Unsteady Flow Simulation
The most commonly considered sources of unsteadiness in wind turbine flow are yaw and tower interference. Wind turbines can pivot on the tower to face the incoming wind. The control system is designed to correct for non zero yaw deviations, but yaw cannot be maintain to zero at all times. It is therefore important to understand the effect of yaw on the forces and moments in the blades in order to estimate fatigue life of the rotor structure subjected to this low frequency solicitations. For the same reason, it is necessary to estimate the rapid load changes that occur when a blade is passing in front of or behind the tower, as this is both a source of fatigue but also of noise.
For this analysis, we have to take into account the following new requirements:
• the blades have different loading and circulation distributions, Гі=1, j, n, n = 1, 2 or 3 for a two – or threebladed rotor
• the circulation on the vortex sheets varies with both i and j along the vortex lattices
• the blades shed vorticity that is convected downstream with the flow
• the incoming flow contributes to the axial and azimuthal components in the rotating frame.
In other words, the circulation is no longer constant along a vortex filament. For this reason, each small element of the vortex lattice carries vorticity that varies in time and space and its individual contribution can no longer be accumulated with those along the same vortex filament. For a threebladed rotor, the induced мvelocity component is calculated as
jx 1 
jx1 

uj,1 
= 2^ (ri, k+1,1 – ri, k,1)ai, k, j,1 
+ (ri, k+1,2 – ri, k,2)ai, k, j,2 
k=1 
k=1 

jx1 
jx1 

+ 
(ri, k+1,3 – ri, k,3)ai, k, j,3 + 
(Гі+1,k,1 – ri, k,1 )ai, k, j,4 
k=1 
k=2 

jx1 
jx1 

+ 
(Гі+1,k,2 – ri, k,2)ai, k, j,5 + 
(Гі+1,k,3 – ri, k,3)ai, k, j,6 
k=2 
k=2 

jx1 
jx1 
+ r1,k,2ak, j,2 + ^ r1,k,3ak, j,3 (Ш.96) k=2 k=2 
where the first three terms correspond to the trailed vorticity influence of the vortex sheets of blades 1, 2 and 3 on blade 1, the next three terms are the contributions of the shed vorticity, and the last two tilde “~” terms correspond to the influence of the lifting lines 2 and 3 on blade 1. Similar formulae hold for blades 2 and 3 due to the symmetry in the geometry of the blades and vortex sheets. Similarly, the шcomponents influence coefficients are stored in arrays ci, j, k,n, n = 1,…, 6.
Note that in this approach, the vortex sheets are not displaced or distorted by yaw or tower interference, which limits the yaw angle to approximately 20° maximum. The vortex sheet is that which corresponds to zero yaw and no tower modelization and is called the “base helix”. It is depicted in Fig. 10.18.
One of the nice features of this approach is that the flow becomes periodic shortly after the initial shed vorticity has crossed the Trefftz plane, that is for a dimensionless time T > TXi— or a number of iterations n > ту xiAAt = .
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