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Potential flow theory for hydrodynamic load determination (from [50])

Assuming the flow is constant, incompressible, non viscous, irrotational and homogeneous, the potential flow theory described in Hydrodynamics of offshore floating structures) is applied. These hypotheses imply that the floater is considered as a large structure, meaning that the wavelength is negligible compared to the floater diameter. Two conditions are added to determine hydrodynamic loads on the float:

• No leak condition on the float:

The wall of the structure is considered as impermeable and water can not come into the float. At any given position on the floater wall, the following equation is verified:

\[ \nabla \Phi \cdot \mathbf{n}=\mathbf{u} \cdot \mathbf{n} \]

where \(\mathbf{n}\) is the local normal vector to the surface and \(\mathbf{u}\) the local velocity of the floater wall.

• No perturbation at infinity condition:

The radiated waves are completely dissipated far from the float.

The wave velocity potential, which is assumed linear, is decomposed as follows:

\[ \Phi(x,y,z,t)=\Phi_{0}+\Phi_{7}+\Phi_{1-6} \]

• \(\Phi_{0}\): The incident wave potential: determined without the structure, focusing only on the non-disturbed incoming wave.
• \(\Phi_{7}\): The diffracted wave potential: resulting from the action of waves around the structure considered as fixed. This is represented by the "restrained in waves" term.
• \(\Phi_{1-6}\): The radiated wave potential: resulting from the motion of a body in water without incoming waves. This corresponds to the "oscillation in still water" term.

Linear decomposition of hydrodynamic loads.

In the potential flow theory, the hydrodynamic force \(F_{hydrodynamic}\) is thus decomposed following the previous decomposition of the velocity potential.

\[ \mathbf{F_{hydrodynamic}}=\mathbf{F_{incoming \, wave}}+\mathbf{ F_{diffracted}}+\mathbf{F_{radiation}} \]

in which: * \(\mathbf{F_{incoming wave}} + \mathbf{F_{diffracted}}\). This force is obtained considering a fixed floater submitted to waves. The incoming force is known as the Froude-Krilov force resulting from the integration of pressure on the surface of the body in the undisturbed waves. The diffracted term corrects Froude-Krilov term to take into account the impact of the floater on wave field. The sum of these two forces is known as the 1st order wave excitation force \(\mathbf{F_{excitation}}\). * \(\mathbf{F_{radiation}}=M_{a} \mathbf{a}+B_{radiation}\mathbf{v}\). \(M_{a}\) is the added mass matrix and $ B_{radiation}$ the radiation damping matrix, both obtained by considering a moving floater in still water.

Thus the motion equation becomes:

\[ \tag{1} (M+M_{a})\mathbf{a}+ (B_{radiation})\mathbf{v}+(K_{stiff}+K_{anchor})\mathbf{x}=\mathbf{F_{excitation}} \]

The equation of motion (1) is resolved in the frequency domain.

Resolution of the motion equation in the frequency domain

The resolution in the frequency domain aims at getting information about the motion of the floater in waves, with strong linearity assumptions. (1) can be solved for each degree of freedom.

Here, to provide an example and for the sake of simplicity, one degree of freedom is considered: the heave response. The heave motion response to a regular wave is described resolving (1) in the frequency domain. The incoming wave elevation is defined by \(\eta=|\eta|e^{-i\omega t}\). z is the coordinate of the system along \((Oz)\) axis and defined by \(z=|z|e^{-i \omega t}\). Knowing that the excitation force is proportional to the incoming wave, the following equation is obtained:

\[ - \omega^{2}(M+M_{a}(\omega)) z -\omega (B_{rad}(\omega))z+Kz=F_0 \eta \]

where \(K_{stiff}+K_{anchor}=K\) and \(F_0\) is the linear excitation force complex amplitude per wave height.

The motion Response Amplitude Operator (RAO) can then be introduced as the ratio between \(z\) and \(\eta\):

\[ RAO=\frac{z}{\eta}=\frac{F_0}{-(M+M_{a}(\omega))-(B_{rad}(\omega))+K} \]

This relation is verified in the frequency domain.

Second order loads: Quadratic Transfer Function (QTF)

In the case of irregular waves (often modelled as the sum of regular Airy waves), various wave frequencies can interact. Their difference or sum leads to low frequency or high frequency loads, the so-called second order wave loads. These second order loads are also computed in the framework of the potential flow theory. Quadratic-transfer-functions, so-called QTF, are derived. \(F_{excitation}\) in (1) becomes the sum of first order and second order wave loads.