Morison equation

Flow forces according to the Morison equation for a body placed in a harmonic flow, as a function of time.

Blue line: drag force; red line: inertia force; black line: total force according to the Morison equation. The

horizontal direction is the time direction, increasing from left to right. Note that the inertia force is in front

of the phase of the drag force: the flow velocity is a sine wave, while the local acceleration is a cosine wave

as a function of time.

In fluid dynamics the Morison equation is a semi-empirical equation for the inline force on a body in

oscillatory flow. It is sometimes called the MOJS equation after all four authors — Morison, O'Brien,

Johnson and Schaaf — of the 1950 paper in which the equation was introduced.[1]

The Morison equation is

used to estimate the wave loads in the design of oil platforms and other offshore structures.[2][3]

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 force in an oscillatory flow.

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.[4][5]

Description

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:[6]

where

F(t) is the total inline force on the object,

the inertia force , is the sum of the Froude-Krylov force and the

hydrodynamic mass force

the drag force ,

Cm = 1 + Ca is the inertia coefficient, and Ca 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 . As a result, F(t) is the

total force per unit cylinder length:

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:[6]

where the total force contributions are:

a: Froude–Krylov force,

b: hydrodynamic mass force,

c: drag force.

Note that the added mass coefficient Ca is related to the inertia coefficient Cm as Cm = 1 + Ca.

Limitations

The Morison equation is a heuristic formulation of the force fluctuations in an oscillatory flow. The

first assumption is that the flow acceleration is more-or-less uniform at the location of the body. For

instance, for a vertical cylinder in surface gravity waves this requires that the diameter of the

cylinder is much smaller than the wavelength. If the diameter of the body is not small compared to

the wavelength, diffraction effects have to be taken into account.

Second, it is assumed that the asymptotic forms: the inertia and drag force contributions, valid for

very small and very large Keulegan–Carpenter numbers respectively, can just be added to describe

the force fluctuations at intermediate Keulegan–Carpenter numbers. However, from experiments it is

found that in this intermediate regime — where both drag and inertia are giving significant

contributions — the Morison equation is not capable to describe the force history very well.

Although the inertia and drag coefficients can be tuned to give the correct extreme values of the

force.

Third, when extended to orbital flow which is a case of non uni-directional flow, for instance

encountered by a horizontal cylinder under waves, the Morison equation does not give a good

representation of the forces as a function of time