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Morison forces

Morison formulae are based on three empirical coefficients, namely the drag Cd, the inertia Cm and the added mass coefficients Ca. These coefficients are associated with an equivalent outer diameter \(d_\xi\) to calculate the different forces generated by a fluid on a line. A lift coefficient might be added for non-circular shapes.

For a cylinder, two different sets of coefficients are usually defined corresponding to the generated forces along the pipe axis (indice t ) and orthogonal (inde n ) to the pipe.

For more complex shapes, polar coefficients may be used. \(Cd(\theta)\) for \(\theta\) in 0:360° is derived with linear interpolation from the polar coefficient values defined in the model according to the angle of attack.

Convention used for polar coefficients.

They may be divided into four kinds of loadings:

  • So called linear Added mass force due to the line acceleration, proportional to the line acceleration which somehow accounts for the radiation force.
\[ \newcommand{\arraystretch}{2} \begin{array}{|c|c|} \hline \hspace{0.5cm} \text{Normal added mass force} \hspace{0.5cm} & \hspace{0.5cm} \vec F_{in} = - \frac {\pi d_{\xi}^2} 4 \rho_{fluid} C_{a_n} \ddot {\vec {x_n}} \hspace{0.5cm} \\ \hline \text{Tangent added mass force} & \vec F_{it} = - \frac{\pi d_{\xi}^2} 4 \rho_{fluid} C_{a_t} \ddot {\vec {x_t}}\\ \hline \end{array} \]
  • The linear Inertia force due to the fluid acceleration which somehow represents the wave excitation loads.
\[ \begin{array}{|c|c| } \hline \hspace{0.5cm} \text{Normal inertia force} \hspace{0.5cm} & \hspace{0.5cm}\vec F_{(wave)n} = \frac {\pi d_{\xi}^2} 4 \rho_{fluid} C_{m_n} {\vec a_{n(wave)}} \hspace{0.5cm} \\ \hline \text{Tangent inertia force} & \vec F_{(wave)n} = \frac {\pi d_{\xi}^2} 4 \rho_{fluid} C_{m_t} {\vec a_{t(wave)}}\\ \hline \end{array} \]

Note

For cylinders, the added mass term Ca and the inertia terms are linked by the following relationship: $$ C_m = C_a + 1 $$

  • The linear Drag force due to the fluid relative velocity combing fluid velocity (wave and current in sea or wind in air) and the line speed:
\[ \Delta \vec {\nu} = \vec {\nu}_{fluid} - \dot {\vec{x} }_{line} \]
\[ \begin{array}{|c|c| } \hline \hspace{0.5cm} \text{Normal drag} \hspace{0.5cm} & \hspace{0.5cm}\vec F_{d} = - \frac {1} 2 d_{\xi} \rho_{fluid} C_{d_n} \lVert \Delta \vec{\nu}_{n} \rVert \Delta \vec{\nu}_{n} \hspace{0.5cm} \\ \hline \text{Tangent drag} & \vec F_{t} = - \frac {1} 2 d_{\xi} \rho_{fluid} C_{d_t}\lVert \Delta \vec{\nu}_{t}\rVert \Delta \vec{\nu}_{t} \hspace{0.5cm}\\ \hline \end{array} \]
  • The linear Lift force perpendicular to the incident fluid velocity for non-symmetrical profile:
\[ \begin{array}{|c| } \hline \hspace{0.5cm} \text{Lift force} \hspace{0.5cm} & \hspace{0.5cm}\vec F_{d} = - \frac {1} 2 d_{\xi} \rho_{fluid} C_{lift} \lVert \Delta \vec{\nu}_{n} \rVert ^2 \vec{n} \hspace{0.5cm} \\ \hline \end{array} \]

Note

By default, the Morison formulation implemented in DeepLinesTM considers the projection of the relative velocity (normal or tangent) in both terms (modulus and vector) to compute the drag and lift forces.

Two specific keywords may change the formulation for selected groups of elements:

  • *DRAGABSOLU: In that case, only the velocity of the structure is used and the fluid velocity is disregarded; as a consequence, the drag term becomes a pure quadratic damping term.

  • *DRAGNORM : The total fluid velocity is used in the modulus term, only the vector velocity is projected in the normal and tangent directions.

Note

Since version V5R5, the Morison forces are automatically modified in case of a pipe-in-pipe contact for the inner pipe:

  • The wave/current speeds and wave acceleration are disregarded ;

  • Drag force is computed considering the speed of the outer pipe as a curent speed; As a consequence the drag force is a function of the relative velcoity of the outer pipe vs the inner pipe.

If both pipes are in contact, they have the same speed and no drag force is generated; If the outer pipe is fixed, drag force acts as a damping force on the inner pipe.

It is possible to remove this automatic process with the keyword *TYPCONTACT