Practical Ship Hydrodynamics

86 Practical Ship Hydrodynamics
ž Open-boundary condition: waves generated by the ship pass unreflected any
artificial boundary of the computational domain.
ž Equilibrium: the ship is in equilibrium, i.e. trim and sinkage are changed
in such a way that the dynamical vertical force and the trim moment are
counteracted.
ž Bottom condition (shallow-water case): no water flows through the sea
bottom.
ž Side-wall condition (canal case): no water flows through the side walls.
ž Kutta condition (for catamaran/SWATH): at the stern/end of the strut the
flow separates. The Kutta condition describes a phenomenon associated with
viscous effects. Potential flow methods use special techniques to ensure that
the flow separates. However, the point of separation has to be determined
externally ‘by higher insight’. For geometries with sharp aftbodies (foils),
this is quite simple. For twin-hull ships, the disturbance of the flow by
one demi-hull induces a slightly non-uniform inflow at the other demi-hull.
This resembles the flow around a foil at a very small angle of incident. A
simplified Kutta condition suffices usually to ensure a realistic flow pattern
at the stern: Zero transverse flow is enforced. This is sometimes called the
‘Joukowski condition’.
The decay condition substitutes the open-boundary condition if the boundary of
the computational domain lies at infinity. The decay condition also substitutes
the bottom and side wall condition if bottom and side wall are at infinity,
which is the usual case.
Hull, transom stern, and Kutta condition are usually enforced numerically
at collocation points. Also a combination of kinematic and dynamic condition
is numerically fulfilled at collocation points. Combining dynamic and kinematic
boundary conditions eliminates the unknown wave elevation, but yields
a non-linear equation to be fulfilled at the apriori unknown free surface
elevation.
Classical methods linearize the differences between the actual flow and
uniform flow to simplify the non-linear boundary condition to a linear condition
fulfilled at the calm-water surface. This condition is called the Kelvin
condition. For practical purposes this crude approximation is nowadays no
longer accepted.
Dawson proposed in 1977 to use the potential of a double-body flow
and the undisturbed water surface as a better approximation. Double-body
linearizations were popular until the early 1990s. The original boundary
condition of Dawson was inconsistent. This inconsistency was copied by
most subsequent publications following Dawson’s approach. Sometimes this
inconsistency is accepted deliberately to avoid evaluation of higher derivatives,
but in most cases and possibly also in the original it was simply an oversight.
Dawson’s approach requires the evaluation of terms on the free surface
along streamlines of the double-body flow. This required either more or less
elaborate schemes for streamline tracking or some ‘courage’ in simply applying
Dawson’s approach on smooth grid lines on the free surface which were
algebraically generated.
Research groups in the UK and Japan proposed in the 1980s non-linear
approximations for the correct boundary condition. These non-linear methods
should not be confused with ‘fully non-linear’ methods that fulfil the correct