hf-limit-wdf-new(1)

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Quadratic Transfer function for Wave Drift Forces 1

Assignment for OE4630-3 (as part of MTM1410)

The assignment is an individual assignment and will be treated as such. The assignment wil be deliveredas a hard copy to mrs Gazendam at the latest January 21 rst 16.00 hours. All documents delivered afterthat date will not be considered as a formal fullfilment of the assignment.The assignement consist of two main questions, a few sub questions and a bonus question

Question I: Derivation High Frequency wave limit Wave drift forces to a Sphere (40pts)

Below you will find a recap of the derivation of the high frequency wave limit of the wave drift forcesto a wall.You are asked to derive the high frequency wave limit of the wave drift force to a sphere.

Consider a wave with an angle ofdegrees to a wall. In our example of a vertical wall we take= 0

The wave description without a wall would look like:

(t) =acos(x t) with =2

g for deep water

Description of the wave with the presence of a wall (total reflection) amounts to:

(t) = 2acos(x)cos(t)

Analogous one finds for the velocity potential (1) :

(1)(x,z,t) =2ag

exp(z)cosx cost

The force on the wall , integration of the wave pressure (Bernuolli)

F(t) =

(t)

(t+ gz+1

22)nxdL, note that nx =-1


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Derive the mean force on the wall.Split the force into first and second order components.Only retaining the second order parts one arrives at:

F(2)(t) =

(t)0

(t+ gz)dz

0

1

22zdz

( I have deleted the contribution from the second order potential (2) since it does not contribute tothe mean force.So:

F(2)(t) = (tz+1

2z2)

(t)

0

0

1

22zdz

Which leads to:

F(2)(t) = +1

2g2(t)

0

1

2

2ag

exp(z) sin(t)

2dz

If we now take the mean of the second order wave force expression we arrive at:

F =F(2)(t) = +1

2g2(t)

0

1

2

2ag

exp(z)sin(t)

2dz

Substitute the expression for (t)we find:

F=g2a1

2g2a =

1

2g2a

So the high frequency wave drift force on a wall is 12g2a per meter width of the wall.

Wave Drift Force for a Sphere

Question I A Now derive the expression for the mean wave drift force on the wall for wave angles that are not zero. (15 pts)Question I B From that result derive the high frequency limit of the wave drift force to the sphere.(25 pts)

Question II: Computation of the wave drift forces in regular and bi-chromatic waves(60

pts)

A ship is moored in a spread mooring system in deep water, which is a system with a low horizontalnatural frequency. The mooring system is linearized. The environmental condition is always beam on.Additional information on the case can be found on the next pages and on accompagning documentsthat are included in the assignment folder on BB. The wave conditions regular 1 and regular 2 arealso defined in the excel sheet in the assignment folder.

Task 1: Mooring system stiffness and natural frequency

1.1) Calculate the sway natural period of the FPSO . (5pts)

Task 2: Wave Drift Force


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2.1) Calculate the high frequency limit of the wave drift force in a beam on two meter waveamplitude assuming all waves will be reflected and the vessel is not moving. (5 pts)

2.2) Calculate the mean wave drift force in beam waves due to wave condition Regular 1using the provided P and Q matrices.(15 pts)

2.3) Make a sketch of the wave envelope of a bi-chromatic wave consisting of the two regularwaves Regular 1 and Regular 2. What is the frequency of the wave envelope? (5pts)

2.4) How large is the mean wave drift load if the two waves would both be two times ashigh? How large is the maximum wave drift load if the two regular waves are only half thespecified height? (10 pts)

2.5) Calculate the wave group spectrum associated with the provided sea state spectrum.(10 pts)

2.6) Calculate the mean wave drift force and the wave drift force spectrum in this irregularwave. (15 pts)

Bonus question : Breaking Waves 10 pts

We assume that the waves can break and that can be described by an adapted Free Surface Condition:

tt+ t+ gz = 0, on z=0, the free surface

In which is the positive dissipation coefficient.Derive how a plane progressive wave would decay as a function of distance. (Tip derive the dispersionrelation)

appendix

The Ship Fixed coordinate system and corresponding sign convention are shown in the figure below andis in accordance with that of OCIMF. Ship Fixed coordinate system and sign convention The motions

are positive in the following directions:

- Positive surge, (x) : towards the bow


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- Positive sway (y) : towards port side

- Positive heave (z) : upwards

- Positive roll straboard down

- Positive pitch bow down

- Positive yaw towards port isde

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