of a Production Utilities Quarters Compression (PUQC) platform in the Rong Doi oil field, offshore Vietnam (see
Oil megaprojects (2010)). The Morison equation is the sum of two force components: an
inertia force in phase with the local flow
acceleration and a
drag force proportional to the (signed)
square of the instantaneous
flow velocity. The inertia force is of the functional form as found in
potential flow theory, while the drag force has the form as found for a body placed in a steady flow. In the
heuristic approach of Morison, O'Brien, Johnson and Schaaf these two force components, inertia and drag, are simply added to describe the inline force in an oscillatory flow. The transverse force—perpendicular to the flow direction, due to
vortex shedding—has to be addressed separately. The Morison equation contains two empirical
hydrodynamic coefficients—an inertia coefficient and a
drag coefficient—which are determined from experimental data. As shown by
dimensional analysis and in experiments by Sarpkaya, these coefficients depend in general on the
Keulegan–Carpenter number,
Reynolds number and
surface roughness. The descriptions given below of the Morison equation are for uni-directional onflow conditions as well as body motion.
Fixed body in an oscillatory flow In an oscillatory flow with
flow velocity u(t), the Morison equation gives the inline force parallel to the flow direction: :F\, =\, \underbrace{\rho\, C_m\, V\, \dot{u}}_{F_I} + \underbrace{\frac12\, \rho\, C_d\, A\, u\, |u|}_{F_D}, where • F(t) is the total inline force on the object, • \dot{u} \equiv \text{d}u/\text{d}t is the flow acceleration, i.e. the
time derivative of the flow velocity u(t), • the inertia force F_I\, =\, \rho\, C_m\, V\, \dot{u}, is the sum of the
Froude–Krylov force \rho\, V\, \dot{u} and the hydrodynamic mass force \rho\, C_a\, V\, \dot{u}, • the drag force F_D\, =\, {\scriptstyle \frac12}\, \rho\, C_d\, A\, u\, |u| according to the
drag equation, • C_m=1+C_a is the inertia coefficient, and 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 A=D and the cylinder volume per unit cylinder length is V={\scriptstyle\frac{1}{4}}\pi{D^2}. As a result, F(t) is the total force per unit cylinder length: :F\, =\, C_m\, \rho\, \frac{\pi}{4} D^2\, \dot{u}\, +\, C_d\, \frac12\, \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.
Moving body in an oscillatory flow In case the body moves as well, with velocity v(t), the Morison equation becomes: : F = \underbrace{\rho\, V \dot{u}}_{a} + \underbrace{\rho\, C_a V \left( \dot{u} - \dot{v} \right)}_{b} + \underbrace{\frac12 \rho\, C_d A \left( u - v \right) \left| u - v \right|}_{c}. where the total force contributions are: •
a:
Froude–Krylov force, due to the pressure gradient at the body's location induced by the fluid acceleration \dot{u}, •
b: hydrodynamic mass force, •
c:
drag force. Note that the added mass coefficient C_a is related to the inertia coefficient C_m as C_m=1+C_a. ==Limitations==