# VAWT, wind induction factor

I'm trying to model a vertical axis wind turbine with the double-multiple streamtube model (with Matlab), where wind speed is dampened by an induction factor so that the output is different for upwind and downwind parts of rotation.

The trouble is that for the downwind part, the induction factor seems to have a peak in the middle, which screws up the system. I was expecting the upwind and downwind angle of attack curves to be the same shape, but of different heights, but the downwind part gets two separate minima or even changes sign (see attached images). This problem occurs at lower wind speeds, but hints of it can also be noted at higher speeds.

I'm wondering if this is just a feature of the DMS model or if there is something wrong with my model.

For the downwind part, the iterative equation I use for solving the induction factor is

$$a(1-a) = \frac{Nc}{8\pi r} \left(\frac{v_R}{v_a}\right)^2 \left(-C_N + \frac{C_T}{\tan{\theta}}\right) \,,$$

where $$v_R$$ is the effective wind speed, $$v_a$$ is the dampened wind speed after the upwind part of the rotor and $$\theta$$ is the azimuthal angle of the turbine blade, when the wind comes from the top of an xy-plane.

Any ideas?