Actuator Disc Theory:
1. Inﬁnitely thin disc of area A which oﬀers no resistance to air passing through it.
2. Purely 1-D analysis
3. Thrust loading and velocity are uniform over the disk.
4. Far upstream and far downstream the pressure is freestream static pressure.
5. Viscous eﬀects are not considered (no drag, no momentum diﬀusion)
6. Incompressible (compressibility correction can be made)
In a helicopter, lift isn’t created by the wings, but by rotating blades. This enables helicopters to hover. In case of hovering flight, the thrust produced by the rotor blades T must be equal to the weight W. But how can we calculate this thrust? One way to do this is by using the actuator disk theory, which has already been treated in the Aerodynamics A course. We will briefly repeat that theory here. Suppose we have a set of rotating blades, called the actuator disk. We can consider 4 points now. Point 0 is infinitely far to the left of the disk, while point 1 is just a little bit to the left of the disk. Identically, point 2 is just to the right of the disk and point 3 is infinitely far to the right. If we assume the velocity is evenly distributed over the actuator disk, and that the flow is incompressible and irrotational, we can derive a few simple relations. Let’s define the induced velocity Vi as Vi =
V1 –V0. We can now show that V2 = V1 and also V3 –V2 = Vi. So therefore the total velocity increment of the air is V3 – V0 = 2Vi. We can also show that the thrust now is
where m is the mass flow of air that goes through the disk and r is the actuator disk radius.
Blade Element Theory
In the blade element theory we look at an infinitely small piece of a rotor blade and derive the lift dL and drag dD acting on that piece. Using those forces we can integrate to find the thrust and drag of the full rotor blades.
So let’s look at an infinitely small piece of a rotor blade, at a distance r from the center. Such a piece is shown in figure 1. Since the rotor rotates with an angular velocity of Ω, the small piece of rotor blade has a velocity component to the left of Ωr. Due to the induced velocity; there is a downward pointing velocity component of Vi. Together they form the resultant velocity Vr. Since this velocity is not directed horizontally, the air has a certain inflow angle f. Also a pitch angle θ and an angle of attack α can be distinguished.
The small bit of thrust dT acting on the rotor blade part now is
There is a simple physical meaning behind this rotor solidity. It’s the part of the rotor disk that is filled with blades. The surface area of the rotor disk is simply _R2, while the surface area of the blades is nRc.
Drag is the force that opposes the motion of an aircraft through the air. Total drag produced by an aircraft is the sum of the profile drag, induced drag, and parasitedrag. Total drag is primarily a function of airspeed. The airspeed that produces the lowest total drag normally determines the aircraft best-rate-of-climb speed, minimum rate-of-descent speed for autorotation, and maximum endurance speed.
The following picture illustrates the different forms of drag versus airspeed:
Profile drag is the drag incurred from frictional resistance of the blades passing through the air. It does not change significantly with angle of attack of the airfoil section, but increases moderately as airspeed increases.
Induced drag is the drag incurred as a result of production of lift. Higher angles of attack which produce more lift also produce increased induced drag. In rotary-wing aircraft, induced drag decreases with increased aircraft airspeed. The induced drag is the portion of the total aerodynamic force which is oriented in the direction opposing the movement of the airfoil. Think of it as lift which is in the wrong direction.
Parasite drag is the drag incurred from the nonlifting portions of the aircraft. It includes the form drag and skin friction associated with the fuselage, cockpit, engine cowlings, rotor hub, landing gear, and tail boom to mention a few. Parasite drag increases with airspeed.
Curve “A” shows that parasite drag is very low at slow airspeeds and increases with higher airspeeds. Parasite drag goes up at an increasing rate at airspeeds above the midrange.
Curve “B” shows how induced drag decreases as aircraft airspeed increases. At a hover, or at lower airspeeds, induced drag is highest. It decreases as airspeed increases and the helicopter moves into undisturbed air.
Curve “C” shows the profile drag curve. Profile drag remains relatively constant throughout the speed range with some increase at the higher airspeeds.
Curve “D” shows total drag and represents the sum of the other three curves. It identifies the airspeed range, line “E”, at which total drag is lowest. That airspeed is the best airspeed for maximum endurance, best rate of climb, and minimum rate of descent in autorotation.
Hovering is the term applied when a helicopter maintains a constant position at a selected point, usually a few feet above the ground (but not always, helicopters can hover high in the air, given sufficient power). For a helicopter to hover, the main rotor must supply lift equal to the total weight of the helicopter. With the blades rotating at high velocity, an increase of blade pitch (angle of attack) would induce the necessary lift for a hover. The forces of lift and weight reach a state of balance during a stationary hover. Hovering is actually an element of vertical flight. Assuming a no-wind condition, the tip-path plane of the blades will remain horizontal. If the angle of attack of the blades is increased while their velocity remains constant, additional vertical thrust is obtained. Thus, by upsetting the vertical balance of forces, helicopters can climb or descend vertically.
Airflow during hovering
At a hover, the rotor tip vortex (air swirl at the tip of the rotor blades) reduces the effectiveness of the outer blade portions. Also, the vortexes of the preceding blade severely affect the lift of the following blades. If the vortex made by one passing blade remains a vicious swirl for some number of seconds, then two blades operating at 350 RPM create 700 long-lasting vortex patterns per minute. This continuous creation of new vortexes and ingestion of existing vortexes is a primary cause of high power requirements for hovering.
During hover, the rotor blades move large volumes of air in a downward direction. This pumping process uses lots of horsepower and accelerates the air to relatively high velocities. Air velocity under the helicopter may reach 60 to 100 knots, depending on the size of the rotor and the gross weight of the helicopter. The air flow pattern of a hovering helicopter is illustrated here:
Note how the downwash (induced flow) of air has introduced another element into the relative wind which alters the angle of attack of the airfoil. When there is no induced flow, the relative wind is opposite and parallel to the flight path of the airfoil. In the hovering case, the downward airflow alters the relative wind and changes the angle of attack so less aerodynamic force is produced. This condition requires the pilot to increase collective pitch to produce enough aerodynamic force to sustain a hover. Although this does increase the lift, it also increases the induced drag, and so total power required is higher.
The high power requirement needed to hover out of ground effect is reduced when operating in ground effect. Ground effect is a condition of improved performance encountered when operating near (within 1/2 rotor diameter) of the ground. It is due to the interference of the surface with the airflow pattern of the rotor system, and it is more pronounced the nearer the ground is approached. Increased blade efficiency while operating in ground effect is due to two separate and distinct phenomena.
First and most important is the reduction of the velocity of the induced airflow. Since the ground interrupts the airflow under the helicopter, the entire flow is altered. This reduces downward velocity of the induced flow. The result is less induced drag and a more vertical lift vector. The lift needed to sustain a hover can be produced with a reduced angle of attack and less power because of the more vertical lift vector:
The second phenomena are a reduction of the rotor tip vortex:
When operating in ground effect, the downward and outward airflow pattern tends to restrict vortex generation. This makes the outboard portion of the rotor blade more efficient and reduces overall system turbulence caused by ingestion and recirculation of the vortex swirls.
Rotor efficiency is increased by ground effect up to a height of about one rotor diameter for most helicopters. This figure illustrates the percent increase in rotor thrust experienced at various rotor heights:
At a rotor height of one-half rotor diameter, the thrust is increased about 7 percent. At rotor heights above one rotor diameter, the thrust increase is small and decreases to zero at a height of about 1 1/4 rotor diameters.
Maximum ground effect is accomplished when hovering over smooth paved surfaces. While hovering over tall grass, rough terrain, revetments, or water, ground effect may be seriously reduced. This phenomenon is due to the partial breakdown and cancellation of ground effect and the return of large vortex patterns with increased downwash angles.
Two identical airfoils with equal blade pitch angles are compared in the following figure:
The top airfoil is out-of-ground-effect while the bottom airfoil is in-ground-effect. The airfoil that is in-ground-effect is more efficient because it operates at a larger angle of attack and produces a more vertical lift vector. Its increased efficiency results from a smaller downward induced wind velocity which increases angle of attack. The airfoil operating out-of-ground-effect is less efficient because of increased induced wind velocity which reduces angle of attack.
If a helicopter hovering out-of-ground-effect descends into a ground-effect hover, blade efficiency increases because of the more favorable induced flow. As efficiency of the rotor system increases, the pilot reduces blade pitch angle to remain in the ground-effect hover. Less power is required to maintain however in-ground-effect than for the out-of-ground-effect hover.
In accordance with Newton’s law of action and reaction, the helicopter fuselage tends to rotate in the direction opposite to the rotor blades. This effect is called torque. Torque must be counteracted and or controlled before flight is possible. In tandem rotor and coaxial helicopter designs, the rotors turn in opposite directions to neutralize or eliminate torque effects. In tip-jet helicopters, power originates at the blade tip and equal and opposite reaction is against the air; there is no torque between the rotor and the fuselage. However, the torque problem is especially important in single main rotor helicopters with a fuselage mounted power source. The torque effect on the fuselage is a direct result of the work/resistance of the main rotor. Therefore torque is at the geometric center of the main rotor. Torque results from the rotor being driven by the engine power output. Any change in engine power output brings about a corresponding change in torque effect. Furthermore, power varies with the flight maneuver and results in a variable torque effect that must be continually corrected.
Compensation for torque in the single main rotor helicopter is accomplished by means of a variable pitch anti-torque rotor (tail rotor) located on the end of a tail boom extension at the rear of the fuselage. Driven by the main rotor at a constant ratio, the tail rotor produces thrust in a horizontal plane opposite to torque reaction developed by the main rotor. Since torque effect varies during flight when power changes are made, it is necessary to vary the thrust of the tail rotor. Anti-torque pedals enable the pilot to compensate for torque variance. A significant part of the engine power is required to drive the tail rotor, especially during operations when maximum power is used. From 5 to 30 percent of the available engine power may be needed to drive the tail rotor depending on helicopter size and design. Normally, larger helicopters use a higher percent of engine power to counteract torque than do smaller aircraft. A helicopter with 9,500 horsepower might require 1,200 horsepower to drive the tail rotor, while a 200 horsepower aircraft might require only 10 horsepower for torque correction.