. Effect of Yaw

The procedure to calculate the circulation at the blades is unchanged. Once the induced velocities are obtained, the local incidence of the blade is found and the steady viscous polar searched for lift and drag. The local blade element working conditions now depend on the yaw angle в and the azimuthal angle f measured to be zero when blade 1 is in the vertical position. The lift provides the circulation according to the Kutta-Joukowski lift theorem

1

Г1, j,1 = 2 qj,1CjClj,1 (10.97)

where qj1 is the local velocity magnitude, Cj the chord and Qjj the local lift coefficient for blade 1.

The new equation that governs the unsteady flow physics is the convection of the circulation along the wake

+ (1 + 2u) = 0 (10.98)

d t dx

where 2u is the average axial velocity at the Trefftz plane as given by the actuator disk theory and is used from the rotor disk, but could be made to vary as is done for the pitch of the helix. This equation is solved with a two-point semi-implicit scheme:

This scheme is unconditionally stable for 9 > 1 and reduces to Crank-Nicolson scheme for 9 = 2. Here we use 9 = 5. The index v represents the inner iteration loop, needed to satisfy, within one time step, both the circulation equation inside the blades and the convection equation along the vortex sheets to a prescribed accuracy e = 10-5. Let a = (1 + 2u)At/(xi – xi-1) denote the Courant-Friedrich-Lewy (CFL) number. The distribution of points along the vortex sheets is stretched from the trailing edge of the blade, where Ax1 = 2.0 x 10-3 to a constant step Ax = (1 + 2u)At from approximately x = 1 to the Trefftz plane, such that the CFL number is one along that uniform mesh region. This feature, together with 9 = 1 provides the nice property that the circulation is convected without dissipation or distortion all the way to the Trefftz plane (perfect shift property) [14].