So I see that you have determined the swept area A in the correct way, so this is no simple explanation for the high maximum Cp values which you have measured. If your measurements have been performed correctly then the result is what it is. But I have no idea how you have realised that the lift generates the major part of the torque. If the major part of the torque is generated by drag, a maximum Cp of 0.26 is simply impossible.
The problem which I have to understand the functioning of the lift part of your rotor also exists if I want to understand the functioning of a normal Savonious rotor with buckets made out of half hollow cylinders. Also for that rotor type, I find very large negative angles of attack when the leading edge of a cup is at the front side of the rotor and when this rotor is rotating at a tip speed ratio of 0.9. So I doubt if a normal Savonious rotor is really a combination of a drag and a lift machine like it is generally accepted. If it is only a drag machine, the high measured Cp values of about 0.2 can only be caused by wind tunnel blockage. I have the same doubts for the VAWT with many cambered blades mounted close to each other.
I have done my best to make my tests as accurate as I can, and I've brought in professionals to check my work, as much as my budget would allow. I had hoped to take that turbine to the large wind tunnel in Ottawa, (where it would be less than 5% of the tunnel swept area), as results measured there would seem to be incontrovertible, however the company which owned the patent was forced to sell it, and the new owners became antagonistic with each other, and me, so it's unlikely that it will move forward in the near future. I still have the # 16 turbine blade, however I no longer have the test equipment, or suitable vehicle for to be run on. Sadly most of my other prototypes were destroyed by the new owners of the patent.
This was fairly crushing to my mental health and I'm not entirely over it. It's been my opinion for 5-6 years, since 2013 anyway, that even if somehow we've over calculated the Cp, and the design were only used in Kinetic Hydro, it would still unlock many MW of untapped sustainable energy, and do so with "low tech", that doesn't require billions of dollars of infrastructure producing high purity silicon and other feed products PV requires.
I'm more sensitive than is healthy for me, and the craziness of modern business people can be shocking. Modern business people are the ideal example of the old saw "experts are people who know more and more about less and less", and nowhere is this more true than in MBA programs. Money is their only lense.
The EPRI report from 2010, a survey of Kinetic Hydro devices in development and their requirements and theoretical outputs, vs resource availability in North America was between 15,000 and 150,000MW of untapped available resource. It's been 10 years and there is still no real solution to accessing this resource, although some success in very large ocean based systems has happened, Canada just abandoned it's latest system. Virtually all these failed attempts take HAWT systems and immerse them, but the lack of uniformity of flow though their blade area destroys them.
I believe the Cp of my #16 turbine has been fairly accurately measured, within +or - 10%, so I've got no problem seeing it as max measured Cp of 0.31, or 0.26, however I suspect that there is quite a bit of room in the design for optimization. Here's my thinking.
As I see it, there's some significant design issues with the turbine #16 as tested, and I will discuss them here, in part because they will help to see why I've taken the direction I have, with my new design.
A mistake I made early on with this design was to not listen to some engineers, in specific engineer Ryan Nichol, a principle of the company DSA in Victoria and Halifax, who argued with me about my method for caculating TSR while doing some early testing of a rotor.
In an eggbeater/tropinskein type Darius, or a tapering Savonius type, like mine, it does not seem as simple.
Normally TSR is simple to calculate, with HAWT's and H rotor Darius designs, it is the outer most part of the turbine vs the wind input speed. In my #16 tapering design, the outer most part of the turbine is not functional blade, but the rim at the bottom, which is unlikely to be contributing positively, other than as a fence to keep the lower portions of the blade from sheding flow in the wrong direction. Even the lowest part of the tubine blade, with the large bump in it's centre, is unlikely to be making large positive aerodynamic contributions. It's my thinking that while all the parts of the turbine may be working together, the centre section, where it's truest to what I imagined as the ideal cross section, is likely providing most of the positive reaction. This section has little taper, and if we take it's average diameter, and calculate tip speed from it, we get a very different number. In my testing using the standard calculation, we found ideal TSR to be about 1.2 -1.3, and that the turbine could run unloaded to about 1.5 or 1.6, but at any significant wind speed TSR's of 1.4- 1.5 or above would create a strong 2 pulse per revolution aerodynamic vibration and our test system was not robust enough for us to try to explore to it's highest possible unloaded speed. This unloaded vibration was strong, and would occur at various RPM's whenever TSR went too high. If we revise the TSR calculation, using the average of the best working area (the middle) then our max Cp came at TSR's of closer to .8-.9 than 1.2-1.3.
I now agree with Ryan, that this is a more accurate way to think about the turbine in terms of TSR.
In the testing of #16 we didn't spend a lot of time characterizing the performance at the low TSR's where the Savonius may function maximally, however it was clear that even at low TSR's, during startup, the turbine produced plenty of torque, and would function even when loaded well past ideal speed. This makes me think that the turbine may have two operating modes, a "drag" mode, similar to the Savonius, and another mode, which I'm reluctant to call a lift based mode, and would rather call a boundary layer mode. The only other boundary layer turbine that I'm familiar with is the "bladeless" Tesla disk turbine.
I don't wish to be too controversial, however I hope without too much trouble readers can accept that there are more than one way to calculate the effects of flow over surfaces, the historic, and accurate for most airplane wings and propellors, high pressure vs low pressure, lift and drag methodology, and a more complex way of thinking that's the foundation of most modern CFD software, that lift is generated by changes in vector and velocity in the medium a foil is traveling through, or mass flow reaction theory, via Newtonian physics. That airfoils are flow turning devices, and the changes in flow direction and velocity result in torque or force on the rotor.
https://www.grc.nasa.gov/www/k-12/airplane/wrong1.htmlI don't consider myself qualified to speak to either, or calculate in either, so I bring them up only to note that both have been used to successfully mathematically model real airfoils performance in a variety of circumstances. My suspicion is that specifically in very low Reynolds numbers, the mass flow method may be more useful.
The mass flow thinking, in my mind is easier to think through, and it allows a more direct comparison between my work, and Tesla's boundary layer turbine. I note that the Tesla turbine is an interesting and contraversial device, as is almost anything connected to Tesla. To the best of my knowledge, as a turbine it's found little market as the best measured efficiency seems to have been less than 50%. It's been theorized that larger versions might do better, however the physics of scaling them up requires large diameter, flat disks that can handle massive centripetal loads, and heat differentials, without any warping. The only real market they have found is in oil pipelines where they are apparently used as pumps and turbines in high solids contaminated flows.
The interesting thing about the Tesla Turbine is that it also operates at very low Reynolds numbers, and it's ideal TSR is 1, or just below. The idea is that the flow encountering the rim of the disk should be very close to the disks speed, so that the flow attaches with minimal turbulence, and as the flow spirals inward, while the RPM stays the same, the disk's surface speed reduces, and so the boundary layer of the fluid remains attached by transferring torque to the spinning disk, as it continues to do until it reaches the ports near the disks centre, where it turns 90 deg and exits. The turbines power transformation is limited, the energy available must be less than the difference between it's input and output speeds, just as with wind turbines. It has other issues, like turbulence in the exhaust, which mean that it's real efficiency must be much less than the input and output speed difference, however it does operate, and does so by boundary layer friction, or at least I've never seen any research that postulates another theory of action.
In the Savonius turbine, as considered by Benesh and Co, one of the principle observations from their A-B testing of an airfoil shaped cup, vs a hemicylindrical cup, is that they believed they observed a 50% increase in Cp between the two rotors, claiming Cp 0.2 for the Blackwell, and Cp 0.3 for their flattened rotor (similar in shape to the Hi Power rotor). Later, Rahai, also with a rotor where the "depth" of the cup had been reduced, also in some AB testing found some apparent benefit (though his methodology was so flawed it may be that no good observations can be made.
Also, in my hydro testing, I found some interesting results. I tested a non helical version of my cross section, and while it had very positive torque in some angles, it was not enough to keep it in rotation, it would rapidly and vigorously"flip" between stable states. The same cross section formed helically, had strong positive torque in all positions.
It's my understanding, from Ian Ross's wind tunnel work, and old NRC canada, real world testing from the 70's that the Savonius turbine's ideal TSR is around .2 or .3, but that unloaded it may reach 1.1 - 1.2, which was also my experience in testing a Blackwell dimensioned rotor of about .5m diameter, and that unloaded at the higher tip speeds without the top plate, it also vibrates at 2 pulses per rev.
From the lift vs drag perspective, the Savonius, at TSR .2-.3 and Cp of between 0.05 and 0.10 is a drag machine, however at TSR 1, it's likely something more is going on.
This something more, could be seen as the flow attaching to the outer most edge of the bucket, matching flow velocity, and having flow adhere, to create the tension in the boundary layer, pulling the turbine blade along.
I think it's worth keeping in mind that the characteristic difference between high Reynolds number modelling and low Reynolds number modelling is that at high Reynolds numbers inertia dominates the flow characteristics, each air molecule's path is dominated by it's own inertia, instead of being "stuck" it is to it's neighbors. At high Reynolds numbers air is more like a series of BB's fired through the air, they each interact with the foil almost independently of the surrounding air. In contrast, at low Reynolds numbers, the "stickiness" or viscosity between the molecules dominates and the flow is dominated by that stickiness.
In the classic Savonius, which even at TSR 1 is a low Reynolds number turbine. In it, the change in blade surface velocity is determined simply by the curve of the bucket, and it seems to me that it is going to be strongly non linear. If instead of a bucket, it was a flatter rotating sheet, like the popular "flip" signs I've seen at my local ice cream store, the the change in surface speed rate would be linear as flow attaches at an edge and proceeds inward toward the centre. This aligns with the experimental work of the Benesh/Modi team's A-B tests of a Blackwell Savonius vs a flattened Savonius. From a lift perspective, perhaps the classic Savonius can be thought about, that while the front edge is generating lift, the depth of the foil is too tall, and flow separation occurs too early. There is a mismatch between the change in rate of flow that can remain attached, and the geometry of the blade.
However, as I found, a flatter foil, without top and bottom plate, is a very poor turbine, so while this explation may fit both the high TSR and low TSR operation of the classic Savonius, and the innovations and improvements of Benesh and team, it does not explain the improvements I seem to be measuring.
A principle difference between the helical and non helical Savonius type turbine is that in the helical turbine, as in the Tesla turbine, the surface area available for the flow reduces as the flow moves toward the centre of the turbine. The helical shape, if cut and flattened into sections of a disk, would show that the outer most edge has a much longer path than the centre, and that flow attached to that edge, and moving along the blade, will have less surface to remain attached to, and this is happening at the same time as the flow is transferring energy to the surface through boundary layer friction. These three geometries, the curve of the blade, the angle of the twist, and the tip speed ratio, can be optimized so that the flow encountering the blade is closely matched to the surface speed of the blade, and with the right combination of twist and flattened profile, will remain attached and sheding velocity to the blade via friction, as long as the geometry will allow, or until it reaches close to the centre, where the flow velocity and surface velocity of the turbine are close to 0, and if the TSR is correct, this should happen within 180 degree's rotation, allowing new, energized flow, into the front of the turbine, and some of this new incoming energy is lost to sweeping out the exhaust.
A further match of the data and theory might be that the 2x per revolution vibration, felt in my turbines 15 and 16, and in earlier hydro versions of the turbine, as well as the Savonius tested without a top plate, all occurred at unloaded or lightly loaded high TSR conditions. In this instance the turbine is encountering the flow with a good blade speed match, so energy is transferred to maintain the high speed condition, but the surface speed drop off is not occurring at the ideal rate, and as the mismatch increases, the flow separates, and separated flow, having been slowed partially, forms an energetic vortex which is shed intermittently 2x per revolution. It's the departure of this vortex that causes the vibration I've noted, and has been noted by others testing Savonius turbines at high speeds and light loads (noted by Mother Earth news in their plans).
I write this all with the confidence that while none of you may find this convincing, you'll hopefully take the time to consider it as an alternative to the conventional Lift vs Drag approach for Savonius type turbines.
I of course have ignored a large part of the cycle of the turbine, the upwind side in this explanation, which I have considered, but am less confident in and do not have more time to write about now. As well, I think the whole thing will be much clearer when I get around to making some animations and illustrations.
Best Wishes,
Drew