Diffraction-Radiation Theory and Morison's Equation |
 
|
Diffraction-Radiation Theory and Morison's Equation |
|
Broadly speaking, two different modelling techniques are generally used to model hydrodynamic loads on a floating structure, Diffraction-Radiation theory and Morison’s Equation. Diffraction-radiation theory is based on the evaluation of pressure integrals around the body surface, and is applicable in situations where the floating body is relatively large compared to the wave length of the incident seastate, as the wave field is disturbed by the presence of the submerged structure. This type of model is quite accurate in the sense that it will capture wave excitation forces (including diffraction) and radiation loads (including added mass and damping). However it does have some limitations:
•It does not account for viscous drag loading on the structure resulting from flow separation.
•Radiation and diffraction potentials are often solved in the frequency domain assuming a linearised boundary for reasons of computational efficiency. This technique assumes that displacements of the free surface and the floating body away from their mean positions remain relatively small, which simplifies the wave-structure interactions significantly, but the downside is that this simplification results in forces which vary linearly with wave amplitude.
•When used in conjunction with structural codes like Flexcom, global hydrodynamic loads are typically concentrated at a single location such as the centre of gravity, rather than being distributed over the wetted surface area of the body. Radiation-diffraction codes readily provide the total loads in terms of coefficient matrices so this is the most convenient means of transferring loads to the structural model. In theory it is possible to apply distributed loads, but this requires more detailed information regarding the load distribution, and the subsequent mapping of these forces to relevant locations on the structural model.
Diffraction-radiation theory is widely accepted industry for marine renewables, particularly for operational sea states where wave heights are moderate.
Morison’s equation is normally used for smaller structures. For example, for cylindrical structures in regular waves, it is generally deemed acceptable if the diameter is less than one-fifth of the wave length. Morison’s equation is an empirically derived hydrodynamic loading model which includes the wave excitation force, and added mass term, and viscous drag forces. Although relatively simplistic in that it completely ignores wave field disturbance, it has some advantages in that:
•Viscous drag forces may be modelled.
•Applied loads may be computed based on the instantaneous location and orientation of the structure, and wave forces may readily be integrated up to the instantaneous free surface elevation using a wave stretching algorithm.
•Applied loading is automatically distributed to various locations around the floating body, given that it is modelled using a mesh of slender beam elements, and this allows the spatial nature of applied loads to be modelled.
These factors result in the application of higher order non-linear loads on a floating body.
Note also that one of the main challenges associated with the Morison approach is the selection of appropriate hydrodynamic coefficients. Guidelines are available based on Reynolds number and surface roughness, but these relate to simple cylinders only. If you are using Morison to model a more complex shape, or indeed a collection of cylinders (e.g. a semi-submersible platform is effectively comprised of a series of columns, pontoons and cross-braces), then the choice of appropriate hydrodynamic coefficients is far from trivial. It may be possible to fine-tune selected values by benchmarking a Morison simulation with a corresponding potential flow model or experimental data.
The applicability of diffraction-radiation theory or Morison’s equation is generally assessed via dimensionless parameters, such as the Keulegan-Carpenter number, the Reynolds number and the diameter-to-wavelength ratio. These parameters define the relative importance of inertia, diffraction, and drag for different flow regions. There is a large body of research material available in this area, and a quick internet search for “regions of validity of potential theory and morison’s equation” will reveal some further helpful background reading.
Many popular simulation tools such as Flexcom allow users to adopt a combined approach, whereby the radiation and diffraction components are modelled using diffraction-radiation theory, which is then supplemented by the inclusion of viscous drag terms derived from Morison’s equation.
The next article on Floating Body Modelling Detail is related to this one. It discusses relatively simple (concentrated loads) and more complex approaches (distributed loads) for physically modelling the floating structure.