A mathematical description of the Far-field Approach for the derivation of Mean Drift Forces

1. Mean Drift Forces
Far-field Approach

2. Mean Drift Forces
2
Far-field Approach
• Control Volume: Ω
• Sb : Under water body
surface
• Sfs : Water free surface
• S∞ : Cylindrical surface
surrounding control
volume at infinite
distance
• Bottom neglected as
infinite water depth is
assumed

3. Mean Drift Forces
3
Far-field Approach
Rate of Change of Linear Momentum:
• 𝑉 ∶ The fluid velocity inside the control volume Ω (u,v,w)
• 𝑈 𝑛: The velocity of the elementary surface dS projected on the unit vector

4. Mean Drift Forces
4
Far-field Approach
Euler’s Equation (incompressible, inviscid):
Conservation of Mass:

5. Mean Drift Forces
5
Far-field Approach
Combining Eq. (1) (2) (3) and using Gauss’ theorem:

6. Mean Drift Forces
6
Far-field Approach
At the free surface:
• gz vanishes as it does not contribute to the Linear Momentum in x-
and y- direction
• No contribution from the free surface at the rate of change of Linear
Momentum

7. Mean Drift Forces
7
Far-field Approach
On the body surface:
• This is the force we are looking for (FX, Fy)

8. Mean Drift Forces
8
Far-field Approach
At the surface of the cylinder at infinity:
• This contribution is unknown and has to be calculated

9. Mean Drift Forces
9
Far-field Approach
Combining Eq. (5) (6) (7) with (4):

10. Mean Drift Forces
10
Far-field Approach
Averaging over one period:
• The average force does not contains the first order forces as
they average zero over one period
• The mean drift force can be estimated by deriving only a
potential solution at infinity
• Solving for the potential at infinity is the next task so as to
estimate the fluid velocity and pressure at infinity

11. Mean Drift Forces
11
Far-field Approach
Green’s Theorem for potentials:
• The potential φj(x,y,z) can be radiation or diffraction potential
at any point (x,y,z) of the domain
• Knowing the potentials on the body surface allows to us to
calculate the potential everywhere in the domain
• G is the Green’s function which dictates how potentials are
transferred throughout the domain as a result of the presence
of the body (see also Offshore Hydromechanics reader p.7-42)
• Green’s function satisfies all the boundary conditions and
conservation of mass
• Now it is needed to derive an approximation of Green’s
function at infinity

12. Mean Drift Forces
12
Far-field Approach
Green’s function approximated at infinity:
• R is the horizontal distance between a point on the body surface
and the point of the domain at which we want to estimate the
function
• ζ is the elevation of a point on the body surface
• The derivation of the above approximation can be found in
Newman’s paper (Blackboard)

13. Mean Drift Forces
13
Far-field Approach
Substituting Eq. (11) in (10) result to the potential at
infinite distance from the body :
• If distance R0 is rather large then the approximation
error is rather small
• H(π+θ) is the complex Kochin function corresponding to
radial direction π+θ
• The potential φj is complex and only space dependent
• For every radial direction (θ) around the body the
Kochin function is unique for every geometry
• The Kochin function is a directional function for
transferring the potential from the body surface to a
large distance R0 away from the body

14. Mean Drift Forces
14
Far-field Approach
Kochin function:
• ξ is the x-coordinate of a point on the body surface
• η is the y-coordinate of a point on the body surface
• The Kochin function contains all the information regarding
the body geometry
• There is a Kochin function for every radiation or diffraction
problem
• Kochin function in numerical simulations always comes with
a radial resolution

15. Mean Drift Forces
15
Far-field Approach
Bernoulli Equation:
Fluid Velocities in polar coordinates:

16. Mean Drift Forces
16
Far-field Approach
Combining Eq. (13) (14) (15) and substituting in Eq. (9) results to:
• A is the wave amplitude
• The detailed mathematical steps can be found in Newman’s paper
• Eq. (16) is only valid in deep water conditions
• Newman’s paper also derives the Mean Yaw Moment and you are
encouraged to study it
• Can you verify in Eq. (16) the quadratic relation to wave amplitude?

17. Mean Drift Forces
17
Far-field Approach
Connection to assignment:
• NEMOH calculates the Kochin functions using the same principles
• The mathematical formulation of mean drift forces with NEMOH is
different in terms of scaling
• NEMOH calculates the Kochin functions unscaled
• In Newman’s paper the diffraction associated Kochin function is
already scaled with wave amplitude
• The radiation associated Kochin function in Newman is already
scaled with the velocity amplitude