View source for Morison equation

{{Short description|Equation for force on an object in sea waves}}
[[File:Morison equation.svg|thumb|right|512px|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. 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, [[Morrough Parker O'Brien|O'Brien]], Johnson and Schaaf—of the 1950 paper in which the equation was introduced.<ref>{{citation | title=Force on a circular cylinder in viscous oscillatory flow at low Keulegan–Carpenter numbers | first=T. | last=Sarpkaya | journal=Journal of Fluid Mechanics |year=1986 | volume=165 | pages=61–71 | doi=10.1017/S0022112086002999 | doi-broken-date=29 November 2024 | postscript=<!--none--> |bibcode = 1986JFM...165...61S | hdl=10945/62176 | s2cid=122046406 | url=https://calhoun.nps.edu/bitstream/10945/62176/1/Sarpkaya_Force_on_a_circular_cylinder.pdf }}</ref> The Morison equation is used to estimate the [[ocean surface wave|wave]] loads in the design of [[oil platform]]s and other [[offshore construction|offshore structures]].<ref>{{citation | first1=Ove T. | last1=Gudmestad | first2= Geir | last2=Moe | title=Hydrodynamic coefficients for calculation of hydrodynamic loads on offshore truss structures | journal=Marine Structures | volume=9 | issue=8 | year=1996 | pages=745–758 | doi=10.1016/0951-8339(95)00023-2 | bibcode=1996MaStr...9..745G | postscript=<!--none--> }}</ref><ref>{{cite web | url=http://www.dnv.no/Binaries/WECguideline_tcm28-181675.pdf | archive-url=https://wayback.archive-it.org/all/20090224222306/http://www.dnv.no/Binaries/WECguideline_tcm28-181675.pdf | url-status=dead | archive-date=2009-02-24 | title=Guidelines on design and operation of wave energy converters | publisher=[[Det Norske Veritas]] | date=May 2005 | accessdate=2009-02-16 }}</ref>
==Description==
[[File:Jacket Wave.ogv|thumb|right|300px|Wave loading on the [[oil platform|steel jacket structure]] 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 (physics)|drag]] force proportional to the (signed) [[square (algebra)|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 [[hydrodynamics|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]].<ref>{{citation | last=Sarpkaya | first=T. | year=1976 | contribution=Vortex shedding and resistance in harmonic flow about smooth and rough circular cylinders | title=Proceedings of the International Conference on the Behavior of Offshore Structures, BOSS '76 | volume=1 | pages=220–235 | postscript=<!--none--> }}</ref><ref>{{citation | last=Sarpkaya | first=T. | year=1977 | title=Vortex shedding and resistance in harmonic flow about smooth and rough cylinders at high Reynolds numbers | id=Report No. NPS-59SL76021 | publisher=Naval Postgraduate School | publication-place=Monterey | postscript=<!--none--> }}</ref>
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]] <math>u(t)</math>, the Morison equation gives the inline force parallel to the flow direction:<ref name=Sumer_Fredsoe_131>Sumer & Fredsøe (2006), p. 131.</ref>
:<math>F\, =\, \underbrace{\rho\, C_m\, V\, \dot{u}}_{F_I} + \underbrace{\frac12\, \rho\, C_d\, A\, u\, |u|}_{F_D},</math>
where
* <math>F(t)</math> is the total inline force on the object,
* <math>\dot{u} \equiv \text{d}u/\text{d}t</math> is the flow acceleration, i.e. the [[time derivative]] of the flow velocity <math>u(t),</math>
* the inertia force <math>F_I\, =\, \rho\, C_m\, V\, \dot{u}</math>, is the sum of the [[Froude–Krylov force]] <math>\rho\, V\, \dot{u}</math> and the hydrodynamic mass force <math>\rho\, C_a\, V\, \dot{u},</math>
* the drag force <math>F_D\, =\, {\scriptstyle \frac12}\, \rho\, C_d\, A\, u\, |u|</math> according to the [[drag equation]],
* <math>C_m=1+C_a</math> is the inertia coefficient, and <math>C_a  </math> 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 <math>A=D</math> and the cylinder volume per unit cylinder length is <math>V={\scriptstyle\frac{1}{4}}\pi{D^2}</math>. As a result, <math>F(t)</math> is the total force per unit cylinder length:
:<math>F\, =\, C_m\, \rho\, \frac{\pi}{4} D^2\, \dot{u}\, +\, C_d\, \frac12\, \rho\, D\, u\, |u|.</math>
Besides the inline force, there are also oscillatory [[lift (force)|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 <math>v(t)</math>, the Morison equation becomes:<ref name=Sumer_Fredsoe_131/>
:<math>
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}.
</math>
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 <math>\dot{u}</math>,
* ''b'': hydrodynamic mass force,
* ''c'': [[drag force]].
Note that the added mass coefficient <math>C_a</math> is related to the inertia coefficient <math>C_m</math> as <math>C_m=1+C_a</math>.
==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 [[ocean surface wave|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.<ref>{{citation | title=Compliant Offshore Structures | last1=Patel | first1=M.H. | last2=Witz | first2=J.A. | isbn=9781483163321 | year=2013 | publisher=Elsevier | pages=80–83}}</ref>
*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.<ref>{{harvtxt|Sarpkaya|2010|pages=95–98}}</ref>
*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.<ref>{{citation | last=Chaplin | first=J. R. | year=1984 | title=Nonlinear forces on a horizontal cylinder beneath waves | journal=Journal of Fluid Mechanics | volume=147 | pages=449–464 | doi=10.1017/S0022112084002160 | doi-broken-date=29 November 2024 | postscript=<!--none-->|bibcode = 1984JFM...147..449C | s2cid=122421362 }}</ref>
== References ==
<references />
== Further reading ==
{{refbegin}}
*{{citation
| last1=Morison
| first1=J. R.
| last2=O'Brien
| first2=M. P.
| last3=Johnson
| first3=J. W.
| last4=Schaaf
| first4=S. A.
| year=1950
| title=The force exerted by surface waves on piles
| journal=Petroleum Transactions
| publisher=[[American Institute of Mining Engineers]]
| volume=189
| issue=5
| pages=149–154
| doi = 10.2118/950149-G
| postscript=<!--none-->
| doi-access=free
}}
*{{citation
| title=Wave Forces on Offshore Structures
| last=Sarpkaya
| first=T.
| isbn=9780521896252
| year=2010
| publisher=Cambridge University Press
}}
*{{citation
| last1=Sarpkaya
| first1=T.
| last2=Isaacson
| first2=M.
| year=1981
| title=Mechanics of wave forces on offshore structures
| publisher=Van Nostrand Reinhold
| place=New York
| isbn=0-442-25402-4
| postscript=<!--none-->
}}
*{{citation
| title=Hydrodynamics around cylindrical structures
| first1=B. M.
| last1=Sumer
| first2=J.
| last2=Fredsøe
| publisher=World Scientific
| series=Advanced Series on Ocean Engineering
| volume=26
| edition= revised
| year=2006
| isbn=981-270-039-0
| postscript=<!--none-->
}}, 530 pages
{{refend}}
[[Category:Fluid dynamics]]
[[Category:Equations of fluid dynamics]]
[[Category:Water waves]]
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[[Category:Marine engineering]]