Morison equation

In an oscillatory flow with flow velocity ${\displaystyle u(t)}$, the Morison equation gives the inline force parallel to the flow direction:[6]

${\displaystyle F\,=\,\underbrace {\rho \,C_{m}\,V\,{\dot {u}}} _{F_{I}}+\underbrace {{\frac {1}{2}}\,\rho \,C_{d}\,A\,u\,|u|} _{F_{D}},}$

where

• ${\displaystyle F(t)}$ is the total inline force on the object,
• ${\displaystyle {\dot {u}}\equiv {\text{d}}u/{\text{d}}t}$ is the flow acceleration, i.e. the time derivative of the flow velocity ${\displaystyle u(t),}$
• the inertia force ${\displaystyle F_{I}\,=\,\rho \,C_{m}\,V\,{\dot {u}}}$, is the sum of the Froude–Krylov force ${\displaystyle \rho \,V\,{\dot {u}}}$ and the hydrodynamic mass force ${\displaystyle \rho \,C_{a}\,V\,{\dot {u}},}$
• the drag force ${\displaystyle F_{D}\,=\,{\scriptstyle {\frac {1}{2}}}\,\rho \,C_{d}\,A\,u\,|u|}$ according to the drag equation,
• ${\displaystyle C_{m}=1+C_{a}}$ is the inertia coefficient, and ${\displaystyle C_{a}}$ the added mass coefficient,
• A is a reference area, e.g. the cross-sectional area of the body perpendicular to the flow direction,
• V is volume of the body.

For instance for a circular cylinder of diameter D in oscillatory flow, the reference area per unit cylinder length is ${\displaystyle A=D}$ and the cylinder volume per unit cylinder length is ${\displaystyle V={\scriptstyle {\frac {1}{4}}}\pi {D^{2}}}$. As a result, ${\displaystyle F(t)}$ is the total force per unit cylinder length:

${\displaystyle F\,=\,C_{m}\,\rho \,{\frac {\pi }{4}}D^{2}\,{\dot {u}}\,+\,C_{d}\,{\frac {1}{2}}\,\rho \,D\,u\,|u|.}$

Besides the inline force, there are also oscillatory lift forces perpendicular to the flow direction, due to vortex shedding. These are not covered by the Morison equation, which is only for the inline forces.

In case the body moves as well, with velocity ${\displaystyle v(t)}$, the Morison equation becomes:[6]

${\displaystyle F=\underbrace {\rho \,V{\dot {u}}} _{a}+\underbrace {\rho \,C_{a}V\left({\dot {u}}-{\dot {v}}\right)} _{b}+\underbrace {{\frac {1}{2}}\rho \,C_{d}A\left(u-v\right)\left|u-v\right|} _{c}.}$

where the total force contributions are:

• a: Froude–Krylov force,
• b: hydrodynamic mass force,
• c: drag force.

Note that the added mass coefficient ${\displaystyle C_{a}}$ is related to the inertia coefficient ${\displaystyle C_{m}}$ as ${\displaystyle C_{m}=1+C_{a}}$.