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  #1  
Old 12-01-2011, 02:56 PM
scsmith scsmith is offline
 
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Default Science of formation drag reduction

OK, now that we got everyone excited, I thought I would write just a bit on the science of what is going on, and especially, how it is that the lead plane benefits in a V formation.

The first thing to understand is simply that there is a circulating field around an airplane because of the trailing vortices. There is generally upwash outboard of each trailing vortex, downwash inboard, and sidewash above and below. Mathematically, we can compute what the induced velocity is, based on the distance from the vortex -- for a vortex that starts at one point and trails downwind to infinity (they almost do!) the tangential velocity drops off as 1/r, r being the perpendicular distance from the vortex. But also significant is that the effect extends significantly upwind of the origin point of the vortex. As you would expect, the farther upwind of the origin point you go, the weaker the induced flow is, but it is important to understand that it does extend upwind - the tangential flow can't just abruptly stop at some point, it decays slowly with distance. If you want to look up the actual math equation for the tangential flow, its called the Biot-Savart Law.

Next, why does flying in an induced upwash field reduce the drag? Mathematically there are couple of different ways to illustrate this, each gives the same answer. For those of you that have some technical background, it comes from the Kutta-Joukowski theorem that describes the force acting on a vortex in cross-flow. Just as the lift comes from rho x U x gamma, induced drag (or thrust) comes from rho x W x gamma. So if W is positive upward, you get thrust on the bound vortex. For lay folks, perhaps easiest to understand is to consider a glider flying along a ridge where the wind is turned upward by the ridge, and the glider is able to fly level along the ridge because of the upward flow. You could say that the glider is still descending through the air at its normal sink rate, but the whole parcel of air, with the glider in it, is being carried upward at the same rate. The glider doesn't know that it is flying level, it thinks it is decending through the air. So, a powerplane flying in an upwash field can reduce the amount of power needed to fly level at the same speed, because it thinks it is descending through the flow.

OK, now to formation flight. One interesting case is line-abreast formation. In this case, each airplane feels some upwash from its neighbors. There is a superposition effect. The tip vortex from my neighbor's nearest tip is producing a lot of upwash for me, but the vortex from my neighbor's far tip is producing some downwash for me. But it is farther away, so it is not as strong; there is a net upwash. If there is another airplane beyond him, I feel weak effects from those vortices too - each airplane in the line adds some upwash from its near tip, and less downwash from its far tip. The guy in the middle is feeling net upwash from every single airplane in the formation, and he gets the most benefit in the formation.

Another interesting case is a very pointy V formation, where the 'sweep angle' of the V is large. In this case, the lead plane feels very little benefit, because it is too far upstream of its neighbors. Each plane is flying in the strong upwash of the neighbor in front of it, and getting a MUCH weaker benefit from the neighbor behind it. The planes near the tail of the V are feeling the accumulated upwash of the whole family of trailing vortices, and get the most benefit.

Now here is where it gets cool -- at least I think its cool: There is an optimum V angle, in between the two cases I described above, where the accumulated benefit of all the vortices produces EQUAL benefit for each plane in the formation. The V angle is flat enough so that there is enough upwind effect from each airplane to benefit the neighbor in front of it just enough so the net benefit is equal for each plane. This result, mathematically optimized, was first published by Peter Lissemann in 1970 in a science journal. You might recognize Peter Lissemann's name as one of the co-founders of Aerovironment along with Paul McCready. So this explains why we measured a substantial benefit for the leader.

Anyway, the cool thing about this ideal V angle is that it is self-seeking. In a flock of birds, if the ones near the middle of the formation are stronger, they pull ahead, making the V angle steeper, thus benefiting the birds out toward the tails of the V so they can catch up. If the birds near the middle of the formation get tired and drop back, making the V angle flatter, then they, near the middle, feel more benefit, so they can rest. So the V angle is stable -- birds naturally fall into the right angle that allows all the birds to keep up the same speed.

As far as the actual fuel saving....one wild card in our method is fuel-air mixture. Carburetors and fuel injection don't necessarily maintain constant fuel-air mixture as you change throttle setting. On my Bendix FI, if I lean to peak EGT at cruise power, the mixture lever is not as far back as if I lean to peak at idle. We got very significant reductions in manifold pressure in formation, and I think on some of the airplanes, the fuel flow reductions were similar. In some of the planes, the fuel flow reduction seemed less than I expected based on manifold pressure reduction. What I can say is that the actual fuel flow measurements from the instruments in the West Coast Ravens airplanes were extremely accurate. We were down to counting pulse-widths and averaging over significant lengths of time.

The other variable, of course, is how well each airplane stayed in its "sweet spot". The best position actually has some wingtip overlap, which is a position that the formation guys are not used to. Although the flow is smooth in that position (not the difficult task of holding in perfect trail), it is fairly dynamic - the roll moment and side force are changing as you move around in the vortex. And of course, there are throttle excursions all the time to hold position, which tend to offset some of the benfit.

All told, I was really pleased that we got the results we did, about 3-5% benefit for each plane. When we did the two-ship F-18 test, with some cockpit display aids using differential GPS to help hold the optimum position, we saw more like 12% (and there are some good stories about those tests too!)

Anyway, glad you all enjoyed the show. Not RV-related, but I also worked on an upcoming episode called Fireworks Man #2, so watch for it.
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Last edited by scsmith : 12-01-2011 at 02:58 PM.
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  #2  
Old 12-01-2011, 03:17 PM
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One of the most interesting and informative posts I've seen in VAF. Good stuff here for the tech and lay person alike. Thanks!!
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  #3  
Old 12-01-2011, 03:38 PM
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kentb kentb is offline
 
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Default Thanks, Steve

I never heard this before and it makes sense.

Now all we need is a way to slave the autopilot to the optimal spacing in the formation and we will save big bucks when going to fly-in's.

Kent
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  #4  
Old 12-01-2011, 03:41 PM
rgmwa rgmwa is online now
 
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Fascinating, and very well explained. Thanks!
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  #5  
Old 12-01-2011, 03:48 PM
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So this is why the Jedi says, "May the FORCE be with you!" [also an old engineering school joke]

Seriously, Steve, I thank you for posting enough of the science to make it interesting and also enough of the practical to help us understand and accept it.
Please post again.
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  #6  
Old 12-01-2011, 03:49 PM
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Excellent post Steve! Now here is a question I have always wondered about (but never heard enough to look it up). Geese have eyes on the side of their heads - not the front, like humans. SO their normally "relaxed" look angle is somewhere to the side of straight ahead. I have always wondered if their "relaxed" look angle corresponds to the angle of the V - and if it does, and that angle is the same as the optimum angle (which as you point out is naturally stable), it sure would be an interesting argument in the direction of evolutionary optimization.

I guess we need an ornithologist to chime in here....

Paul
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  #7  
Old 12-01-2011, 04:01 PM
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flyboy1963 flyboy1963 is offline
 
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Default aerodynamics for dummies?

Steve, great explanation, even without pictures and diagrams!

some day I hope to be able to make up stuff like "....Kutta-Joukowski theorem that describes the force acting on a vortex in cross-flow. Just as the lift comes from rho x U x gamma,.....
that is sweet!...

I like Paul's idea ....what was it? ....that as our eyes evolved to the front of our faces, our ability to fly tight formation has diminished? No wonder I washed out of air cadets....too far evolved!!!

..something like that, or, I could be completely wrong here, eh?
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  #8  
Old 12-01-2011, 04:04 PM
scsmith scsmith is offline
 
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Default Bird's field of view

I remember seeing drawings depicting the view angles of various animals, including birds. They have a wide enough field of view that they do have some stereo vision straight ahead (which makes sense, it would be hard to land without depth perception, let alone pluck a kernel of corn off the ground). If the relaxed focal axis of their eye orientation just happened to coincide with the optimum V angle for formation, that would be just AMAZING, but not impossible, whether you think of it as the blind watchmaker of evolution or the wisdom of a creator.
But of the various tasks that birds need to use their vision for, watching the red-head in the lead plane is probably farther down on the list than, say, finding food. So I kind of doubt there is a connection.
Intriguing idea though!

By the way, I should have mentioned that the optimum V angle varies with the number of members of the formation. Although, for populations above 5 or 7, it pretty much converges to a certain angle. With RV's, we could not safely fly at the optimum V angle with the optimum overlap, because it put wingtips too close to horizontal tails. So we flew a bit more steep V than would be best. Nice that the lead still showed benefit.


Quote:
Originally Posted by Ironflight View Post
Excellent post Steve! Now here is a question I have always wondered about (but never heard enough to look it up). Geese have eyes on the side of their heads - not the front, like humans. SO their normally "relaxed" look angle is somewhere to the side of straight ahead. I have always wondered if their "relaxed" look angle corresponds to the angle of the V - and if it does, and that angle is the same as the optimum angle (which as you point out is naturally stable), it sure would be an interesting argument in the direction of evolutionary optimization.

I guess we need an ornithologist to chime in here....

Paul
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  #9  
Old 12-01-2011, 05:59 PM
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Quote:
Originally Posted by scsmith View Post
watching the red-head in the lead plane is probably farther down on the list than, say, finding food.
It is???
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  #10  
Old 12-01-2011, 06:26 PM
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Quote:
Originally Posted by scsmith View Post
...But also significant is that the effect extends significantly upwind of the origin point of the vortex...
But that only holds for subsonic speeds, right? Not that it has much relevance to the RV community, but that must have had some bearing on the F-18 application.

Thanks, Bob K.
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