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Morison Equation

Fluids Engineering

Morison Equation

The computation of the action on a cylindrical object (a member) caused by waves, currents or a combination of waves and currents depends on the ratio of the wave length to the member diameter. When this ratio is large (> 5), the member does not significantly modify the incident wave. The action can then be computed as the sum of a hydrodynamic drag action and a hydrodynamic inertia action, as given in Equation 1.

Equation 1
F = Fd + Fi = Cd · (1/2) · ρw · U · |U| · A + Cm · ρw · V · ( δU / δt )

Where:

F = local action vector per unit length acting normal to the axis of the member;
Fd = vector for the drag action per unit length acting normal to the axis of the member in the plane of the member axis and U;
Fi = vector for the inertia action per unit length acting normal to the axis of the member in the plane of the member axis and ( δU / δt )
Cd = hydrodynamic drag coefficient;
ρw = mass density of water;
A = effective dimension of the cross-sectional area normal to the member axis per unit length (= D for circular cylinders);
V = displaced volume of the member per unit length (= π D2/4 for circular cylinders);
D = effective diameter of a member (a circular cylinder), including marine growth;
U = component of the local water particle velocity vector (due to waves and/or current) normal to the axis of the member;
|U| = modulus (the absolute value) of U;
Cm = hydrodynamic inertia coefficient;
δU / δt = component of the local water particle acceleration vector normal to the axis of the member.

Morison Waves

The Morison equation, as stated here, ignores the contribution from the Froude-Krylov action associated with the convective acceleration component. This is not always appropriate for inertia dominated structures, small wave heights, steep waves or very large diameter components. The equation also excludes actions due to hydrodynamic lift, slamming or slapping actions and axial Froude-Krylov actions.

When the size of a structural object or component is sufficiently large compared to the wave length, the incident waves are scattered or diffracted. This “diffraction regime” is usually considered to occur when the component width exceeds a fifth of the incident wave length. In such cases, diffraction theory, which computes the pressures acting on the structure due to both the incident wave and the scattered wave, shall be used instead of the Morison equation to determine the local action caused by waves. If a structure has some components in the Morison regime and others in the diffraction regime, or a component is adjacent to another (component of a) structure that is in the diffraction regime, the effect of wave diffraction on the wave kinematics to be used for the Morison component should be considered.

NOTE: Depending on its diameter, a free-standing caisson structure can be in the diffraction regime, particularly for the lower sea states associated with fatigue conditions.

Wave Theory
Cd
Cm
 Comments Reference
Linear Theory
1.0
0.95
Mean values for ocean wave data on 13-24in cylinders Wiegel, et al (1957)
1.0- 1.4
2.0
Recommended design values based on statistical analysis of published data Agerschou and Edens (1965)
Stokes 3rd order
1.34
1.46
Mean Values for oscillatory flow for 2-3in cylinders Keulegan and Carpenter (1958)
Stokes 5th order
0.8- 1.0
2.0
Recommended values based on statistical analysis of published data Agerschou and Edens (1965)


Related:

Source:

  • Petroleum and natural gas industries - Fixed steel offshore structures, ISO 19902
  • Hydrodynamics for Ocean Engineers Prof. A.H. Techet - Massachusetts Institute of Technology

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