In
fluid dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) an ...
the Morison equation is a semi-
empirical
Empirical evidence for a proposition is evidence, i.e. what supports or counters this proposition, that is constituted by or accessible to sense experience or experimental procedure. Empirical evidence is of central importance to the sciences and ...
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. The Morison equation is used to estimate the
wave
In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (res ...
loads in the design of
oil platform
An oil platform (or oil rig, offshore platform, oil production platform, and similar terms) is a large structure with facilities to extract and process petroleum and natural gas that lie in rock formations beneath the seabed. Many oil platfor ...
s and other
offshore structures.
Description
The Morison equation is the sum of two force components: an
inertia
Inertia is the idea that an object will continue its current motion until some force causes its speed or direction to change. The term is properly understood as shorthand for "the principle of inertia" as described by Newton in his first law ...
force in phase with the local flow
acceleration
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by the ...
and a
drag force proportional to the (signed)
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...
of the instantaneous
flow velocity
In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...
. The inertia force is of the functional form as found in
potential flow
In fluid dynamics, potential flow (or ideal flow) describes the velocity field as the gradient of a scalar function: the velocity potential. As a result, a potential flow is characterized by an irrotational velocity field, which is a valid appr ...
theory, while the drag force has the form as found for a body placed in a steady flow. In the
heuristic
A heuristic (; ), or heuristic technique, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be optimal, perfect, or rational, but is nevertheless sufficient for reaching an immediate, ...
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
In fluid dynamics, vortex shedding is an oscillating flow that takes place when a fluid such as air or water flows past a bluff (as opposed to streamlined) body at certain velocities, depending on the size and shape of the body. In this flow, v ...
—has to be addressed separately.
The Morison equation contains two empirical
hydrodynamic
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) and ...
coefficients—an inertia coefficient and a
drag coefficient
In fluid dynamics, the drag coefficient (commonly denoted as: c_\mathrm, c_x or c_) is a dimensionless quantity that is used to quantify the drag or resistance of an object in a fluid environment, such as air or water. It is used in the drag equ ...
—which are determined from experimental data. As shown by
dimensional analysis
In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric current) and units of measure (such as m ...
and in experiments by Sarpkaya, these coefficients depend in general on the
Keulegan–Carpenter number
In fluid dynamics, the Keulegan–Carpenter number, also called the period number, is a dimensionless quantity describing the relative importance of the drag forces over inertia forces for bluff objects in an oscillatory fluid flow. Or simi ...
,
Reynolds number
In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be domi ...
and
surface roughness
Surface roughness, often shortened to roughness, is a component of surface finish (surface texture). It is quantified by the deviations in the direction of the normal vector of a real surface from its ideal form. If these deviations are large, ...
.
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
In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...
, the Morison equation gives the inline force parallel to the flow direction:
[Sumer & Fredsøe (2006), p. 131.]
:
where
*
is the total inline force on the object,
*
is the flow acceleration, i.e. the
time derivative
A time derivative is a derivative of a function with respect to time, usually interpreted as the rate of change of the value of the function. The variable denoting time is usually written as t.
Notation
A variety of notations are used to denote th ...
of the flow velocity
* the inertia force
, is the sum of the
Froude–Krylov force
In fluid dynamics, the Froude–Krylov force—sometimes also called the Froude–Kriloff force—is a hydrodynamical force named after William Froude and Alexei Krylov. The Froude–Krylov force is the force introduced by the unsteady pressure ...
and the hydrodynamic mass force
* the drag force
according to the
drag equation
In fluid dynamics, the drag equation is a formula used to calculate the force of drag experienced by an object due to movement through a fully enclosing fluid. The equation is:
F_\, =\, \tfrac12\, \rho\, u^2\, c_\, A
where
*F_ is the drag force ...
,
*
is the inertia coefficient, and
the
added mass In fluid mechanics, added mass or virtual mass is the inertia added to a system because an accelerating or decelerating body must move (or deflect) some volume of surrounding fluid as it moves through it. Added mass is a common issue because the obj ...
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
and the cylinder volume per unit cylinder length is
. As a result,
is the total force per unit cylinder length:
:
Besides the inline force, there are also oscillatory
lift
Lift or LIFT may refer to:
Physical devices
* Elevator, or lift, a device used for raising and lowering people or goods
** Paternoster lift, a type of lift using a continuous chain of cars which do not stop
** Patient lift, or Hoyer lift, mobil ...
forces perpendicular to the flow direction, due to
vortex shedding
In fluid dynamics, vortex shedding is an oscillating flow that takes place when a fluid such as air or water flows past a bluff (as opposed to streamlined) body at certain velocities, depending on the size and shape of the body. In this flow, v ...
. 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
, the Morison equation becomes:
[
:
where the total force contributions are:
* ''a'': ]Froude–Krylov force
In fluid dynamics, the Froude–Krylov force—sometimes also called the Froude–Kriloff force—is a hydrodynamical force named after William Froude and Alexei Krylov. The Froude–Krylov force is the force introduced by the unsteady pressure ...
,
* ''b'': hydrodynamic mass force,
* ''c'': drag force.
Note that the added mass coefficient is related to the inertia coefficient as .
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
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tro ...
. If the diameter of the body is not small compared to the wavelength, diffraction
Diffraction is defined as the interference or bending of waves around the corners of an obstacle or through an aperture into the region of geometrical shadow of the obstacle/aperture. The diffracting object or aperture effectively becomes a s ...
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 of describing 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.
References
Further reading
*
*
*
*, 530 pages
{{refend
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