Practical Ship Hydrodynamics

Ship manoeuvring 167
accuracy, but the experience published so far is insufficient to establish this as
state of the art.
The simplest approach to body force computations is the use of regression
formulae based on slender-body theory, but with empirical coefficients found
from analysing various model experiments, e.g. Clarke et al. (1983). The next
more sophisticated approach would be to apply slender-body methods directly,
deriving the added mass terms for each strip from analytical (Lewis form) or
BEM computations. These approaches are still state of the art in ship design
practice and have been discussed in the previous chapters.
The application of three-dimensional CFD methods, using either liftgenerating
boundary elements (vortex or dipole) or field methods (Euler or
RANSE solvers) is still predominantly a matter of research, although the
boundary element methods are occasionally applied in practical design. The
main individual CFD approaches are ranked in increasing complexity:
ž Lifting surface methods
An alternative to slender-body theory, applicable to rudder and hull (separately
or in combination), is the lifting surface model. It models the inviscid
flow about a plate (centre plane), satisfying the Kutta condition (smooth
flow at the trailing edge) and usually the free-surface condition for zero Fn
(double-body flow). The flow is determined as a superposition of horseshoe
vortices which are symmetrical with respect to the water surface (mirror
plane). The strength of each horseshoe vortex is determined by a collocation
method from Biot–Savart’s law. For stationary flow conditions, in the
ship’s wake there are no vertical vortex lines, whereas in instationary flow
vertical vortex lines are required also in the wake. The vortex strength in
the wake follows from three conditions:
1. Vortex lines in the wake flow backwards with the surrounding fluid
velocity, approximately with the ship speed u.
2. If the sum of vertical (‘bound’) vortex strength increases over time
within the body (due to larger angles of attack), a corresponding negative
vorticity leaves the trailing edge, entering into the wake.
3. The vertical vortex density is continuous at the trailing edge.
Except for a ship in waves, it seems accurate enough to use the stationary
vortex model for manoeuvring investigations.
Vortex strengths within the body are determined from the condition that
the flow is parallel to the midship (or rudder) plane at a number of collocation
points. The vortices are located at 1/4 of the chord length from
the bow, the collocation points at 3/4 of the chord length from the bow.
This gives a system of linear equations to determine the vortex strengths.
Transverse forces on the body may then be determined from the law of
Kutta–Joukowski, i.e. the body force is the force exerted on all ‘bound’
(vertical) vortices by the surrounding flow.
Alternatively one can smooth the bound vorticity over the plate length,
determine the pressure difference between port and starboard of the plate,
and integrate this pressure difference. For shallow water, reflections of the
vortices are necessary both at the water surface and at the bottom. This
produces an infinite number of reflections, a subset of which is used in
numerical approximations. If the horizontal vortex lines are arranged in the
ship’s centre plane, only transverse forces depending linearly on v and r are
generated. The equivalent to the non-linear cross-flow forces in slender-body