Long Wavelength Approximations
Very frequently the length of ambient waves is large compared to the dimension of floating bodies. For example the length of a wave with period is . The beam of a ship with length can be as is the case for the diameter of the leg of an offshore platform.
GI Taylor's formula
Consider a flow field given by
In the absence of viscous effects and to leading order for :
Derivation using Euler's equations
An alternative form of GI Taylor's formula for a fixed body follows from Euler's equations:
If the body is also translating in the x-direction with displacement then the total force becomes
Often, when the ambient velocity is arising from plane progressive waves, and is omitted. Note that does not include disturbance effects due to the body.
Applications of GI Taylor's formula in wave-body interactions
So Archimedes' formula is a special case of GI Taylor when there is no flow. This offers an intuitive meaning to the term that includes the body displacement.
Regular waves over a circle fixed under the free surface
So the horizontal force on the circle is:
We can derive the vertical force along very similar lines. It is simply out of phase relative to with the same modulus.
Horizontal force on a fixed circular cylinder of draft
This case arises frequently in wave interactions with floating offshore platforms.
Here we will evaluate on the axis of the platform and use a strip wise integration to evaluate the total hydrodynamic force.
The differential horizontal force over a strip at a depth becomes:
The total horizontal force over a truncated cylinder of draft becomes:
This is a very useful and practical result. It provides an estimate of the surge exciting force on one leg of a possibly multi-leg platform as
Horizontal force on multiple vertical cylinders in any arrangement
The proof is essentially based on a phasing argument. Relative to the reference frame,
Expressing the incident wave relative to the local frames by introducing the phase factors,
Then relative to the i-th leg,
Ignoring interactions between legs, which is a good approximation in long waves, the total exciting force on an n-cylinder platform is
The above expression gives the complex amplitude of the force with given in the single cylinder case.
The above technique may be easily extended to estimate the Sway force and Yaw moment on n-cylinders with little extra effort.
Surge exciting force on a 2D section
If the body section is a circle with radius ,
So in long waves, the surge exciting force is equally divided between the Froude-Krylov and the diffraction components. This is not the case for Heave!
Heave exciting force on a surface piercing section
In long waves, the leading order effect in the exciting force is the hydrostatic contribution
where is the body water plane area in 2D or 3D. is the wave amplitude. This can be shown to be the leading order contribution from the Froude-Krylov force:
Using the Taylor series expansion,
It is easy to verify that .
The scattering contribution is of order . For submerged bodies, .
This article is based on the MIT open course notes and the original article can be found here