Wind Energy

262 S. Ivanell et al.
Assuming that all wake vorticity is included in the tip and the root vortex,
the circulation at tip and root should approximately correspond to the maximum
bound circulation of the blade. This is of course an idealization, since
the roll-up of the vortex sheets is expected to form a concentrated vorticity
distribution in the far wake [7]. The computed circulation at the blades is
given in Fig. 48.3.
Figure 48.3 shows some wiggles that start at about 500 ◦ behind the blades
for the root vortex and about 800 ◦ behind the blade for the tip vortex. These
are most likely due to integration errors when integrating the circulation.
At these positions the vortex core starts to be smeared out and it therefore
becomes difficult to identify a good integration path.
The results, however, correspond fairly well to classical theories from
Helmholtz. The tip vortex leaves the blade with a circulation value close to
the maximum bound circulation at the blades. The root vortex leaves the
blade with a much lower value of the circulation, but it increases rapidly and
reaches values at the same order as the bound circulation. The reason why
the root vortex is smeared out earlier than the tip vortex is partly because the
vortex cores are closer at the root and partly because of the radial gradient of
the circulation. The result is in good agreement with classical theorems from
Helmholtz, from which it follows that the wake tip vortex has the same circulation
as the maximum value of the bound circulation on the blade. A likely
explanation for this good agreement is that the symmetry of the flow domain
prohibits the roll-up of the vortex sheet from the blades.
Circulation
0.14
0.13
0.12
0.11
0.1
0.09
0.08
0.07
0.06
0.05
96 Tip,eps =1 Re1
96 Tip,eps =1 Re2
96 Tip,eps =1 Re3
96 Rot,eps =1 Re1
96 Rot,eps =1 Re2
96 Rot,eps =1 Re3
Maximum circulation at blade
0 200 400 600 800 1000 1200
Deg. behind blade
Fig. 48.3. The figure shows the circulation distribution in the wake at three different
Reynolds number compared with the maximum circulation at the blade [8]