Application of pressure head theory to determine the aerodynamic characteristics of the helicopter main rotor

A method is proposed for calculating main aerodynamic characteristics of the main rotor of the Mi-26 helicopter: a lift, a reactive moment, a required power of the propulsion system, a speed of the inductive air flow, a throttle response coefficient, a speed of the horizontal flight. This method based on the pressure propeller theory. The parameters of the Mi-26 helicopter and the propeller of the experimental setup calculated according to the pressure theory of the main rotor are experimentally confirmed. The percentage of the discrepancy between the statistical data on the aerodynamic characteristics of the Mi-26 helicopter rotor calculated with the pressure theory does not exceed 1.5%.


Introduction
At present, the aerodynamic calculation of the helicopter main rotor is theoretically described by the classical theory of the helicopter main rotor, developed back in the fifties of the last century by M. Mil and B. Yuriev. However, this theory has some assumptions and does not provide an accurate explanation of the dependence of the rotor thrust on the kinetic energy and the speed of the inductive flow, it includes many approximate coefficients that are refined after experimental verification. The pressure head theory of the propeller (called in this text as the PHTP theory) appeared as a result of combining the experimental and statistical data of the aerodynamic characteristics of the helicopter main rotor of three types by Mi-2, Mi-24, Mi-26 helicopters and the propeller of an experimental aerodynamic installation. The pressure-applied method of determining the main aerodynamic characteristics of a helicopter blade, based on the PHTP theory, is a simple, reliable and fairly accurate method for calculating the main parameters of the helicopter main rotor of the Mi-26 helicopter. The PHTP theory is a good addition to the vortex, impulse and classical theories of helicopter main rotor and for a designer who designs any aircraft that has helicopter main rotor, it is a good help in his creative work. The PHTP theory can be used not only to determine the aerodynamic characteristics of a helicopter propeller, but also to determine the parameters of the propeller of a subsonic aircraft.

Theoretical research
To consider, the PHTP theory, we will start from the idealized representation of the velocity head of the blade (called in this text as VHB), in which all parameters and characteristics of the helicopter main rotor are calculated in the most energy-intensive mode of its operation -hovering, which is associated with its safety in all modes work [1]: where: Qiideal velocity head of the blade, N ρmass density of air at the calculated height, кg/m 3 ; nspeed of rotation helicopter main rotor, s -1 ; Rbdistance from the axis of rotation of the helicopter main rotor to the end part of the blade, m; Rmthe distance from the axis of rotation of the helicopter main rotor to the tail part of the blade, m. According to the PHTP theory, the velocity head of the blade is determined by the formula: where: Qrreal velocity head of the blade, N/ m 2 Uklcircumferential velocity of the end of blade, s -1 . Velocity head of the blade is a component of the lift force, that we can calculate the value of the lift force of the helicopter main rotor [1,2]: where: ‫ܥ‬the mean value of the lift coefficient of the blade of the helicopter main rotor; ܵarea of the feathered part of the blade, m 2 .
To verify the validity of formula (2), we calculate the lift of the helicopter main rotor of the Mi-26 helicopter using fragmentary method (Fig. 1).
To begin with, we divide the feathered part of the blade of the helicopter main rotor into 4 parts, with the length of each fragment being 3.325 m with an area of 2.7 m 2 . Then the distance from the axis of rotation of the helicopter main rotor to the middle of the feathered part of the first fragment will be 14.3 m, the second is 11 m, the third is 7.7 m, the fourth is 4.4 m. The distance to the center of feathered part of the blade is 9.35 m. Length feathered blades and 13.3 m. Chord of the blade over the entire length is 0.8 m. Frequency of rotation of the helicopter main rotor is 2.2 s -1 .
The mean value of the lift coefficient Cl mean for each fragment are 0.5; 0.55; 0.85; 0.95 respectively [2,16]. The error of 2.9% is quite small, so it can be ignored, because you can set more accurate values of the Cl for each fragment of the blade.
The maximum take-off weight of the Mi-26 helicopter is 56000 kgf, which means that each of the eight blades must have a lifting force of 7000 kgf. The difference between the calculated and statistical data arose from the fact that the calculation of the lifting force of the blade was made according to the formula for determining the thrust using the ideal velocity head of the blade (see formula 1), which does not take into account the loop and end loss of lift force. The value of the coefficient of loop and end losses is in the range of 0.85…0.9 [1].
Take the value X = 0.89, then: which almost completely coincides with the statistical value of the lifting force of one blade. Velocity head of the blade according to the formula (1) allows the calculation of lift force helicopter main rotor due to the fact that the distance from the axis of rotation of helicopter main rotor to the middle of the feathered part of the helicopter's blade is 0.56-0.58 to helicopter main rotor radius. If excluded from the calculation of lift force value X=0,85 ÷ 0.9, then it becomes possible to calculate the value (0.5 Ukl ) which is the peripheral speed of rotation of the middle radius of the blade, while ‫ܥ‬ is determined by the angle of attack of the blades in the middle of the feathered part of the blade, taking into account their geometrical twist. Now, we will determine the main aerodynamic characteristics of the main rotor of the Mi-26 helicopter according to the classical theory and the PHTP theory and conduct a comparative analysis. For the convenience of calculating the main aerodynamic characteristics of the main rotor of the Mi-26 helicopter according to the PHTP theory, it is proposed to use the real blade velocity head, which is determined by the formula (1).
1. Lift force of the helicopter main rotor (thrust of the helicopter main rotor). Traditionally, the lift force of the helicopter main rotor (thrust) T is determined by the formula (3) [2-4, 10,11]: where, Сtthrust coefficient of the helicopter main rotor; ܵ ுெோarea of swept surface of the helicopter main rotor; ߩair density; ωRthe peripheral speed of the blade tip. In the formula (3), Сt is determined as the rotor solidify ratio of the helicopter main rotor.
С .the lift coefficient of the blade in the section as 0.7ܴ of the main rotor.
According to the PHTP theory, the helicopter main rotor thrust is determined as: where, ‫ܥ‬mean value of the lift coefficient of the helicopter main rotor blade; ‫ܥ‬ = 0,71; ܵarea of the feathering part of the blades of the helicopter main rotor; ܵ = 85,12 m 2 .
According to the PHTP theory, the ‫ܥ‬ coefficient is determined using tables of aerodynamic profiles for the angle of attack of the blade through the angle of its installation in the middle of the feathering part [3,10]: where: ߙmean the angle of attack of the helicopter main rotor blade; φinstallation angle in the middle of the feathered part of the helicopter main rotor blade; ݂conversion factor from the installation angle to the angle of attack (݂ = 0,55 ÷ 0,75); Then, the thrust of the helicopter main rotor by Мi-26 is: • 85,12 = 55864,33 ݂݇݃.
2. Reactive moment. The reactive moment of the helicopter main rotor according to the classical theory of the helicopter main rotor is defined as the torque of the power plant with the opposite sign [3][4][5][12][13][14]: where: ܴ ுெோradius of the helicopter main rotor; ݉blade activity factor, determined by the formula: According the PHTP theory, the reactive moment of the helicopter main rotor is determined as: where: ‫ܥ‬ ௗmean profile drag coefficient of the helicopter main rotor blade; ‫ܥ‬ ௗ =0,072; ‫ܥ‬ ௗ of the blade is determined using tables of aerodynamic profiles by the angle of attack of the blade through the angle of its installation in the middle of the feathered part. then: ‫ܯ‬ ோ = 0,072 0,125(0,5 • 221) ଶ 2 • 85,12 • 16 = 74832 ݂݇݃ • ݉ 3. Required engine power. The required power of the engine ܰ ோ in hover mode according to the classical theory of the helicopter main rotor is determined through the coefficient of the engine torque [3][4][5]11]: According to the PHTP theory, the required power of the engine is determined as [5]: where: Tthe lift force of the helicopter main rotor (thrust) expressed in Newtons, then: ܰ ோ = 560000 · 221 · 0,072 = 8910,720 (kW) 4. The speed of the inductive air flow. The speed of the inductive air flow (Vi1) in the plane of rotation of the helicopter main rotor in the hover mode according to the classical theory is determined as [3][4][5]: where: Хroot and tip losses factor. According to the PHTP theory, the speed of the inductive air flow in the plane of rotation of the helicopter main rotor is determined as [6,7]: where: ߪ ଵ − load factor main rotor blade, defined as the ratio of the load on one blade to the load of the main rotor. The speed of the inductive air flow behind the helicopter main rotor is determined as twice speed of the inductive air flow in the plane of rotation: Because coefficient ߪ ଵ is constant any conditions, then in the PHTP theory speed of the inductive air flow (Vi1) can be determined through the rotor solidify ratio (σ) of the helicopter main rotor as [8]: The speed of the inductive air flow behind the helicopter main rotor is determined also: 5. Thrust pick-up ratio. Thrust pick-up ratio (Zt) of the propeller and the helicopter main rotor is a value that determines the relationship between the weight air flow rate and the thrust developed by the propeller at the selected operating mode. Thrust pick-up ratio is a variable value that changes when the propeller operating mode changes. Thrust pick-up ratio to decrease its value with increasing load on the propeller, i.e [4]: where: SHMR -area of the swept surface of the helicopter main rotor; SR -area of the swept surface of the propeller.
The lower the load on the engine power, the lower the thrust pick-up ratio:

T/NHMR < T/ NPR → ZtHMR < ZtPR
Since the value of Zt is variable, then for the characteristics of the helicopter main rotor, this value must be indicated as extremely small, in this case the relationship between the maximum thrust of the helicopter main rotor and its power-to-weight ratio will be seen. The higher the power-to-weight ratio of the helicopter, the lower its thrust pick-up ratio.
According to the PHTP theory, the thrust pick-up ratio is determined as [6]: where: θmweight air flow of the helicopter main rotor. Its determined as: where: γdensity of the air; γ = 1,25kg/m 3 ; θvvolumetric air flow rate of the helicopter main rotor. Its determined as: θv = SНMR · Vi2 We will calculate the thrust pick-up ratio of the helicopter main rotor for the Mi-26 helicopter.
Data source [3,11]: 6. The speed of the horizontal flight. The speed of the horizontal flight ܸ of the helicopter is determined by the classical method using the formula [3,4]: where: ܰ -effective power of the power plant; Gweight of the helicopter, L/Dlift/drag ratio, ξthe utilization rate of power, ηefficiency of the propeller. According to the PHTP theory, the speed of the horizontal flight is determined by the formula:  Table 1.

Сonclusions
The proposed calculation method based on the pressure head theory of the propeller allows, with an error of no more than 1.5%, to determine the main aerodynamic characteristics of the Mi-26 helicopter main rotor (lift, jet torque, required power of the power plant, inductive air flow velocity, thrust response coefficient, speed horizontal flight). And it can be used when calculating the aerodynamic characteristics of any propellers.