Practical Ship Hydrodynamics

Resistance and propulsion 89
resistance problem, the time-domain approach seems unnecessarily expensive
and is rarely used in practice. Norwegian researchers tried to reduce
the computational domain by matching the panel solution for the near-field
to a thin-ship-theory solution in the far-field. However, this approach saved
only little computational time at the expense of a considerably more complicated
code and was subsequently abandoned. The ‘staggered grid’ technique
is again an elegant alternative. Without further special treatment, waves leave
the computational domain without reflection.
Most methods integrate the pressure on the ship’s surface to determine the
forces (especially the resistance) and moments. ‘Fully non-linear’ methods
integrate over the actually wetted surface while older methods often take the
CWL as the upper boundary for the integration. An alternative to pressure
integration is the analysis of the wave energy behind the ship (wave cut analysis).
The wave resistance coefficients should theoretically tend to zero for low
speeds. Pressure integration gives usually resistance coefficients which remain
finite for small Froude numbers. However, wave cut analysis requires larger
grids behind the ship leading to increased computational time and storage.
Most developers of wave resistance codes have at some point tried to incorporate
wave cut analysis to determine the wave resistance more accurately.
So far the evidence has not yet been compelling enough to abandon the direct
pressure integration.
Most panel methods give as a direct result the source strengths of the panels.
A subsequent computation determines the velocities at the individual points.
Bernoulli’s equation then gives pressures and wave elevations (again at individual
points). Integration of pressures and wave heights finally yields the
desired forces and moments which in turn are used to determine dynamical
trim and sinkage (‘squat’).
Fully non-linear state of the art codes fulfil iteratively an equilibrium condition
(dynamical trim and sinkage) and both kinematic and dynamic conditions
on the actually deformed free surface. The differences in results between ‘fully
non-linear’ and linear or ‘somewhat non-linear’ computations are considerable
(typically 25%), but the agreement of computed and measured resistances
is not better in ‘fully non-linear’ methods. This may in part be due to the
computational procedure or inherent assumptions in computing a wave resistance
from experimental data (usually using a form factor method), but also
due to computational errors in determining the resistance which are of similar
magnitude as the actual resistance. One reason for the unsatisfactory accuracy
of the numerical procedures lies in the numerical sensitivity of the pressure
integration. The pressure integration involves basically subtracting forces of
same magnitude which largely cancel. The relative error is strongly propagated
in such a case. Initial errors stem from the discretization. For example,
integration of the hydrostatic pressure for the ship at rest should give zero
longitudinal force, but usual discretizations show forces that may lie within
the same order of magnitude as the wave resistance. Still, there is consensus
that panel methods capture the pressure distribution at the bow quite accurately.
The vertical force is not affected by the numerical sensitivity. Predictions for
the dynamical sinkage differ usually by less than 5% for a large bandwidth of
Froude numbers. Trim moment is not predicted as well due to viscous effects
and numerical sensitivity. This tendency is amplified by shallow water.