Practical Ship Hydrodynamics

Ship seakeeping 119
for each strip can be solved analytically or by panel methods, which are
the two-dimensional equivalent of the three-dimensional methods described
below. The analytical approaches use conform mapping to transform semicircles
to cross-sections resembling ship sections (Lewis sections). Although
this transformation is limited and, e.g., submerged bulbous bow sections
cannot be represented in satisfactory approximation, this approach still yields
for many ships results of similar quality as strip methods based on panel
methods (close-fit approach). A close-fit approach (panel method) to solve
the two-dimensional problem will be described in section 7.4, Chapter 7.
Strip methods are – despite inherent theoretical shortcomings – fast, cheap
and for most problems sufficiently accurate. However, this depends on many
details. Insufficient accuracy of strip methods often cited in the literature is
often due to the particular implementation of a code and not due to the strip
method in principle. But at least in their conventional form, strip methods
fail (as most other computational methods) for waves shorter than perhaps
1
3
of the ship length. Therefore, the added resistance in short waves (being
considerable for ships with a blunt waterline) can also only be estimated by
strip methods if empirical corrections are introduced. Section 4.4.2 describes
a linear strip method in more detail.
ž Unified theory
Newman (1978) and Sclavounos developed at the MIT the ‘unified theory’
for slender bodies. Kashiwagi (1997) describes more recent developments
of this theory. In essence, the theory uses the slenderness of the ship hull
to justify a two-dimensional approach in the near field which is coupled
to a three-dimensional flow in the far field. The far-field flow is generated
by distributing singularities along the centreline of the ship. This approach
is theoretically applicable to all frequencies, hence ‘unified’. Despite its
better theoretical foundation, unified theories failed to give significantly and
consistently better results than strip theories for real ship geometries. The
method therefore failed to be accepted by practice.
ž ‘High-speed strip theory’ (HSST)
Several authors have contributed to the high-speed strip theory after the
initial work of Chapman (1975). A review of work since then can be
found in Kashiwagi (1997). HSST usually computes the ship motions in an
elementary wave using linear potential theory. The method is often called
2 1
2
dimensional, since it considers the effect of upstream sections on the
flow at a point x, but not the effect of downstream sections. Starting at the
bow, the flow problem is solved for individual strips (sections) x D constant.
The boundary conditions at the free surface and the hull (strip contour) are
used to determine the wave elevation and the velocity potential at the free
surface and the hull. Derivatives in longitudinal direction are computed as
numerical differences to the upstream strip which has been computed in the
previous step. The computation marches downstream from strip to strip and
ends at the stern resp. just before the transom. HSST is the appropriate tool
for fast ships with Froude numbers Fn > 0.4. For lower Froude numbers,
it is inappropriate.
ž Green function method (GFM)
ISSC (1994) gives a literature review of these methods. GFM distribute
panels on the average wetted surface (usually for calm-water floating position
neglecting dynamical trim and sinkage and the steady wave profile) or