Practical Ship Hydrodynamics

228 Practical Ship Hydrodynamics
similarly employ a Kutta condition, but this is often omitted. If a Kutta condition
is employed in frequency-domain computations, the wake will oscillate
harmonically in strength. This can be modelled by discrete dipole elements
of constant strength, but for high frequencies this approach requires many
elements. The use of special elements which consider the oscillating strength
analytically is more efficient and accurate, but also more complicated. Such a
‘Thiart element’ has been developed by Professor Gerhard Thiart of Stellenbosch
University and is described in detail in Bertram (1998b), and Bertram
and Thiart (1998). The oscillating ship creates a vorticity. The problem is
similar to that of an oscillating airfoil. The circulation is assumed constant
within the ship. Behind the ship, vorticity is shed downstream with ship speed
V. Then:
� ∂
∂t
�
∂
V
∂x
⊲x, z, t⊳ D 0
is the vortex density, i.e. the strength distribution for a continuous vortex
sheet. The following distribution fulfils the above condition:
⊲x, z, t⊳ D Re⊲ Oa⊲z⊳ Ð e i⊲ωe/V⊳⊲x xa⊳ Ð e iωet ⊳ for x xa
Here Oa is the vorticity density at the trailing edge xa (stern of the ship).
We continue the vortex sheet inside the ship at the symmetry plane y D 0,
assuming a constant vorticity density:
⊲x, z, t⊳ D Re⊲ Oa⊲z⊳ Ð e iωet ⊳ for xa x xf
xf is the leading edge (forward stem of the ship). This vorticity density is
spatially constant within the ship.
A vortex distribution is equivalent to a dipole distribution if the vortex
density and the dipole density m are coupled by:
D dm
dx
The potential of an equivalent semi-infinite strip of dipoles is then obtained
by integration. This potential is given (except for a so far arbitrary ‘strength’
constant) by:
�� � xf zo
8⊲x, y, z⊳ D Re
1 zu
with r D p ⊲x ⊳ 2 C y 2 C ⊲z ⊳ 2 and:
Om⊲ ⊳ y
�
d d eiωet D Re⊲ϕ⊲x, y, z⊳ Ð e 3
r iωet
⊳
�
xf for xa xf
Om⊲ ⊳ D
V �
i⊲ωe/V⊳⊲ xa⊳ 1 e
iωe
� C ⊲xf xa⊳ for 1 xa