The second order velocity potential for diffraction of waves by fixed offshore structures.
It is well known that second order effects may in many
cases be important for the nonlinear hydrodynamic problems arising
in ocean engineering. Despite considerable efforts having been
made in the past in calculating second order unsteady forces,
similar studies are rare for the actual second order velocity
potential itself, which is important for the understanding of wave
kinematics. A mathematical model has been developed for the
calculation of the second order sum frequency diffraction potential
for fixed bodies in waves.
It is believed that a first step towards the solution of
the second order problem is the accurate evaluation of the first
order quantities. By the use of Green's second identity, the first
order problem can be cast into the form of a Fredholm integral
equation and then solved by the Boundary Element Method. Some new
developments based on this technique have been undertaken in this
work, and as a result, there is a major improvement in the accuracy
of the first order analysis.
For the second order problem, the solution procedures are
similar to those used for the first order problem except that
special techniques have been developed to calculate efficiently the
additional free surface integral which decays slowly to infinity in
a highly oscillatory manner. In addition, an effective method has
also been implemented to calculate the second derivative term in
the free surface integral. From the numerical results presented, a
number of interesting findings are illustrated.
A closed form expression for a vertical circular cylinder
has also been developed which not only furnishes a valuable check
on the general numerical model but also provides some physical
explanation for the second order phenomena. Moreover, it has been
used to investigate some theoretical problems which (in the past)
have caused confusion and error in the second order analysis. They
are mainly associated with the troublesome nonhomogeneity presented
in the free surface boundary condition.