**3. Unsteady flow model**

The unsteady fluid force includes the inertia, the damping, and the stiffness effects.

The inertia effect is assumed to be independent of the flow velocity and can be measured experimentally, or computed using the potential-flow theory. In order to utilize this model, fluid force coefficients are required. The first attempt to obtain these force Coefficients experimentally was done by Tanaka and Takahara (Tanaka & Takahara, 1981). The damping and Stiffness effects were expressed in terms of fluid force amplitudes (C) and phase angles (φ) between fluid forces and tube oscillation. To validate the CFD results with experimental data, a simpler variant of the unsteady flow theory as presented by Tanaka and Takahara (Tanaka & Takahara, 1981) is used. The formulation is presented briefly here, however a complete description and derivation is available in (Tanaka & Takahara, 1981; Omar, 2010).

Fig. 1. In-line square tube geometry a) as given in experiments of Tanaka and Takahara (Tanaka & Takahara, 1981; Omar, 2010; Hassan et al, 2010), and b) close-up of moving tube (shaded) and related dimensions.

Referring to the square array in Figure 1a, and considering the forces and displacements in *x* as well as *y*, and assuming the effects of all surrounding tubes can be summed linearly (and that only the four tubes immediately adjacent the center tube 1 have significant influence) the total force expected on tube 1 can be expressed as

The inertia effect is assumed to be independent of the flow velocity and can be measured experimentally, or computed using the potential-flow theory. In order to utilize this model, fluid force coefficients are required. The first attempt to obtain these force Coefficients experimentally was done by Tanaka and Takahara (Tanaka & Takahara, 1981). The damping and Stiffness effects were expressed in terms of fluid force amplitudes (C) and phase angles (φ) between fluid forces and tube oscillation. To validate the CFD results with experimental data, a simpler variant of the unsteady flow theory as presented by Tanaka and Takahara (Tanaka & Takahara, 1981) is used. The formulation is presented briefly here, however a complete description and derivation is available in (Tanaka & Takahara, 1981; Omar, 2010).

L

*x y* 

Flow Direction Monitor points

2

*d* 

(a)

1

*P* 

5

*1 2*

(b)

*P* 

Referring to the square array in Figure 1a, and considering the forces and displacements in *x* as well as *y*, and assuming the effects of all surrounding tubes can be summed linearly (and that only the four tubes immediately adjacent the center tube 1 have significant influence)

Fig. 1. In-line square tube geometry a) as given in experiments of Tanaka and Takahara (Tanaka & Takahara, 1981; Omar, 2010; Hassan et al, 2010), and b) close-up of moving tube

15 d

W

The unsteady fluid force includes the inertia, the damping, and the stiffness effects.

**3. Unsteady flow model** 

(shaded) and related dimensions.

the total force expected on tube 1 can be expressed as

$$F\_{\mathbf{x}} = \frac{1}{2} \rho \mathcal{U} I\_{\text{sup}}^2 \sum\_{j=1,2,3,4,5} \left( \mathbf{C}\_{\text{X}\circ\mathbf{X}} p\_j + \mathbf{C}\_{\text{X}\circ\mathbf{Y}} q\_j \right) \tag{5a}$$

$$F\_y = \frac{1}{2}\rho \mathcal{L} I\_{gwp}^2 \sum\_{j=1,2,3,4,5} \left(\mathcal{C}\_{Y\bar{\chi}X} p\_j + \mathcal{C}\_{Y\bar{\chi}Y} q\_j\right) \tag{5b}$$

where *CX X*1 and *CY Y*1 are fluid force coefficient amplitudes. The fluid forces ( *FX X*<sup>1</sup> , *FY Y*<sup>1</sup> ) acting on tube j lead the displacement of tube j ( *Xj* ,*Yj* ) by phase angles (*XjX* and *YjY* ). The three suffixes of the coefficients represent the direction of the force, the tube index, and the direction of vibration, respectively. For example, the lift fluid force component acting on tube 1 due to the motion of tube 4 in the drag direction (Y ) is expressed as *FX Y*<sup>4</sup> . Therefore, the total lift fluid forces consists of 10 different components corresponding to the lift motion effect (*CX X*<sup>1</sup> ...*CX X*<sup>5</sup> ) and to the drag motion (*CX X*<sup>1</sup> ...*CX X*<sup>5</sup> ). Similarly, the overall drag force comprises of 10 components (*CY X*<sup>1</sup> ...*CY X*<sup>5</sup> , *CY Y*<sup>1</sup> ...*CY Y*<sup>5</sup> ). As the centre tube (1) oscillates in Y direction, *FXjY* is equal to zero except for *FX Y*4 and *FX Y*<sup>5</sup> . Similarly, as the centre tube oscillates in X direction, the *FYjX* is equal zero except of *FY X*4 and *FX Y*<sup>5</sup> .

The reader can refer to (Tanaka & Takahara, 1981; Omar, 2010) for a description. Based on these simplifications, Eq. 5 can be reduced to:

$$F\_X = \frac{1}{2} \rho L l\_{gap}^2 \left( \mathbb{C}\_{X \ge 1} p\_1 + \mathbb{C}\_{X \ge 4X} \left( p\_4 + p\_5 \right) + \mathbb{C}\_{X \ge 4Y} \left( q\_4 + q\_5 \right) \right. \tag{6a}$$
 
$$+ \mathbb{C}\_{X \ge 2X} p\_2 + \mathbb{C}\_{X \ge 3X} p\_3 \text{ }$$

$$F\_y = \frac{1}{2} \rho L l\_{g\text{up}}^2 \left( \mathbf{C}\_{Y1Y} q\_1 + \mathbf{C}\_{4LX} \left( p\_4 + p\_5 \right) + \mathbf{C}\_{Y4Y} \left( q\_4 - q\_5 \right) \right. \tag{6b}$$

$$+ \mathbf{C}\_{Y2Y} q\_2 + \mathbf{C}\_{Y3Y} q\_3 \big)$$
