In the coaxial twin-rotor helicopter, the main rotors are positioned on a single axis — one above and the other below. Such a helicopter has certain operational characteristics. The area swept by the two main rotors is equal to the area swept by a single rotor
where Fc is the area swept by the system of coaxial rotors;
F^ is the area swept by a single rotor.
In this case, we have assumed that the diameters of the upper and lower /34
rotors are the same.
Let us examine the system of air jets passing through the areas swept by the upper and lower rotors (Figure 26). Increase of the distance between the hubs of the upper and lower rotors degrades the operating conditions of the lower rotor and complicates the construction of the entire system, while reduction of this distance leads to the danger of collision of the rotor blades and increases helicopter vibration. This distance is h = 0.08D = 0.8m in the Ka-15 and Ka-18 helicopters. At this distance, the lower rotor has no effect on the operation of the upper rotor. The jet from the upper rotor contracts, and in the plane of rotation of the lower rotor its radius is 0.7R,
where R is the rotor radius. In this case, the lower rotor blade tips
operate under the same conditions as those of the upper rotor and draw additional air in from the side.
On this basis, we shall estimate the effective area of the entire system through which the air flows, just as for an isolated rotor in the hovering regime.
From the area swept by the upper rotor, we must subtract the root loss area (of radius 0.25R). Under conditions similar to those in the hovering regime, only the tips of the lower rotor blades operate. The area swept by these tips is
F ігтеЯ* _ тгО.72/?3.
Consequently, the effective area of both rotors through which the stream flows, as in the case of hovering of an isolated rotor, is found from the formula
Fc = *R2 _ *0.2S2/?2 + кГГ – – *0.7=i42 == 7гД* (1 _10.06 +
. +1 -0.49) = 1.45г,. ‘ ,
That portion of the lower rotor which operates in the jet of the upper rotor has lower efficiency. The angles of attack of the lower rotor blade elements are reduced as a result of the induced velocity of the upper rotor (see Figure 23b), which leads to reduction of the thrust. To reduce this effect, the incidence angles of the lower rotor blades are made 2-3° larger than for the upper rotor, but this does not eliminate entirely the harmful influence of the upper rotor on the lower. In the presence of this influence, the efficiency of the central portion of the lower rotor, which is in the jet from the upper rotor is reduced by a factor of two, in comparison with the efficiency of the tip area outside the jet from the upper rotor.
The swept area of the lower rotor, operating in the jet from the upper /35 rotor, is found from the formula
F± = kPFOJ2 – */?20.252 = = ^20.43-0.43Л.
Since its efficiency is less than that of the upper rotor by a factor of two, the additional effective area of the lower rotor is
F = 0.43^0.5 = 0.22Fi.
e. 1 • –
The effective area of the entire
system isFe ^=lA5Fi–Q.22Fi = l.67Fi •
This formula shows that the thrust of
two coaxial rotors under the same conditions is greater than the thrust of an isolated main rotor of the same diameter by a factor of 1.67.
If the thrusts of the coaxial system and the isolated rotor are the same, then less power is required to create the thrust of the coaxial rotor system, which follows from ideal rotor momentum theory.
The power required to turn the ideal rotor is entirely converted into kinetic energy of the jet, i. e., N = TV^.
If we use Tc, Vc> F^, respectively, to denote the thrust, induced velocity, and effective area of the coaxial system of two rotors, and T^, V^, F^ to denote the thrust, induced velocity, and swept area of the isolated
rotor, then we have T = Tn.
We know that
Fc = 1.67/7!.
Hence, we find
T„ 2pFtVf V
Vc~ 2p 1.677?! = 1.67 ’
In order to obtain thrust on a system of coaxial rotors equal to the thrust of an isolated rotor of the same diameter, the induced velocity of the coaxial system must be less than the induced velocity of the isolated rotor.
Since the ideal rotor power required is proportional to 1Л, less power is required to obtain the same thrust for the coaxial system than for the /36
isolated rotor. This is the advantage of the coaxial system. The number
0. 78 ‘v is called the aerodynamic advantage coefficient, and is denoted by Using this coefficient, we express the power required for the coaxial system in terms of the power required of an isolated ideal rotor
This implies that for the same power the coaxial rotor system provides 13-15% more thrust than the isolated main rotor. Therefore, the helicopter with coaxial rotors has smaller dimensions than the single-rotor helicopter.
However, to date only light helicopters have been built using this scheme because of structural complexity and other problems.
Twin-rotor helicopters of other arrangements, for example, with the rotors placed longitudinally and with intermeshing rotors, also have an aerodynamic advantage in the axial flow regime. The aerodynamic advantage coefficient of these systems approaches closer to 0.8, the less the distance between the main rotor hub axes.