Figure 2: Convergence of the circulation around the shown hydrofoil
section as predicted by the low-order panel method, with and
without the ``saw-tooth'' effects.
The foil is at an angle 
and the analytical non-dimensional value
of the circulation is equal to 0.8844. From Kinnas and Hsin (1994).
Figure 3: Schematic showing the ``saw-tooth'' perturbation
potential at the trailing edge of a hydrofoil. From Kinnas and Hsin (1994).
The low-order boundary element method has been found to work very well
for most hydrofoil or propeller blade geometries[5].
In particular, it has
been found that the potential-based formulation can handle very well even
very thin hydrofoil sections (with as low as 
thickness to chord ratio)
when low-order velocity-based formulations have been found to fail[5].
However, in the case of sections with moderate to high camber
an error associated with the
treatment of the flow at the sharp trailing edge of a hydrofoil has been
found to slow the convergence of the method with number of panels, drastically
[6]. This may be seen in Figure 2 where the predicted
circulation distribution for a typical hydrofoil section
(shown at the top of the figure)
is shown to be in relative error even when 120 panels are used around the
hydrofoil. The error was found to be due to the omission of the ``saw-tooth''
effects, as shown in Figure 3. We call ``saw-tooth'' the difference
of the continuous potential distribution from the piece-wise constant distribution
which is employed in the case of a low-order method.
These effects would have been included
automatically if a linear (or higher order)
instead of constant panel distribution had been
used on each panel. Alternatively, the ``saw-tooth'' effects may be included
within the constant strength panel method by correcting the right-hand side
of the matrix equation (8) as follows:
where 
is the matrix of the saw-tooth coefficients
(linear potential distributions over each panel with unit slope and zero value at
the panel mid-point), and 
is the vector
with N components the values of the slopes 
of the potential with respect to the arc-length. These slopes are determined
from the values of the potentials at the panel mid-points using a second-order
accurate finite difference scheme. The second term in the right-hand side of
equation (9) is updated with iteration number i, until convergence
of the potential 
.
Usually 2-3 iterations suffice. The results from applying the saw-tooth correction
are shown in Figure 2 for the same hydrofoil. Notice the remarquable
improvement in the convergence of the constant panel method when the saw-tooth
correction is included. Apparently, this method can be extended to include higher
than linear saw-tooth corrections. However, for most hydrofoil and propeller
applications the linear saw-tooth correction has been found to be sufficient.
The main advantage of the ``saw-tooth'' correction method over the ``direct''
high-order methods is the fact that high-order saw-tooth corrections can be
easily included in the right-hand side of the equations without having to
change the matrix that multiplies the solution. In addition, the ``saw-tooth''
method applies the BEM always
at the panel mid-points, thus avoiding the need of
special treatment for the corners. This feature is what prompted the application
of this method into the structural BEM which will be presented in later sections.
