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Explain the method

3D flow problem is replaced by several 2D problems-

- 2D flows in an earth fixed transverse cross planes
- Development of the 2D flow in time is computed
- Variation of the flow in longtudinal direction is neglected

Applicable for slender ships and high froude numbers


Boundary conditions 3D

Laplace equation:
ø_xx + ø_yy + ø_zz = 0

Free surface boundary condition:
Dø/dt = 1/2(ø_x^2+ø_x*2+ø_z^2)-gz
Dx/dt = ø_x, Dy/dt = ø_y, Dz/dt = ø_z

Body boundary condition:
ø_n=v_b*n = U*n_x (no flux)

2D is the same without the x-components


Calm water boundary conditions

- Hull: normal velocity ø_n=U*n_x known
- Outer boundaries: ø_n =0 known
- Free surface: potential ø known


Computation of the time development

Initial conditions y0, z0 and ø0 on the free surface assumed. Free surface boundary conditions are used for time stepping (MEL)

- For steady problems only a single section needs to be considered
- Different time instants are assigned different locations along the hull to obtain teh steady flow


Typical vertical force distribution for planning vessels obtained by the 2D+t method (Calm water)

- increases almost linearly from the bow towards the stern
- Reaches its maximum at the chine-wetted position
- Increases again after separation towards the stern


Why can neglected 3D effects cause errors? (Calm water)

Pressures at the transom stern tend to be overpredicted.


What are the components of the BEM equation?

Flow is computed numerically by BEM which is the integral alpha(x)*ø(x) = int(....ln(r) - ø_n*ln(r))
The first part describes the potential of the dipols and the second part describes the potential of the sources.

This means that the sources and the dipols are integrated over the 2D surface to find the flow.

BEM method is very efficient, but has high computational effort.


What has to be considered in rough water?

Heave and pitch


What can be accounted for in this method?

Some of the most important nonlinear effects of the hydrodynamics of high speed ships


What is a weakness of the method?

Neglected 3D effects.