Morison equation

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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, 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]

Description[edit]

Wave loading on the 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 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]

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[edit]

In an oscillatory flow with flow velocity u(t), the Morison equation gives the inline force parallel to the flow direction:[6]

F\, =\, \underbrace{\rho\, C_m\, V\, \dot{u}}_{F_I} + \underbrace{\frac12\, \rho\, C_d\, A\, u\, |u|}_{F_D},

where

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[edit]

In case the body moves as well, with velocity v(t), the Morison equation becomes:[6]

   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:

Note that the added mass coefficient C_a is related to the inertia coefficient C_m as C_m=1+C_a.

Limitations[edit]

See also[edit]

Notes[edit]

  1. ^ Sarpkaya, T. (1986), "Force on a circular cylinder in viscous oscillatory flow at low Keulegan–Carpenter numbers", Journal of Fluid Mechanics 165: 61–71, Bibcode:1986JFM...165...61S, doi:10.1017/S0022112086002999 
  2. ^ Gudmestad, Ove T.; Moe, Geir (1996), "Hydrodynamic coefficients for calculation of hydrodynamic loads on offshore truss structures", Marine Structures 9 (8): 745–758, doi:10.1016/0951-8339(95)00023-2 
  3. ^ "Guidelines on design and operation of wave energy converters". Det Norske Veritas. May 2005. Retrieved 2009-02-16. 
  4. ^ Sarpkaya, T. (1976), "Vortex shedding and resistance in harmonic flow about smooth and rough circular cylinders", Proceedings of the International Conference on the Behavior of Offshore Structures, BOSS '76 1, pp. 220–235 
  5. ^ Sarpkaya, T. (1977), Vortex shedding and resistance in harmonic flow about smooth and rough cylinders at high Reynolds numbers, Monterey: Naval Postgraduate School, Report No. NPS-59SL76021 
  6. ^ a b Sumer & Fredsøe (2006), p. 131.
  7. ^ Chaplin, J. R. (1984), "Nonlinear forces on a horizontal cylinder beneath waves", Journal of Fluid Mechanics 147: 449–464, Bibcode:1984JFM...147..449C, doi:10.1017/S0022112084002160 

References[edit]

  • Morison, J. R.; O'Brien, M. P.; Johnson, J. W.; Schaaf, S. A. (1950), "The force exerted by surface waves on piles", Petroleum Transactions (American Institute of Mining Engineers) 189: 149–154, doi:10.2118/950149-G 
  • Sarpkaya, T.; Isaacson, M. (1981), Mechanics of wave forces on offshore structures, New York: Van Nostrand Reinhold, ISBN 0-442-25402-4 
  • Sumer, B. M.; Fredsøe, J. (2006), Hydrodynamics around cylindrical structures, Advanced Series on Ocean Engineering 26 (revised ed.), World Scientific, ISBN 981-270-039-0 , 530 pages