Mean Drift ForcesFar-field Approach

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Mean Drift ForcesFar-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

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Mean Drift ForcesFar-field ApproachRate of Change of Linear Momentum:

• The fluid velocity inside the control volume Ω (u,v,w)• : The velocity of the elementary surface dS projected on the unit vector

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Mean Drift ForcesFar-field ApproachEuler’s Equation (incompressible, inviscid):

Conservation of Mass:

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Mean Drift ForcesFar-field ApproachCombining Eq. (1) (2) (3) and using Gauss’ theorem:

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Mean Drift ForcesFar-field ApproachAt 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

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Mean Drift ForcesFar-field ApproachOn the body surface:

• This is the force we are looking for (FX, Fy)

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Mean Drift ForcesFar-field ApproachAt the surface of the cylinder at infinity:

• This contribution is unknown and has to be calculated

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Mean Drift ForcesFar-field ApproachCombining Eq. (5) (6) (7) with (4):

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Mean Drift ForcesFar-field ApproachAveraging 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

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Mean Drift ForcesFar-field ApproachGreen’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

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Mean Drift ForcesFar-field ApproachGreen’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)

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Mean Drift ForcesFar-field ApproachSubstituting 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

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Mean Drift ForcesFar-field ApproachKochin 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

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Mean Drift ForcesFar-field ApproachBernoulli Equation:

Fluid Velocities in polar coordinates:

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Mean Drift ForcesFar-field ApproachCombining 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?

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Mean Drift ForcesFar-field ApproachConnection 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