2.3.10. Forces on slender elements

When viscous effects may be advanced to have a significant effect on the dynamic behavior of an offshore structure, Morison’s equation can be used to evaluate wave loads on slender elements of the structure [9, 10, 11]. In SeaFEM, force corrections due to viscous effects can be also taken into account by using the Morison’s equation. For this purpose, an auxiliary framework structure, associated to a body must be defined. See the SeaFEM user manual for details on how to define the auxiliary framework structure elements using the Tcl interface of SeaFEM.

Based on the information provided by the user, SeaFEM evaluates Morison’s forces per unit length acting on the framework structure. After integration along the different elements, the resultant forces are incorporated to the dynamic solver of the rigid body to which the idealized framework structure has been associated. It is useful to write the Morison’s equation in a vectorial formulation that automatically takes into account the actual orientation of structural elements and force components. Considering a segment of a long slender structural element submerged into water its local orientation is given by a unit vector

(2.655)\[\begin{aligned} & \boldsymbol{l} = l \boldsymbol{i} + m \boldsymbol{j} + n \boldsymbol{k} \end{aligned}\]

being \(l\), \(m\), \(n\) the directional cosines and \((\boldsymbol{l}, \boldsymbol{j}, \boldsymbol{k})\) the unit vectors of the global coordinate system. Similarly, the relative fluid velocity vector and the relative acceleration vector of the submerged body are given by:

(2.656)\[\begin{aligned} & \boldsymbol{v} = v_x \boldsymbol{i} + v_y \boldsymbol{j} + v_z \boldsymbol{k} \end{aligned}\]

(2.657)\[\begin{aligned} & \boldsymbol{a} = \boldsymbol{a}^w - \boldsymbol{a}^b = a_x \boldsymbol{i} + a_y \boldsymbol{j} + a_z \boldsymbol{k} \end{aligned}\]

The relative fluid velocity vector and the relative acceleration vector are evaluated based on the undisturbed wave potential equations. The force per unit length on a slender cylindrical element may be written as the sum of inertia, drag, friction and lift forces:

(2.658)\[\begin{aligned} & \boldsymbol{F} = \boldsymbol{F}_M + \boldsymbol{F}_D + \boldsymbol{F}_V + \boldsymbol{F}_F + \boldsymbol{F}_L \end{aligned}\]

where the inertia force \(\boldsymbol{F}_M\) is oriented along the acceleration vector component normal to the element member, and its magnitude is proportional to the acceleration component. Lift force \(\boldsymbol{F}_L\) is oriented normal to the velocity vector and normal to the axis of the element, and its magnitude is proportional the velocity squared. Drag force \(\boldsymbol{F}_D\) is proportional to the squared velocity component normal to the element and normal to the lift force, while the linear drag force \(\boldsymbol{F}_V\) is proportional to the velocity component normal to the element. Finally, friction force \(\boldsymbol{F}_F\) is aligned along the axis of the element and proportional to the squared velocity components tangential to the element axis. All these are satisfied if the various force components are defined as follows:

(2.659)\[\begin{split}\begin{aligned} & \boldsymbol{F}_M = (1-\Delta _v)(1+C_M) \rho S (\boldsymbol{l} \times \boldsymbol{a}^w \times \boldsymbol{l}) - \rho SC_M (\boldsymbol{l} \times \boldsymbol{a}^b \times \boldsymbol{l}) \\ & \boldsymbol{F}_D = \frac{1}{2} C_D \rho D |\boldsymbol{l} \times \boldsymbol{v}\times \boldsymbol{l}| (\boldsymbol{l} \times \boldsymbol{v} \times \boldsymbol{l}) \\ & \boldsymbol{F}_V = \frac{1}{2} C_V \rho D (\boldsymbol{l} \times \boldsymbol{v} \times \boldsymbol{l}) \\ & \boldsymbol{F}_F = \frac{1}{2} C_F \rho πD |\boldsymbol{l} \cdot \boldsymbol{v}| (\boldsymbol{l} \cdot \boldsymbol{v}) \cdot \boldsymbol{l} \\ & \boldsymbol{F}_L = \frac{1}{2} C_L \rho D |\boldsymbol{l} \times \boldsymbol{v}| (\boldsymbol{l} \times \boldsymbol{v}) \\ \end{aligned}\end{split}\]

where \(D\) is a linear dimension (the diameter in the case of a cylinder), \(S\) is the cross section area, \(C_M\) is the added mass coefficient, \(C_D\) is the non-linear drag coefficient, \(C_V\) is the linear drag coefficient, \(C_F\) is the friction coefficient, \(C_L\) is the lift coefficient, and \(\Delta _v\) takes the value 1 for virtual elements and 0 otherwise. Remark: The first term in the right hand side of the \(F_M\) equation, includes the Froude-Kriloff force (i.e. undisturbed wave pressure force) and the diffraction inertial force, while the second term represents the radiation inertial force. Remark: Note that while \(C_M\), \(C_D\), \(C_F\), \(C_L\) are non-dimensional coefficients, \(C_V\) has dimensions of velocity. As stated above, Eqs. (10-1) - (10-4) can estimate the different components of the force per unit length on a long (slender) structural element. Therefore, they can be integrated along the element axis, to obtain the additional forces and moments acting on the centre of gravity of the associated body.