Morison and Longitudinal Pressure Forces

By considering the total particle kinematics relative to the sub-structure elements, the resulting forces acting on the structures are calculated as the sum of two components:

• Drag and inertia forces using the relative motion form of the Morison equation
• Longitudinal pressure force

as illustrated in Figure 1. The Morison equation is applied as \(\bscalar{F_y}\) and \(\bscalar{F_z}\) at each end point of the structural element \(\bscalar{a}\) and \(\bscalar{b}\). These forces are calculated at the two distinct locations and are applied along the element length \(L\) by assuming a linear distribution, see Figure 1. For a special case like when considering heave plates, the Morison equation is also applied as \(\bscalar{F_x}\) at two end points. On that note, the longitudinal pressure force is always applied as \(\bscalar{F_x}\) at the element end points for all conditions.

Figure 1: Illustration of the force vector applied to structural elements. Left: vectorial representation of the force vector at two element ends, right: linear distribution to apply the load along the element length.

These forces are then used to calculate the sub-structure modal forces. The inertia and drag forces on the support structure are accounted for using the ‘relative motion’ form of the Morison equation:

\[ \begin{flalign} \bscalar{F_x} &= C_{a,x} \rho\frac{\pi D^{2}}{4}{\bscalar{\dot{u}}}_{t,x} + \rho\frac{\pi D^{2}}{4}{\bscalar{\dot{u}}}_{w,x} + \frac{1}{2}C_{d,x}\rho D\bscalar{u}_{t,x} \left| \bvector{u}_{t,x} \right| \\[1ex] \bscalar{F_y} &= \left( C_{m,y} - 1 \right)\rho\frac{\pi D^{2}}{4}{\bscalar{\dot{u}}}_{t,y} + \rho\frac{\pi D^{2}}{4}{\bscalar{\dot{u}}}_{w,y} + \frac{1}{2}C_{d,y}\rho D\bscalar{u_{t,y}} |\bvector{u}_r| \\[1ex] \bscalar{F_z} &= \left( C_{m,z} - 1 \right)\rho\frac{\pi D^{2}}{4}{\bscalar{\dot{u}}}_{t,z} + \rho\frac{\pi D^{2}}{4}{\bscalar{\dot{u}}}_{w,z} + \frac{1}{2}C_{d,y}\rho D\bscalar{u_{t,z}} |\bvector{u}_r| \end{flalign} \]

with \(\bscalar{F_x}\) the axial force and \(\bscalar{F_y}\) and \(\bscalar{F_z}\) representing the components normal to the element axis. Here variable \(\bscalar{D}\) describes the end point diameter at which the force being applied to (at point \(\bscalar{a}\) or \(\bscalar{b}\)) and \(\bscalar{\rho}\) is the water density. The relative velocity vector between the water particle velocity vector (\(\bvector{u_w}\)) and the structural velocity vector at the particular node (\(\bvector{u}_s\)) is described by \(\bvector{u}_t\). Similarly, \(\bvector{\dot{u}}_t\), \(\bvector{\dot{u}}_w\) and \(\bvector{\dot{u}}_s\) represent the time derivative of these velocity components for a particular direction, for example in \(\bscalar{y}\) or \(\bscalar{z}\) axes.

The velocity vectors are defined in the local element frame in the form \(\bvector{u}_i = (u_{i,x}, u_{i,y}, u_{i,z})\). The index \(i\) refers to either the relative \(t\), structural \(s\) or absolute water particle velocity \(w\). For example \(\bvector{u}_t = (u_{t,x}, u_{t,y}, u_{t,z})\) refers to the relative velocity at either end (a or b) of the element as shown in Figure 1. This has magnitude \(|\bvector{u}_t|\). The relative, structural and water particle velocities are related by the following equation \(\bvector{u}_t = \bvector{u}_w - \bvector{u}_s\).

The velocity vector formed by the flow components normal to the element axis is denoted the radial velocity \(\bvector{u}_r = (u_{t,y}, u_{t,z})\). This has a magnitude \(|\bvector{u}_r|\).

The parameters \(\bscalar{C_{m,y}}\) and \(\bscalar{C_{m,z}}\) describe the anisotropic inertia coefficients in \(\bscalar{y}\) and \(\bscalar{z}\) directions, respectively. Similarly, \(\bscalar{C_{d,y}}\) and \(\bscalar{C_{d,y}}\) represent the anisotropic drag coefficients in these two directions. By default, it is assumed that isotropic inertia and drag coefficients are used where \(\bscalar{C_m = C_{m,y} = C_{m,z}}\) and \(\bscalar{C_d = C_{d,y} = C_{d,z}}\), respectively, but they can be specified independently if required. The inertia and drag coefficients may also vary between the element ends. Note again that \(\bscalar{F_y}\) and \(\bscalar{F_z}\) are calculated as the force per unit length and a linear distribution is assumed along the element length, see Figure 1.

By default the heave plates coefficients \(C_{a,x}\) and \(C_{d,x}\) are zero. However a user can specify non-zero values. This will introduce an axial load acting on the structural element. Note that unlike the inertia and drag coefficients defined in the normal directions \(\bscalar{y}\) and \(\bscalar{z}\) only one value can be defined along the entire element axis.

By employing the relationship \(\bvector{\dot{u}}_t = \bvector{\dot{u}}_w - \bvector{\dot{u}}_s\), the Morison equation can be re-written as in Equations \(\eqref{eq_morison_rewritten_1}\) - \(\eqref{eq_morison_rewritten_3}\). This form represents the implementation in Bladed, with the first term being hydrodynamic added mass included in the mass matrix (which here depends on the structural acceleration only) and the second term denoting an applied load as a function of \(\bscalar{C_{m}}\):

\[ \begin{flalign} \label{eq_morison_rewritten_1} \bscalar{F_x} &= - C_{a,x} \rho\frac{\pi D^{2}}{4}{\bscalar{\dot{u}}}_{s,x} + (C_{a,x} + 1) \rho\frac{\pi D^{2}}{4}{\bscalar{\dot{u}}}_{w,x} + \frac{1}{2}C_{d,x}\rho D\bscalar{u}_{t,x} | \bvector{u}_{t,x} |, \\[1ex] \label{eq_morison_rewritten_2} \bscalar{F_y} &= - \left( C_{m,y} - 1 \right)\rho\frac{\pi D^{2}}{4}{\bscalar{\dot{u}}}_{s,y} + C_{m,y}\rho\frac{\pi D^{2}}{4}{\bscalar{\dot{u}}}_{w,y} + \bscalar{\frac{1}{2}C_{d,y}\rho D}\bscalar{u}_{t,y} | \bvector{u}_r |, \\[1ex] \label{eq_morison_rewritten_3} \bscalar{F_z} &= - \left( C_{m,z} - 1 \right)\rho\frac{\pi D^{2}}{4}{\bscalar{\dot{u}}}_{s,z} + C_{m,z}\rho\frac{\pi D^{2}}{4}{\bscalar{\dot{u}}}_{w,z} + \frac{1}{2}C_{d,y}\rho D \bscalar{u}_{t,z} | \bvector{u}_r |. \end{flalign} \]

The wave activated added mass term in Equation \(\eqref{eq_morison_rewritten_1}\) is treated differently. The term

\[ \begin{equation} (C_{a,x} + 1) \rho\frac{\pi D^{2}}{4}{\bscalar{\dot{u}}}_{w,x} = C_{a,x} \rho\frac{\pi D^{2}}{4}{\bscalar{\dot{u}}}_{w,x} + \bscalar{F}_x^p \end{equation} \]

is resolved in two parts. The \(C_{a,x}\) term is computed explicitly while \(\bscalar{F}_x^p\) is instead estimated using the hydrodynamic pressure in the ambient wave field acting at the element ends. For a structure with diameters \(\bscalar{D}\) at the element end point (at point \(\bscalar{a}\) or \(\bscalar{b}\)), the longitudinal force acting on this portion of the structure is described as:

\[ \begin{equation} \bscalar{F}_x^p= \dfrac{1}{4}\pi D^2 p \end{equation} \]

Here \(\bscalar{p}\) denotes the hydrodynamic pressure at each element end (at point \(\bscalar{a}\) or \(\bscalar{b}\)). No pressure force is included where the end of the tubular member passes through the free surface or terminates at the sea-bed.

Last updated 10-09-2024