Here it is the average elevation angle in relation to the entire crosswind eight-figure path which counts to determine the cosine, then the cosine squared if we only count the force of the wind, and finally the cosine cubed if we count the power of the wind.

For my second statement, the cosine is connected to the elevation angle. For example for 30 degrees, cosine is 0.8660. For power, wind speed is cubed, just like the cosine. See the equation (45) which also applies to Magnus cylinders in the concerned publication.

You need to understand how the equation was derived, where the cosine cubed comes from, to understand if it also applies to Magnus-effect based systems. For that the Argatov paper seems like a good start, the Magnus paper isn’t. A different question is if the equation, or model, matches reality, for either system, for that the paper in my last comment for example might be relevant.

No. Gupta’s publication is very well presented. It repeats Argatov’s conclusion, applying it to Magnus cylinders: both wind power and cosine are cubed for both airfoils and Magnus cylinders, as stated in the equation (45) and the text just before. It’s crystal clear.

The purpose of a reference is to say: it’s there I got the result or conclusion that I am using here, look at that if you want to know how the result came to be. So we look at the reference, in this case the Argatov paper.

The work to arrive at the equation was done in the Argotov paper, so we look at that to better understand the equation. The strength of the conclusion, the equation in this case, comes from the work that is described to come to the conclusion, math, arguments, or experiments. To look for the supporting math we go to the original paper, we don’t look at the later paper that only repeats the conclusion.

But to trust it, and to be able to conclude that it also should apply to Magnus-effect based systems, you would need to be able to follow along with the derivation. And again, a different question is how closely the model (equation) matches reality. That is probably going to be a different topic and off-topic here, but here are some references for that, in addition to the one in a previous comment:

I am following this a little. My initial thinking is that a magnus cylinder works like an airfoil wing, except you need to spend a little power to make it rotate. So anything Argatov wrote should still hold right? cosine cubed seems the most obvious correct answer.

So now I am wondering why this is being discussed, what did I miss?

If you see the need, we could open a topic on “cosine loss”, because we are going off topic. But is it really necessary?

From your reference:

The investigation performed by Urbán et al. (2019) shows that yaw misalignment of a turbine in the wake of another turbine can exhibit significant variations in the power–yaw loss exponent. In particular, α depends on the shape of the wake deficit profile, which evolves as it propagates downstream. The wake recovery rate is highly dependent on turbine spacing and ambient turbulence intensity.

“Power–yaw loss exponent” includes, next to cosine losses, “the wake recovery rate” which is “highly dependent on turbine spacing and ambient turbulence intensity”.

To come back to the topic, this would perhaps be a basis to calculate the loss within a farm of Wind Fisher Magnus balloons (or for any farm of any AWES), taking into account a misalignment of units if we think that this would reduce losses by increasing the “the wake recovery rate”.

We are therefore very far from the losses per cosine cubed as indicated by Argatov and taken up by Gupta for Magnus cylinders.

Maybe you can look at the Argotov paper and understand where the cosine cubed in equation (69) comes from. I’m sure it’s correct but I wasn’t able to follow along, and with that can’t determine if it also should apply to Magnus effect-based systems.

The cubic cosine loss is based on a simplistic model. Any accurate
model would probably give slightly different results, I expect mostly more losses than cubic cosine losses.

It has been mentioned before, but cosine losses are not strictly losses in energy terms. I think when you are discussing wake losses we are taking about less usable wind energy available for downwind plants. Cosine losses are about producing less power relative to a theoretical downwind crosswind flight. Only theoretical because sustained production cant happen there without infinite turning speed of the kite [well, actually any AWE rotating in-place could achieve zero cosine losses, like Daisy/Pyramid].

So maybe this is a discussion that is a rabbit hole with very large depth, we will not be able to agree on this one, unless we get more rigid with definitions. And wake losses is a complex matter that probably can’t be advanced through forum discussion but rather real life observations and CFD simulations.