Ok. So this will be a long post. I did some calculations.

So the thing I will be discussing here is a magnus based AWE rig working in Lift mode (Yoyo), but with a energy harvesting device onboard (drag mode turbine). The drag mode turbine will harvest energy that is used to rotate the magnus wings. This is not necessarily a sensible design, just a basis for the calculations. The design has one tether and one wing, but you could make variations with more wings and such, the math is more or less the same anyway.

w - wind speed

G_e - glide number for magnus wing at speed ratio k

G_{e,\Sigma} - glide number of magnus wing and power harvester together

\eta - efficiency of power harvesting, ie losses from creating drag to making magnus wings rotate

C_L - lift coefficient of magnus wing

C_D - drag coefficient of magnus wing

C_{D,\Sigma} - drag coefficient of magnus wing and power harvesting device

l - length of magnus wing

c - chord/diameter of magnus wing

k - speed ratio apparent wind vs speed of magnus wing skin

C_f - friction constant from paper, value 0.007 Experiments on a Flettner rotor at critical and supercritical Reynolds numbers [1]

P - power generated by the rig

P_m - power consumed by motors to rotate magnus wings

D_m - drag force of magnus wing harvesting device

D_\Sigma - Drag force of wing and harvesting device together

n - loss factor to account for cosine loss, roll loss etc, 0 < n < 1

w_a - apparent wind at the kite

We’ll start looking at the apparent wind speed of the kite. I wont get into details, but lets assume it is:

w_a = n G_{e,\Sigma} w

That is, the kite is moving with the wind speed times glide ratio, deducted cosine losses.

The power necessary for the magnus rotation motors are

P_m = \frac{1}{2} C_f \rho (k n G_{e,\Sigma} w)^3 \left( \pi l c \right)

Using power equals to speed times force we get

D_m = \frac{P_m}{w_a}

D_m = \frac{1}{2} C_f k^3 \rho (n G_{e,\Sigma} w)^2 \left( \pi l c \right)

Next, we calculate the total drag of the wing and harvesting device

D_\Sigma = \frac{1}{2} \rho (n G_{e,\Sigma} w)^2 C_D l c + \frac{1}{2} \frac{1}{\eta} C_f \rho k^3 (n G_{e,\Sigma} w)^2 \left( \pi l c \right)

D_\Sigma = \frac{1}{2} \rho (n G_{e,\Sigma} w)^2 \left( C_D + \frac{1}{\eta} C_f k^3 \pi \right) l c

We can see from this equation that the effective drag coefficient is

C_{D,\Sigma} = C_D \left( 1 + \frac{C_f k^3 \pi}{\eta C_D} \right)

G_{e,\Sigma} = \frac{C_L}{C_D \left( 1 + \frac{C_f k^3 \pi}{\eta C_D} \right)}

G_{e,\Sigma} = G_e \frac{1}{ 1 + \frac{C_f k^3 \pi}{\eta C_D} }

Next, we’ll assume that the kite is being reeled out at one third wind speed (proven optimal in theory). Thus the power generated by the rig is

P = \left(\frac{1}{3} w \right) \left(\frac{1}{2} \rho n^2 G_{e,\Sigma}^2 w^2 C_L l c\right)

P = \frac{1}{6} \rho n^2 G_{e,\Sigma}^2 w^3 C_L l c

P = \frac{1}{6} \rho n^2 w^3 C_L G_e^2 l c \frac{1}{\left( 1 + \frac{C_f k^3 \pi}{\eta C_D} \right)^2}

Now to analyze this, we see that the power is equal to the nominal power of the wing without the power harvesting device, multiplied by a harvesting/rotation loss. The [square root of the] denominator of this factor is:

1 + \frac{C_f k^3 \pi}{\eta C_D}

Let’s select some test values. We assume \eta = 0.5, 50% loss in powering the magnus motors. Lift and drag coefficients for different k were «lifted» from the paper [1]. Some values of k give us:

```
____ (50% conv loss) ____ ____ (0% conv loss) ____
k losses [%] C_L^3/C_D^2 losses [%] C_L^3/C_D^2
1.0 7.54152 4.28501 3.88161 4.54195
2.0 31.3559 9.18151 17.877 12.0145
3.0 56.5068 8.53804 36.8272 14.9459
4.0 74.2496 5.79932 54.6728 13.5438
5.0 84.8691 3.50258 68.6289 10.4565
soft kite (*) 6.3
```

*) Reference soft kite with C_L = 1.0, G_e = 2.5

Optimal value of k seems to be around 2.3 for conversion loss 50% and 3.1 without conversion loss.

For this rig because the C_L is huge and G_e is kind of average, the rig produces more power compared to a soft kite for a given projected area. Since the G_e is already quite low, tether drag is relatively less important so it can use relatively longer tethers before stalling.

The second column is the loss due to spinning the magnus wings, while the third column should be proportional to the power generated. The loss is relevant for Betz limit considerations and also the dimensions of the magnus motors.

Then to the bad point: I did some quick simulations (100% unverified, so they may be wrong) but the rig did not respond well to mass. This is because the glide number is poor I guess. So in particular for this rig, where we need both rotating motors and harvesting turbines onboard its going to be a challenge to get this thing working in light winds.

I gather the magnus based turbine is intriguing but not the most promising direction for AWE overall…