of the hydrodynamic coefficients. The higher the Cauchy number, the higher the motions and the risk of collision. Damping estimation is of crucial importance as this effect will prevent instabilities to arise. Since the reduced velocity is high, hydrodynamic damping can be estimated theoretically (see e.g. Blevins, 1990, 2004) by a first order expansion:

$$\zeta_{x,drag} = \frac{1}{4\pi f D} \frac{\rho D^2}{m} Cd, \quad \zeta_{y,drag} = \frac{1}{2} \zeta_{x,drad} \quad (2)$$

for x and y direction, respectively, due to incoming flow. Numerical predictions of damping coefficient are performed from free decay tests, either starting from slightly disturbed static equilibrium, or after a periodic regime has been established. Both methods lead to results similar to (Eq. 2).

Stability analysis is classically performed using these non dimensional numbers, i.e. plotting the reduced velocity versus the Scruton number. As the damping coefficient varies linearly with reduced velocity (Eq. 2), it is therefore expected that the data points representing risers should lie on a straight line in the plot ($A_R, V_R$), as can be seen in figure 1. The results are also compared with existing experimental data for heat exchangers. Indeed, the hydrodynamic phenomenon leading to galloping instabilities, or WIO, has been largely studied for nuclear power plants. It should be remembered that heat exchangers are formed by closely spaced cylinders arrays, the spacing being of the order of one cylinder diameter. Nevertheless, results clearly indicate that riser interference phenomena may arise in deep and ultra-deep water depth, as most configurations are found to be unstable should shielding effect be too strong and spacing between risers too small. This graph explains the recent efforts that have been made to better understand riser interference (see e.g. Sagatum, Herfjord & Holmas, 2002, Allen & Henning, 2003, Kalleklev, Mork & Sodahl, 2003, Blevins, 2004).

<table><caption>Table 1: Main parameters governing WIO.</caption><thead><tr><th>UFR type</th><th>Depth (m)</th><th>OD (m)</th><th>μ</th><th>T<sub>l</sub> (s)</th><th>V<sub>R</sub></th><th>C<sub>Y</sub></th><th>Sc</th></tr></thead><tbody><tr><td>TTR Production</td><td>660</td><td>027</td><td>2.33</td><td>16</td><td>48</td><td>1E3</td><td>28.6</td></tr><tr><td>TTR Drilling</td><td>660</td><td>0.53</td><td>2.21</td><td>22</td><td>34</td><td>5E2</td><td>20.3</td></tr><tr><td>TLP Tendon</td><td>660</td><td>0.61</td><td>0.98</td><td>15</td><td>20</td><td>4E2</td><td>11.8</td></tr><tr><td>SCR Production</td><td>1300</td><td>0.5</td><td>1.34</td><td>95</td><td>150</td><td>17E3</td><td>89.6</td></tr><tr><td>Umbilical gas inj.</td><td>1300</td><td>0.1</td><td>2.5</td><td>58</td><td>567</td><td>13E4</td><td>340</td></tr><tr><td>Composite riser</td><td>2000</td><td>0.32</td><td>1.3</td><td>54</td><td>134</td><td>14E3</td><td>80.5</td></tr><tr><td>TTR Drilling</td><td>2300</td><td>1.4</td><td>1.06</td><td>150</td><td>100</td><td>10E3</td><td>49.6</td></tr></tbody></table>

<figure><img src="image_2.png" /><figcaption>Fig 1 Stability diagram used to design heat exchangers. Symbols indicate stability criteria derived experimentally.</figcaption></figure>

## WIO MODELLING FOR BASIN TESTING: RISERS IN TANDEM

### Mathematical model

WIO are classically studied using strip theory together with modal decomposition, leading to a 2 degrees of freedom equivalent dynamical system. The dynamics of two cylinders elastically supported in tandem is illustrated fig. 2. Using as a basis the model proposed by Blevins (1977, 2004) and Wu (2001) and taking into account different diameters $D_i$, added mass $(\rho\pi D_i^2/4)$ and structural damping coefficients $\zeta_i$ leads to (Eq. 3). Flow effective velocity ahead of the cylinders $V_0$ and wake velocity $U(x)$ downstream of the front cylinder
<figure><img src="image_3.png" /><figcaption>Fig. 2 : Mathematical Model Definitions.</figcaption></figure>

$$ egin{cases} 
\ddot{x}_1 + 2 \zeta_1 \omega_1 \dot{x}_1 + \omega_1^2 x_1 = \frac{\rho V_0^2 D_1}{2(m_1 + ma_1)} Cd_1(X_2 - X_1 + L/D_1) (1 - \frac{\dot{x}_1}{V_0}) Abs(1 - \frac{\dot{x}_1}{V_0}) \\ 
\ddot{x}_2 + 2 \zeta_2 \omega_2 \dot{x}_2 + \omega_2^2 x_2 = \frac{\rho V_0^2 D_2}{2(m_2 + ma_2)} Cd_2(X_2 - X_1 + L/D_1) (1 - \frac{\dot{x}_2}{U}) Abs(1 - \frac{\dot{x}_2}{U}) 
\end{cases} 
\quad (3) $$

$$ \bar{U} = \frac{U}{V_0} = \frac{U(X_2 - X_1 + L/D_1)}{V_0} = \left( 1 - \sqrt{\frac{Cd_1(X_2 - X_1 + L/D_1)}{(X_2 - X_1 + L/D_1) + \frac{4}{Cd_1(X_2 - X_1 + L/D_1)}}} \right) \quad (4) $$

$$ \tau_i = \omega_i t, \quad X_i = \frac{x_i}{D_1}, \quad Ur_i = \frac{V_0}{\omega_i D_1}, \quad a_i = \frac{2(m_i + ma_i)}{\rho D_1^2}, \quad \Delta X_c = \frac{D_2 + D_1}{2 D_1} - X_2 + X_1 - L/D_1 \quad (5) $$

$$ egin{cases} 
\ddot{X}_1 + 2 \zeta_1 \dot{X}_1 + X_1 = \frac{Ur_1^2}{a_1} Cd_1(X_2 - X_1 + L/D_1) (1 - \frac{\dot{X}_1}{Ur_1}) Abs(1 - \frac{\dot{X}_1}{Ur_1}) - \frac{Kc}{K_i} \Delta X_c \\ 
\ddot{X}_2 + 2 \zeta_2 \dot{X}_2 + X_2 = \frac{Ur_2^2}{a_2 D_1} Cd_2(X_2 - X_1 + L/D_1) (1 - \frac{\dot{X}_2}{\bar{U} Ur_2}) Abs(1 - \frac{\dot{X}_2}{\bar{U} Ur_2}) + \frac{Kc}{K_i} \Delta X_c 
\end{cases} 
\quad (6) $$

are considered. $U(x)$ is based on the turbulent wake expression of Schlichting (Eq. 4). This set of equation is transformed by use of the non dimensional parameters given in (Eq. 5). In case of contact between both cylinders, a spring force based on the riser surface elasticity is added in the R.H.S. of both equations (Eq. 6). When the drag coefficient is known as a function of the position, these equations can easily be integrated numerically.

As WIO result from static changes of lift and drag coefficients with respect to position (Eq. 6), special care has to be given to the hydrodynamic loading acting on the riser. The models described by Wu (2001) and Blevins (2004) are based on Schlichting and Huse (1993) analytical formulation of the flow field, which is valid for large L/D. Little information is given about the hydrodynamic loads acting on the cylinders for small spacing, in particular when negative drag is observed on the downstream cylinder. Moreover, drag coefficients $Cd_1(X)$ and $Cd_2(X)$ on cylinders in tandem depend on Reynolds number (see fig. 3 and 4), thus the need to assess a proper scaling procedure.

## Similarity rules

The mathematical model also allows us to derive similitude rules based on conservation of non dimensional number arising in the equations. In order to get the same values of $X_1(t)$ and $X_2(t)$ at full or at model scale, the following parameters shall be kept constant

$$ (Ur_1)_m = (Ur_1)_r \quad (7) $$

$$ (\bar{U}Ur_2)_m = (\bar{U}Ur_2)_r \quad (8) $$

$$ \left( \frac{Ur_i^2}{a_i} Cd_i \right)_m = \left( \frac{Ur_i^2}{a_i} Cd_i \right)_r \quad (9) $$

$$ \frac{Ur_i^2}{a_i} Cd_i = \frac{\rho Vo^2 Cd_i}{2 K_i} \quad (10) $$

and subscripts m and r refer to model and real configuration respectively. Hydroelastic instabilities such as WIO and VIV amplification are studied experimentally in the IFREMER Circulating
Water Channel at Boulogne / Mer on pivoting rods in 2 m water depth, and in the BGO- FIRST Wave-Current tank managed by Oceanide at La Seyne / Mer on cylindrical rods in 6 m water depth. Models are at scale of about 1/10 and 1/100 respectively. In these basins, water velocity ranges from 0.1 to 1 m/s keeping good flow quality. At full scale, Reynolds number typically ranges from 20,000 to 500,000 depending on the riser type, therefore corresponding to a range in between 1,000 to 50,000 at model scale.

## Hydrodynamic coefficients versus Reynolds number.

In order to investigate Reynolds effects, a data base has been collected through a literature survey, (Biermann, 1933, Hori, 1959, Igarashi, 1981, Imaichi, 1973, Price, 1984, Suzuki, 1971, Tanida, 1973, Wardlaw, 1973 and Zdravkovich, 1977). Plots of Cd₁ and Cd₂ as a function of the cylinder spacing L/D and the Reynolds number are presented in fig. 3 and 4. When the spacing gets below a value of around 3 to 4 a significant drop in drag coefficients occurs on both cylinders. Then the upstream cylinder drag increases when the spacing gets reduced while the downstream cylinder drag still decreases to reach negative values, thus the attracting effect from the front riser. Reynolds effect in the range mentioned ahead affects significantly the drag coefficients, at large spacing by factors which may reach 1.4 on the upstream cylinder and 4 on the downstream one.

The mathematical model describing the dynamics of two cylinders in tandem requires as input a mapping of the drag coefficient. Experimental values have therefore been fitted analytically

$$ \begin{cases} Cd = \beta(Re) + \frac{\gamma(Re)}{L/D + \alpha} & L/D > (L/D)_c (Re) \\ Cd = a(Re) + b(Re) & L/D \le (L/D)_c (Re) \end{cases} \quad (11) $$

where $(L/D)_c$ is the critical distance before Cd values drop. Figures 5(a) and 5(b) show the variation of the values of Cd₁ and Cd₂ due to the Reynolds effect at model scale and at full scale. The minimum Reynolds number at full scale corresponds to the minimum one at model scale. Consequently one may have to consider that the drag on the front riser model may be underestimated by a factor of about 0.5/1.3 ≈ 0.4 at large L/D for the minimum Reynolds cases.

## Example of two cylinders in tandem

As an example, figure 6 shows the unsteady motion of two cylinders with equal diameters in tandem configuration. The initial spacing is 15 diameters, first natural period is of the order of 100 s and the current effective velocity increases up to 0.4 m/s by steps of 0.02 m/s. Varying $V_0$ and consequently $Ur$ allows to predict clashing theoretical onset criteria. The front riser moves more backward than the rear one. When $V_0$ reaches about 0.36 m/s the gap between both risers becomes less than 3 diameters and the rear riser is attracted by the front one (since $Cd_2<0$). Contact and bouncing still persist until the current speed drops sufficiently due to the time delay of the front riser motion with respect to the input.

## A model testing case : application of the similarity rules accounting for Reynolds effect.

One considers the riser configuration presented above at a $λ=1/33$ scale in an effective current velocity of 0.225m/s. Taking care of the Reynolds influence between full scale and model scale at large L/D spacing, the natural frequencies and the support stiffness of the experimental pivoting rod are computed using similarity rules defined in Eq. (7) to (10), choosing a selected basin velocity $V_{0m} ≈ 0.3$ m/s. Accounting for Reynolds effect on Cd values for both risers implies some variation of the stiffness $K_i$ between the front and the rear risers according to Eq. (10). The stiffness $K_2$ and the mass criteria $a_2$ of the

<figure><img src="image_4.png" /><figcaption>Fig. 3 Cylinders in tandem : front drag coefficient as a function of the spacing L/D and Re.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 4 Cylinders in tandem : rear drag coefficient as a function of the spacing L/D and Re.</figcaption></figure>

downstream riser are significantly changed to compensate drag coefficient variation ($Cd_2$) due to Reynolds (Table 2). If Reynolds effect is not accounted, $Cd_i$ and $\bar{U}$ can be deleted in equations 8 to 10 and both models have the same stiffness and mass.

Figures 7(a) and 7(b) present the time evolution of the displacements for full scale and model test configurations. In the later case, Reynolds effect as defined previously is shown. Figure 8 presents drag coefficients at full and model scale. Figure 9 summarizes the displacement responses as a function of reduced velocity $Ur$.

As similitude rules are used to scale the experiment, it is important to check if bias are not introduced in the process, leading to non conservative results. At full scale, for the maximum current velocity of 0.225 m/s the front riser moves backward by approximately 14D and the rear one by 5D according to the modeling (Fig. 7(a)). The relative distance between the two riser stays around 6D, above the critical value of 3 to 4D for which the front riser starts to attract the rear one.

At model scale, if Reynolds effect is not accounted, the front riser moves approximately by 12D only and the rear one motion is increased to about 8D (Fig. 7(b)), due to larger values of the drag coefficient at model scale (see Fig. 8). The relative distance stays above 11D and the criticality of the configuration is not exhibited. Instead if Reynolds effect is accounted on the model, the front riser motion remains identical to the full scale one (Fig. 9). The motion of the rear one is reduced to 4D due to the variation of the drag coefficient (see Fig. 8) which drops faster at model scale than at full scale for low L/D. One reaches earlier the stage where the front riser is starting to attract the rear one. Testing is therefore performed on a conservative basis.
<figure><img src="image_6.png" /><figcaption>Fig. 5(a) – Front cylinder. Effect of the test scaling on the drag coefficient.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 5(b) – Rear cylinder. Effect of the test scaling on the drag coefficient.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 6 – Motion and drag coefficients for two risers in tandem in a variable current $Vo(t)$.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 7(a) – Time evolution for the displacement at full scale.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 7(b) – Time evolution for the displacement at model scale, with and without Reynolds correction.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 8 – Predicted spatial evolution of the drag coefficient at full scale and model scale.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 9 – Comparison of the displacements at full scale and model scale with and without Reynolds correction.</figcaption></figure>
<table><caption>Table 2: WIO parameters at full scale and model scale</caption><thead><tr><th></th><th></th><th>PARAMETERS</th><th>FULL SCALE<br>(Re ~ 110 000)</th><th>1/33 MODEL<br>ACCOUNTING<br>FOR REYNOLDS<br>EFFECT.<br>(Re ~ 4400)</th><th>1/33 MODEL<br>WITHOUT<br>ACCOUNTING<br>FOR REYNOLDS<br>CORRECTION</th></tr></thead><tbody><tr><td rowspan="4">UPSTREAM<br>RISER</td><td>Cd<sub>1</sub></td><td>1.135</td><td>1.0</td><td>1.0</td><td></td></tr><tr><td>fn<sub>1</sub></td><td>0.01</td><td>0.48</td><td>0.48</td><td></td></tr><tr><td>K<sub>1</sub>/m</td><td>2.1</td><td>3.4</td><td>3.8</td><td></td></tr><tr><td>a<sub>1</sub></td><td>3.7</td><td>3.3</td><td>3.7</td><td></td></tr><tr><td rowspan="5">DOWNSTREAM<br>RISER</td><td>Cd2</td><td>0.45</td><td>0.71</td><td>0.71</td><td></td></tr><tr><td>U/Vo</td><td>0.75</td><td>0.77</td><td>0.77</td><td></td></tr><tr><td>fn<sub>2</sub></td><td>0.01</td><td>0.49</td><td>0.48</td><td></td></tr><tr><td>K<sub>2</sub>/m</td><td>2.1</td><td>6.0</td><td>3.8</td><td></td></tr><tr><td>a<sub>2</sub></td><td>3.7</td><td>5.6</td><td>3.7</td><td></td></tr></tbody></table>

## EXPERIMENTAL STUDIES OF VIV AMPLIFICATION

Experiments have been performed in BGO-FIRST tank on two closely spaced vertical risers clashing each other in 6 m water depth. Riser diameter is 0.014 m and riser length is 8.6m. Excitation by a slightly sheared current ranging from 0.14 to 0.38 m/s was produced over 1/3 of the riser length. Three accelerometers were used together with visual tracking (in air and underwater) of targets disposed along the riser. Accelerometers were placed on the upstream riser at mid depth, and on the downstream riser at mid and quarter depth from free surface. Acceleration signals were integrated twice and filtered to find displacement. Fig. 10 present time frequency analysis of the raw (acceleration) and reconstructed (displacement) signal indicating well established, statistically steady vibration. During the experiment, averaged dynamical tension varied in the range [160, 200] N for the downstream cylinder and [130, 200] N for the upstream one.

Following Allen and Henning (2003), variations of accelerations and displacements ratios versus the upstream cylinder displacement are plotted in figures 11(a) to (f). The purpose is to identify the thresholds above which amplification of the downstream cylinder motion may occur. These thresholds concern both the in-line and transverse displacements. All variables are calculated as rms quantities over the entire time series excluding the starting and ending transient phases. For this set of data, the gap between the two cylinders is $L/D=3$, therefore in the critical range where drag coefficient on the downstream cylinder rapidly decreases. As a result, clashing was observed, with frequency depending mainly on the intensity of the current. As expected, clashing occur increasingly with the amplitude of the transverse displacement of the upstream cylinder.

In this experimental campaign, the measured transverse displacement of the upstream cylinder does not exceed $D/3$. Thresholds can be identified in terms of the upstream cylinder motion. Hence amplification may start when $x_i/D \approx 0.1$ (fig. 11(a) and 11(e)) and $y_i/D \approx 0.1$ (fig. 11(b) and (c)).

≈ 0.2 (fig. 11(d) and 11(f)). It is worth noting that above these thresholds, the ratios of accelerations (in line or transverse) are roughly 1.5 (fig. 11(a) and (e)). For the displacements the ratio is ranging in [1.5,2] for large displacements of the upstream riser (fig. 11(e) and 11(f)). The dynamic of the downstream cylinder is hence much more complex than that of the upstream cylinder.

Another asymmetry can be underlined by plotting the ratios of the acceleration components in the two planar directions. As illustrated in fig. 11(b) and 11(c) the transverse accelerations of both upstream and downstream cylinders are always greater than their respective in-line acceleration. For the present gap between cylinders, these results agree well with those of Allen and Henning (2003). In particular as the in-line motion of the upstream cylinder increases, the two components of the acceleration are roughly of the same magnitude (fig. 11(b) and (c)).

## CONCLUSIONS & PERSPECTIVES

Typical risers configurations for Deep Water Offshore Drilling and Production have been selected for which similitude parameters related to the frequency response and the damping have been derived. Although this approach is conservative, plotting the non dimensional numbers in a stability diagram for Heat Exchanger pipes at small spacing shows that risers configurations may become unstable due to Wake Induced Oscillation (WIO) phenomena should shielding effect be too strong or spacing be to small.

Mathematical models describing WIO require mapping of hydrodynamic coefficients as a function of the riser spacing, including the near field where negative drag is measured on the rear riser at low spacing. An experimental data base from a literature survey has exhibited a significant influence of the spacing and of the Reynolds number as well. Correlation formula from the data base have been used for the development of a mathematical model of the dynamics of two risers in tandem taking into account relative motion as in Wu (2001) and elastic bouncing when contact occurs. It enables to estimate the current velocity at which contact occurs as shown on an example.

Similarity laws taking into account the Reynolds effect on the wake velocity and the drag coefficients have been derived. For similitude to be satisfied on pivoting rod at initially large spacing, the stiffness and the mass need to be adjusted differently on both riser models. This is shown for a model at 1/33 scale. Then the front riser motion at model scale can be identical to the full scale one and the rear one is on a conservative side as the drag coefficient drops faster at low spacing for low Reynolds numbers. Without accounting for Reynolds effect, the front riser motion is decreased and the rear one increased which gives a too large spacing estimate and a non conservative testing procedure. This has to be considered either in the models design or in the up-scaling process of the basin results.

Experiments have been performed on two closely spaced vertical risers clashing each other in 6 m water depth to assess the effect of VIV amplification. Experimental results indicate that the rms motion of the downstream cylinder can be more than twice as much than that of the front cylinder.

This work is currently carried out within a French JIP (Fontaine et al., 2005). This project is to be continued over the next two years (2005, 2006). Further investigations and model tests are planed to investigate WIO and VIV amplifications on vertical risers, jumpers and SCRs.
## AKNOWLEDGMENTS

This paper and his sister ship presented to the Ultra-deep Offshore Technology Conference, Tokyo, February, 2005, is dedicated to Jean-Pierre MOREL, Head of the IFREMER Applied Hydrodynamics Laboratory, who has been involved on risers array projects (among others) over the last four years. J.P. MOREL who will be retired in April 2005 has influenced young researchers especially in the field of experimental testing and deriving similarity rules.

We also would like to acknowledge the financial support of TOTAL and of the Comité d'Etudes Pétrolières et Marines, CEP&M, under contract CO 3007/04. The experimental campaign in BGO-First is carried out within the network GIS-Hydro, financially supported by Conseil Général du Var.

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<figure><img src="image_13.png" /><figcaption>Fig. 10 – Frequency versus Time for (a) inline reconstructed displacement X, (b) acceleration $d^2X/dt^2$, (c) cross flow reconstructed displacement Y, (d) acceleration $d^2Y/dt^2$ indicating fully developed statistically steady vibrations.</figcaption></figure>
<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /></figure>

<figure><img src="image_16.png" /></figure>

<figure><img src="image_17.png" /></figure>

<figure><img src="image_18.png" /></figure>

<figure><img src="image_19.png" /><figcaption>Fig. 11 : Uniform Flow. (a) Downstream cylinder in-line acceleration / upstream cylinder in-line acceleration versus upstream in-line rms displacement. (b) Upstream cylinder - transverse accelerations / in-line accelerations versus upstream transverse rms displacement. (c) Downstream cylinder - transverse accelerations / in-line accelerations versus upstream transverse rms displacement. (d) Downstream cylinder transverse acceleration / upstream cylinder transverse acceleration versus upstream transverse rms displacement. (e) Downstream cylinder in-line displacement / upstream cylinder in-line displacement versus upstream in-line rms displacement. (f) Downstream cylinder transverse displacement / upstream cylinder transverse displacement versus upstream in-line rms displacement.</figcaption></figure>