Don talks about Contra-Rotating props
From : Don Stackhouse
So, back to contra-rotating props: what do we need to do to get the most from them? First, you need enough power to make it worth the extra weight and complexity.
OK, so the vast majority of models do not satisfy that requirement, but we want to have a contra-rotating system anyway for scale appearance purposes. What should we do to minimize the detriments?
The swirl dissipates through friction with the surrounding air. To recover the maximum of whatever swirl energy is available, the two props should be as close to each other as possible. However, that also worsens the vibrational effects of the blades passing each other. That arrangement also generally requires one of those complex and troublesome gearboxes I discussed above. It's a tradeoff. In the case of the Cessna 337, Dornier Do335 "Pfeil" ("Arrow"), Savoia-Marchetti S-65 racer, etc., they give up some of the possible swirl recovery, and also worsen the inflow environment and efficiency of the aft prop, to eliminate the gearbox. The mechanical simplicity may make it a worthwhile tradeoff.
The real key then to getting the most out of any pusher installation, including one with a tractor up front as well, is to get the airflow into the rear prop as clean as possible. Any fat fuselages, bracing, struts, and especially any flying surfaces or any large bodies that are to one side of the prop's axis can spell serious trouble.
For example, I know of one prominent twin-pusher that had fairly fat nacelles sitting on top of a fat wing root ahead of the props, and a fuselage to one side of the prop disk. The inflow angle over approximately one-fourth of the prop disk was fifteen degrees different than over the other three-fourths of the disk! Imagine what would happen to your glide ratio and the comfort of your passengers if during a max-performance glide you started rapidly and continuously porpoising the nose up and down over a 15 degree range. That's what was happening to the blades of those props. The vibration problems were extremely serious, and the performance fell well short of original projections. They ended up having to go to much more powerful engines (with their attendant increase in fuel burn and other operating costs) to make up the difference.
So, the key is to keep the inflow as undisturbed as possible, and also to have it as symmetrical around the axis of the prop as possible. Anything that creates turbulence is bad, and anything that deflects the airflow to a different angle (so that the angle of attack seen by the blades varies as they sweep around the disk) is even worse. Wings, tail surfaces and deflected control surfaces can be serious problems. A thin, shoulder-mounted wing mounted well ahead of the prop on a slender, smooth fuselage (such as the case for the aft engine on the Voyager) is better than a high wing and a lop-sided fuselage right in front of the prop such as the Cessna 337.
Speaking of the Cessna, some folks like to trot out the fact that it climbs better on the aft engine alone than on the front engine alone as support for their flawed claims that pushers are generally more efficient than tractors. In truth, the aft prop of the 337 is less efficient. However, the lower aft fuselage of the 337 slopes upward at such a steep angle in front of the aft prop that, without the induced flow from the aft prop, the airflow over the aft lower fuselage separates, causing massive amounts of drag. When one powerplant is shut down and feathered, the plane climbs worse on the front engine alone because of the massive increase in fuselage drag due to the poor aft fuselage shape, in spite of the front prop's better efficiency.
OK, so we've learned that pushers are usually a detriment unless you really do your homework, contra rotation is not generally worth the trouble on models, but if we're going to do it anyway, we should try to keep the airflow into both props as clean, smooth and uniform as possible. What's that bit someone else mentioned about different diameters due to "slipstream contraction", and what about the need for different pitches and/or rpm's for the two props?
A prop makes thrust by grabbing chunks of air from in front of it, and accelerating them out behind. About half the acceleration occurs in front of the prop, and the other half behind. The reaction to the force required to accelerate the air's mass shows up as thrust. Because the air has to be accelerated to make thrust, the velocity of the air behind the prop is faster than the velocity in front of the prop.
As the velocity changes, the roughly cylindrical stream of air flowing through the prop has to obey Bernoulli's principle. If its airspeed increases, then the cross-sectional area (and therefore the diameter) of the stream has to decrease in proportion to that in order for the volume of the flow to remain constant. If this were not so, the flow through the prop would violate the law of conservation of mass and energy, which happens to be one of the most inflexible laws in all of Newtonian physics. Thus, the diameter of the inflow to the prop is actually larger than the prop at some point upstream of it, then contracts during that first half of its acceleration until it is equal in diameter to the prop when it reaches the prop disk. It continues to contract after it passes through the prop, during the second half of its acceleration. This is that "slipstream contraction" that some other posters to this thread have mentioned. This means that a second prop, aft of the first one, that is supposed to be working with the slipstream of the first prop, needs to be a little smaller in diameter in order to match the boundaries of the now-contracted slipstream.
Just how much faster (and therefore how much smaller in diameter) depends on a number of factors. For the ratio of slipstream dynamic pressure to free-stream dynamic pressure, Daniel E. Dommasch's "Airplane Aerodynamics" suggests an equation, which with a little algebraic juggling gives us:
Qt = Q + [(4 * T) / (D^2 * Pi)]
Qt = dynamic pressure ("ram air pressure" minus the static pressure) in the fully developed slipstream well aft of a prop
Q is the dynamic pressure in the freestream well ahead of the prop, and outside of the propwash
T = thrust
D = prop diameter
and of course "Pi" is 3.141592...
Dynamic pressure ("Q") is equal to one-half the air density, times the velocity squared. If we plug that back into the formula and do some more algebra, we get:
Vt = SQRT [V^2 + (8T / rho * D^2 * Pi)]
Vt = the velocity in the fully developed freestream in feet per second
"SQRT" means you take the square root of the result of the formula inside the [ ]
V^2 = the freestream velocity squared (velocity in feet per second)
T = thrust in pounds
rho = air density in slugs/ft^3 (.00238 at sea level standard day conditions)
D^2 = prop diameter in feet
Other units will work as well, just make sure that you use the same system of units throughout (no fair mixing feet in one variable with inches in another, or metric units with English, etc.!).
Ok, now that half of you are getting glassy-eyed and most of the rest are running for cover in a mad panic, let's clarify that terrifying blast of algebra with a practical example:
Suppose we have a twin-engined model that weighs 1 pound, and we're planning to modify it into a twin contra-rotating arrangement. Let's also assume that the L/D (essentially the same as the glide ratio) at our expected cruise speed of about 25 mph ( multiply by 22 and divide by 15 to get 36.67 fps) is about 4:1 (I know that sounds low, but remember, typical cruise speeds are higher than best gliding speed, and besides, this airplane has a bunch of extra stuff hanging out in the breeze). This means our drag is equal to the weight divided by the L/D, or 0.25 pounds. In level flight, that is also equal to the total thrust.
Let's also assume the front prop is doing about 55% of the work (0.138 pounds of thrust) to allow for the lower efficiency of the aft prop. We'll define the prop as having a 6" diameter (0.5 feet).
Plugging all of that data into our formula:
Vt = SQRT [36.67^2 + (8 * 0.138 / .00238 * 0.5^2 * 3.1416)]
which is equal to 43.99 feet per second, or 30 mph. That's a velocity ratio of 1.2, or 20% more than the freestream velocity.
This means that if the aft prop is far back enough to sit in the fully developed slipstream from the forward prop, it will need either 20% more pitch (the preferred solution) or 20% more rpm (which opens several other cans of worms). In addition, the slipstream contraction will be SQRT (1/1.2), or 0.913 . That means the aft prop should be 91.3% of the diameter of the forward prop, or just a little less than 5.5" diameter. See, that wasn't so hard, was it?
If you plan to do this a lot, I suggest coding these formulas into your favorite spreadsheet program, such as Excel.
I helped advise a guy recently who scratch-built a VERY giant-scale electric model of the Voyager. As I recall, his original setup used the same size props on both ends. It flew much better when we put a prop with more pitch on the aft motor.
So, that's all there is to it! Just correct for slipstream effects on the rear prop, and keep the inflow into it as clean and undisturbed as possible. You will probably not have as much prop efficiency as a pair of tractor props with nice clean inflow, but it shouldn't be too bad.