Figure 13: Local flow at the blade tip as treated in THPUF-3A.
Figure 12: Local flow at the blade tip with a developed vortex cavity.
The flow at the tip of a blade in the presence of
a developed tip vortex cavity is depicted in Figure 12.
The flow angularity (i.e.
the contraction angle 
with respect to the pure helical direction)
should be included in the dynamic boundary condition
in THPUF-3A. The analysis is an extension of that in [18], and
is also depicted in Figure 13.
Defining as:
: the unperturbed inflow velocity
at radius r with respect to the blade, with 
being the propeller
angular velocity and 
the axial inflow velocity at radius r.
: the angle of the unperturbed flow with respect to the normal
to the blade leading edge line.
: the total velocity on the cavity with components 
and
along the normal and the tangential directions to the blade leading
edge line, respectively.
: the angle between the direction of and the normal to the
leading edge line. It is given from:
Then:
where A is the Lighthill leading edge correction factor applied along the normal to the blade leading edge line:
By using 
we get:
Finally, from 
, we get:
where 
is defined as follows:
Notice that equation (7) is in a similar form to that of equation (44) in [18]
The effect of the angle on the cavity
solution at the tip and on the overall cavity volume was presented
in [1] (pages 26 to 29).
The angle can be estimated from matching the flow angle at the
tip of the blade
to the angle between the axial velocity 
(along the trajectory
of the vortex) and the total velocity direction on the surface of
the tip vortex cavity. By
using the following approximation on the surface of the tip vortex:
and knowing that 
(see [18]) with
, we end up with:
where J is the advance coefficient.
First, the effect of including the tip flow angularity correction is given in Figure 14. Notice that the predicted cavity volume is increased appreciably when the tip effects are included.
Figure 14: Effect of the tip flow angularity on the cavity volume
for the sample run.
