Mean Drift Forces Far-field Approach

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Mean Drift ForcesFar-field Approach

Mean Drift Forces2Far-field Approach

Control Volume: Sb : Under water body surfaceSfs : Water free surfaceS : Cylindrical surface surrounding control volume at infinite distanceBottom neglected as infinite water depth is assumed

Mean Drift Forces3Far-field ApproachRate of Change of Linear Momentum:

Mean Drift Forces4Far-field ApproachEulers Equation (incompressible, inviscid):

Conservation of Mass:

Mean Drift Forces5Far-field ApproachCombining Eq. (1) (2) (3) and using Gauss theorem:

Mean Drift Forces6Far-field ApproachAt the free surface:

gz vanishes as it does not contribute to the Linear Momentum in x- and y- directionNo contribution from the free surface at the rate of change of Linear Momentum

Mean Drift Forces7Far-field ApproachOn the body surface:This is the force we are looking for (FX, Fy)

Mean Drift Forces8Far-field ApproachAt the surface of the cylinder at infinity:This contribution is unknown and has to be calculated

Mean Drift Forces9Far-field ApproachCombining Eq. (5) (6) (7) with (4):

Mean Drift Forces10Far-field ApproachAveraging over one period:

The average force does not contains the first order forces as they average zero over one periodThe mean drift force can be estimated by deriving only a potential solution at infinitySolving for the potential at infinity is the next task so as to estimate the fluid velocity and pressure at infinity

Mean Drift Forces11Far-field ApproachGreens Theorem for potentials:The potential j(x,y,z) can be radiation or diffraction potential at any point (x,y,z) of the domainKnowing the potentials on the body surface allows to us to calculate the potential everywhere in the domainG is the Greens 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)Greens function satisfies all the boundary conditions and conservation of mass Now it is needed to derive an approximation of Greens function at infinity

Mean Drift Forces12Far-field ApproachGreens 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 is the elevation of a point on the body surfaceThe derivation of the above approximation can be found in Newmans paper (Blackboard)

Mean Drift Forces13Far-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 smallH(+) is the complex Kochin function corresponding to radial direction +The potential j is complex and only space dependent For every radial direction () around the body the Kochin function is unique for every geometryThe Kochin function is a directional function for transferring the potential from the body surface to a large distance R0 away from the body

Mean Drift Forces14Far-field ApproachKochin function:

is the x-coordinate of a point on the body surface is the y-coordinate of a point on the body surfaceThe Kochin function contains all the information regarding the body geometryThere is a Kochin function for every radiation or diffraction problemKochin function in numerical simulations always comes with a radial resolution

Mean Drift Forces15Far-field ApproachBernoulli Equation:

Fluid Velocities in polar coordinates:

Mean Drift Forces16Far-field ApproachCombining Eq. (13) (14) (15) and substituting in Eq. (9) results to:

A is the wave amplitudeThe detailed mathematical steps can be found in Newmans paper Eq. (16) is only valid in deep water conditionsNewmans paper also derives the Mean Yaw Moment and you are encouraged to study itCan you verify in Eq. (16) the quadratic relation to wave amplitude?

Mean Drift Forces17Far-field ApproachConnection to assignment:NEMOH calculates the Kochin functions using the same principlesThe mathematical formulation of mean drift forces with NEMOH is different in terms of scalingNEMOH calculates the Kochin functions unscaledIn Newmans paper the diffraction associated Kochin function is already scaled with wave amplitudeThe radiation associated Kochin function in Newman is already scaled with the velocity amplitude