Practical Ship Hydrodynamics

Resistance and propulsion 87
non-linear boundary condition iteratively. They are not much better than
Dawson’s approach and no longer considered state of the art.
The first consistently linearized free surface condition for arbitrary approximations
of the base flow and the free surface elevation was developed in
HamburgbySöding. This condition is rather complicated involving up to
third derivatives of the potential, but it can be simply repeated in an iterative
process which is usually started with uniform flow and no waves. Section 7.3,
Chapter 7, will derive this expression for the wave resistance problem.
Fully non-linear methods were first developed in Sweden and Germany
in the late 1980s. The success of these methods quickly motivated various
other research groups to copy the techniques and apply the methods commercially.
The most well-known codes used in commercial applications include
SHIPFLOW-XPAN, SHALLO, RAPID, SWIFT, and FSWAVE/VSAERO. The
development is very near the limit of what potential flow codes can achieve.
The state of the art is well documented in two PhD theses, Raven (1996) and
Janson (1996). Despite occasional other claims, all ‘fully non-linear’ codes
have similar capabilities when used by their designers or somebody well trained
in using the specific code. Everybody loves his own child best, but objectively
the differences are small. All ‘fully non-linear’ codes in commercial use share
similar shortcomings when it comes to handling breaking waves, semi-planing
or planing boats or extreme non-linearities. It is debatable if these topics should
be researched following an inviscid approach in view of the progress that
viscous free-surface CFD codes have made.
Once the unknown velocity potential is determined, Bernoulli’s equation
determines the wave elevation. In principle, a linearized version of Bernoulli’s
equation might be used. However, it is computationally simpler to use the
non-linear equation. Once the potential is determined, the forces can also be
determined by direct pressure integration on the wetted hull. The wave resistance
may also be determined by an analysis of the wave pattern (wave cut
analysis) which is reported to be often more accurate. The z-force and ymoment
are used to adjust the position of the ship in fully non-linear methods.
Waves propagate only downstream (except for rare shallow-water cases).
This radiation condition has to be enforced by numerical techniques. Most
methods employ special finite difference (FD) operators to compute second
derivatives of the potential in the free surface condition. Dawson proposed
a four-point FD operator for second derivatives along streamlines. Besides
the considered collocation point, the FD operator uses the next three points
upstream. Dawson’s method automatically requires grids oriented along
streamlines of the double-body flow approximate solution. Dawson determined
his operator by trial and error for a two-dimensional flow with a simple Kelvin
condition. His criteria were that the wave length should correspond to the
analytically predicted wave length and the wave amplitude should remain
constant some distance behind the disturbance causing the waves.
Dawson approximated the derivative of any function H with respect to ℓ at
the point i numerically by:
Hℓi ³ CAiHi C CBiHi 1 C CCiHi 2 C CDiHi 3
Hℓi is the derivative with respect to ℓ at point Pi. Hi to Hi 3 are the values
of the function H at points Pi to Pi 3, all lying on the same streamline of the
double-body flow upstream of Pi. Thecoefficients CAi to CDi are determined