47 Helicopters & Vertical Flight


Helicopters are a form of Vertical Take Off and Landing (VTOL) aircraft. They can take off and land from almost anywhere on land or at sea, hover motionless in the air, and then fly in any direction at the pilot’s whim. VTOL is a tremendously valuable capability for an aircraft, sometimes referred to as a runway-independent aircraft. However, there is a significant price to pay for VTOL capability in terms of engineering complexity, limited flight range, endurance, other compromised flight capabilities, as well as much higher costs.

The development of the helicopter, which is also a type of rotorcraft, lagged behind successful airplanes by almost 30 years. The success of helicopters is attributed to the advancement of aeronautical technologies and understanding. Over time, they have evolved into reliable and technologically advanced machines with various capabilities, playing a crucial role in various industries, including civil and military services. The versatility and unique capabilities of helicopters have made them indispensable tools in various missions, such as rescue operations, medical services, surveillance, and defense. Any military service would fail to function without access to the helicopter’s unique capabilities.

Contemporary helicopters take advantage of state-of-the-art aerospace technologies, including advanced aerodynamics, composite materials, fly-by-wire (FBW) flight controls, and health and usage monitoring systems (HUMS).

Modern helicopters utilize cutting-edge aerospace technologies to enhance performance, safety, and efficiency. Advanced aerodynamics provides increased flight performance and lower vibration levels. The use of composite materials helps to reduce weight and improve durability. Fly-by-wire flight control systems allow for greater precision in maneuvering, and HUMS provides real-time monitoring of the helicopter’s condition and performance, enabling proactive maintenance and prolonging the aircraft’s lifespan. These technologies have contributed to the continued success and development of the helicopter industry.

Learning Objectives

  • Know the history and challenges of developing a successful helicopter compared to an airplane.
  • Better understand the factors that affect the hovering flight performance of helicopters and other rotating-wing aircraft.
  • Understand the basic performance characteristics in forward flight and the aerodynamic limitations that arise.
  • Appreciate how a helicopter is controlled during flight, including using blade hinges and cyclic blade pitch.


Jacob_Ellehammer of Denmark was an aeronautical engineer and one pioneer of the helicopter, his coaxial rotor machine being photographed in flight as early as 1913, as shown below. His machine made brief hops off the ground, and short hovering flights but could not fly forward or do anything useful. In France, Paul Cornu and Louis Breguet also made notable attempts to build and fly helicopter concepts between 1906 and 1909, but they were failures. The helicopter concepts built by Oehmichen, DeBothezat and D’Ascanio during the 1920s had somewhat more success, but their capabilities were very marginal. Many other attempts were made to build and fly helicopters in the following decade, but only with incremental success.

A black and white photograph showing a dual blade concept helicopter in silhouette.
One of the first helicopters was designed by Jacob Ellehammer in Denmark in 1913. It only made short hops off the ground.

The first successful rotorcraft concept was not a helicopter but an autogiro with an unpowered rotor called an autogiro. The name “autogiro” comes from the Greek words αὐτός (auto) and γύρος (turning in a circle or forming a disk), essentially meaning a “self-turning” or “autotoating” rotor. The name “Autogiro” is a proprietary name coined by Cierva, but all aircraft in this class are known as “autogiros.” Juan de la Cierva’s fourth attempt to build an autogiro resulted in the C-4, as shown in the photograph below, which first flew in 1923.

A grainy black and white photograph of an aircraft lifting off with one large rotating propeller above the pilot.
Cierva’s C-4 autogiro was the first successful rotating-wing aircraft, but it could not hover because the rotor was unpowered.

His primary innovation included an independent flapping hinge on each blade, allowing the blades to freely move up and down out of the plane of rotation (i.e., to flap) to balance the asymmetric aerodynamic forces over the rotor disk when the machine flew forward. As a result, the autogiro could take off and land at very low airspeeds, but it could not hover, the aircraft always requiring forward and/or downward motion through the air for the rotor to autorotate.

Through the mid-1930s, the autogiro proved its usefulness, but it was a helicopter with the ability to hover that was ultimately desired. Nevertheless, the autogiro formed an engineering platform for the helicopter’s future development. Of significance was the development of flapping and lead/lag hinges to attach the blades and allow movement (articulation) as well as a means of giving it effective flight control using cyclic variations of blade pitch. To this end, notable engineering advancements integrating the successful elements were made by Raoul Hafner with his AR-III gyroplane.

By the outbreak of WW2, the first successful helicopters had started to appear, including the Focke-Wulf Fw-61 and the Sikorsky R-4. Germany used several helicopters during WW2, albeit in limited numbers. However, not until after WW2 did they become a part of the aviation spectrum for military and commercial use. Many advancements in helicopter performance were made during the 1940s and 1950s, much of them in the U.S. The first civil-certified helicopters included the Bell-47, as shown in the photo below, the Bristol Sycamore, the Sikorsky S-55, and the Piasecki HRP-1. The Sikorsky UH-60 Black Hawk, which was introduced in the late 1970s, was to become one of the world’s most successful military helicopters and remains in production today.

Photograph of a helicopter on a grey sky background. The helicopter has mostly visible framework and a clear bubble-style cockpit.
In 1948 the Bell-47 was first used by the U.S. military in the Korean War, but it was also the world’s first civil certified helicopter and was built in many thousands.

Sustained development in areas of helicopter technology over the last half-century has led to many other successful military and civil helicopter designs. Today, helicopters are increasingly made of advanced composite materials and incorporate many current aerospace technologies to improve their performance, reliability, and safety. Indeed, modern helicopters have matured into sophisticated aircraft with extraordinary performance capability and have a unique role within the aviation spectrum provided by no other aircraft.

A modern helicopter, in this case an EC-145, is built primarily of composite materials to give the airframe strength and lightness.

One issue with helicopters is that they are inherently low-speed aircraft with maximum flight speeds of only about 150 knots (173 mph, 278 kph). This speed limitation is partly caused by the rotor system’s conflicting aerodynamic characteristics in that, unlike an airplane, the rotor begins to experience both compressibility effects and stall effects simultaneously. Remember that an airplane will only experience stall at low airspeeds and compressibility effects at high airspeeds. To try to overcome such inherent limitations, some helicopters have been fitted with auxiliary fixed wings or a separate propulsion system, which are known as compound helicopters. These systems can allow them to fly faster, but usually at the expense of more power and fuel, and also carrying less useful load.

Hybrid rotorcraft such as the notorious Bell-Boeing V-22 Osprey and the long-awaited Leonardo/AgustaWestland AW609 (formerly the Bell/Agusta BA609) attempt to combine the vertical takeoff and hover capabilities of helicopters with the increased speed and efficiency offered by airplanes. However, tiltrotors are not as good as helicopters for the things a helicopter does well (e.g., to hover or operate at low airspeeds), and not as good as airplanes for the things airplanes do well (e.g., flying faster over longer distances when carrying a significant payload). Nevertheless, the V-22 Osprey tiltrotor has proven helpful for certain military missions. However, it remains to be seen whether the tiltrotor can succeed in the civil market, partly because its mission is not clear, as well as the much higher costs than an airplane or a helicopter.

A tiltrotor attempts to combine the benefits of the helicopter with those of the airplane, although it falls short in matching the unique capabilities of either helicopters or airplanes.

Many uncrewed drones or Unoccupied Aerial Vehicles (UAV), which are more generally known as Unoccupied Aerial Systems (UAS), use rotors and/or propellers and have been designed specifically for VTOL capability. UAVs are preferred for D3 or “Dull, Dirty & Dangerous” missions. UAVs originated for military applications but their use has expanded exponentially for civil applications. As shown in the photograph below, the quad-rotor or “quadcopter” design has become a prevalent configuration for smaller UAVs. Most UAVs carry cameras, although other sensor packages may be used too. They can be used for reconnaissance, aerial photography, surveys, disaster relief, etc.

An electrically-powered “quadcopter” VTOL aircraft or eVTOL drone can potentially be used for many purposes.

The recent emergence of electrically powered VTOL or eVTOL concepts for Urban Aerial Mobility (UAM) has kindled interest in advancing vertical flight technologies, including batteries and electric motors. Such eVTOL aircraft are envisaged to be part of safe and efficient aviation “air-taxi” UAM infrastructure that will rapidly transport passengers and cargo over short distances at lower altitudes within urban and suburban areas. The eVTOL community has many innovators, but it remains to be seen if these less orthodox aircraft can be made to work technically with the speed, range, and endurance that is needed, as well as integrated within the national airspace system.

Helicopter Configurations

Many of the early helicopter concepts prior to 1930 were of the coaxial or side-by-side (lateral) rotor configuration, as shown in the figure below. Perhaps the simplest idea of using just a single main rotor with a smaller, sideward-thrusting tail rotor to compensate for torque reaction was not used until much later in the development of the helicopter. Nevertheless, the single main rotor/tail rotor configuration has since become the most common, comprising over 95% of all helicopters currently flying. The tandem rotor design is attractive for bigger helicopters, with both rotors providing useful lift, despite the greater mechanical complexity involved in gearing and controlling the two rotors

Types of helicopter. (a) Single main rotor/tail rotor (conventional) configuration. (b) Tandem rotors. (c) Coaxial rotors. (d) Side-by-side rotors. (e) Intermeshing rotors.

Contra-rotating “coaxial”rotors with one rotor above the other on a concentric shaft automatically balances torque reaction on the airframe, a helicopter configuration made famous laterally by the Kamov company. Another advantage is the compact footprint of a coaxial concept, despite the greater mechanical complexity of the two rotors. Side-by-side rotors, especially if the shafts were inclined inwards, gave the early machines somewhat better lateral stability, but the design is uncommon today. One notable example is the enormous Mil MV-12. Again there is a greater level of mechanical complexity associated with this type of design. In the intermeshing design has outward tilted contra-rotating shafts with intersecting rotor disk planes, such as on the Kaman K-Max. Like the coaxial, the advantage of the intermeshing design is a smaller overall footprint.

Basis of Helicopter Flight

An airplane has separate systems for lift, propulsion and control, while a helicopter’s rotor system must perform all three functions. The rotor blades provide lift to overcome the weight of the helicopter, while the engine provides the power to spin the rotor and generate lift. The helicopter’s control surfaces, such as the swashplate, pedals, and collective, allow for control over the helicopter’s direction and altitude. This makes the design and operation of a helicopter more complex compared to an airplane, as the rotor system must be able to perform multiple tasks simultaneously. Nevertheless, the helicopter’s unique ability to hover and fly vertically makes it ideal for many missions where a fixed-wing aircraft is not practical.

As shown in the figure below, the rotor generates a vertical lifting force (called thrust) in opposition to the helicopter’s weight, which is obtained from the collective lift forces on the spinning rotor blades. The generation of a horizontal propulsive force for forward flight is obtained by tilting the rotor disk plane forward to give a component of the rotor thrust to overcome the drag on the helicopter, which is accomplished by the pilot by actuating the flight controls. The means of controlling a helicopter during its flight is discussed later.

The rotor provides lift to overcome weight but also a propulsive force to overcome the drag in forward flight.

The rotor also provides a means of generating forces and moments to help control the attitude and position of the helicopter in three-dimensional space. This behavior is obtained by tilting the orientation rotor disk left and right as well as fore and aft, as shown in the figure below. Tilting the disk requires that the blade lift be modulated to cause flapping about the hinges, which results in a moment being applied through the hub and rotor shaft to the fuselage, and the so the fuselage quickly aligns with the tilted rotor plane.

Flight control is also obtained by tilting the rotor disk. The resulting forces and moments on the fuselage cause it to quickly align with the rotor plane.

For a single rotor helicopter, an anti-torque and yaw control system is necessary and is typically achieved with a tail rotor that generates sideward thrust. The tail rotor thrust can be modulated by the pilot for yaw control. In other rotor configurations such as coaxial, tandem, and side-by-side, a tail rotor is not needed as the net torque reaction is already balanced. However, a small residual torque reaction can be balanced through differential tilts of the two rotor disks.


An anti-torque system is needed to balance the torque reaction on the airframe from the main rotor.

Rotor Flow Environment

The lifting capability of a lifting surface is related to its local angle of attack and flow velocity, specifically the dynamic pressure. In the case of a fixed (non-rotating) wing then the free-stream velocity is uniform and the lift is also fairly uniform along its span. However, in the case of a rotor, the flow velocity varies linearly along the span because of the rotation, the consequences being that the aerodynamic loads over the rotor disk are much more biased toward the blade tip, as shown in the figure below.

The differences between a non-rotating wing and rotating wing is that the former needs to be moving forward to create lift whereas a rotor will create lift without any forward speed.

A fixed-wing must constantly be moving forward to create lift. However, with a rotating wing or rotor blade, which for now can be assumed to be in hovering  (non-translating forward) flight, then because of the rotation of the blade the lift can be generated without any “free-stream” flow or forward motion. There will be no flow velocity at the rotational axis but the flow velocity will increase linearly along the span of the blade. The velocity will reach a maximum, V_{\rm tip} = \Omega \, R, at the blade tip, where R is the rotor radius and \Omega is the angular velocity of the rotor. The consequence of the foregoing is that a wing will have a reasonably uniform lift distribution, but a helicopter rotor will have a lift distribution that is much more biased toward the blade tips.

When a rotor flies forward such that now there is a component of a “free stream,” the flow over the rotor blades will no longer be axisymmetric about the center of the rotor, as shown in the schematic below. To explain why, the blade position can be defined in terms of an azimuth angle, \psi, which is defined as zero when the blade is pointing downstream, as shown in the figure below. Now a component of the free stream, V_{\infty}, adds to or subtracts from the rotational velocity at each part of the blade, i.e., for the tip section then

(1)   \begin{equation*} V_{\rm{tip}} =\Omega R + V_{\infty} \sin \psi \end{equation*}

and at any radial position a distance y from the rotational axis then

(2)   \begin{equation*} V (y, V_{\infty}) =\Omega y + V_{\infty} \sin \psi \end{equation*}

The asymmetries in the flow velocities at a rotor in forward flight are the source of substantially different aerodynamic environments over the rotor disk.

The consequence of this effect is that now the right-side (advancing) of the rotor disk where the blades advance into the free-stream flow will see a higher overall flow velocity and dynamic pressure, so higher lift. On the left-side (retreating) side of the rotor disk, where the dynamic pressure is much lower, the blades will experience much less lift. This is the problem, originally identified by Juan de la Cierva, often referred to as  the “dissymmetry in lift,” as explained in the schematic below. Consequently, with blades that are rigidly attached to the rotor shaft, there will be substantial aerodynamic rolling moments on the rotor (turning the rotor toward its retreating side) that will make a rotorcraft of any kind impossible to fly.

A solution to the “dissymmetry in lift” problem between the advancing and retreating sides of the rotor disk is to use a flapping hinge on each blade to allow it to respond naturally to the changing flow velocities.

Cierva’s solution to this problem was to modulate the asymmetry in blade lift between the two sides of the rotor disk by using a pin-joint or “flap hinge” at the root of each blade. This hinge then allowed each blade to freely move up and down in response to the changing aerodynamic lift, as shown in the schematic below. A “lead-lag” hinge was also used to alleviate the in-plane inertia loads caused by Coriolis accelerations as the blade flaps up and down and its center of gravity (or radius of gyration) moves inward and outward relative to the rotational axis.

A rotor blade will have a flapping and lead/lag hinge. A feathering bearing allows for changes in blade pitch, which can be modulated through a control system.

The effects of blade flapping change the angles of attack on the blades in a favorable way to help balance out the distribution of aerodynamic loads over the rotor disk. On the advancing side of the rotor disk, where there is an excess of lift because of the higher dynamic pressure, then the blade flaps up about the hinge, which decreases the effective angle of attack and so reduces the lift, as shown in the schematic below. On the retreating side, where there is a much lower dynamic pressure, the blade flaps down, which will correspondingly increase the lift there. The net effect, therefore, is that a flapping hinge allows the lift forces and moments created on the advancing and retreating sides to be better balanced overall and also over the entire rotor disk. Indeed, Cierva refers to his invention of the flapping hinge as his “secret of success.”

As a blade flaps up and down about the hinge, the changes in angles of attack lift act to oppose flapping.

By further modulating the blade pitch angle or “feathering” angle, the lift on the blades can be changed so that the blades will flap up or flap down at the appropriate location around the rotor azimuth, thereby causing the rotor disk plane as a whole to tilt left-and-right or fore-and-aft. By tilting the rotor disk it gives a basis of controlling the orientation of the rotor thrust vector and so the forces and moments acting on the helicopter as a whole. The hinges and feathering bearing can be arranged in different orders, with one example of a fully articulated rotor being shown in the photograph below.

A fully articulated rotor system using flap hinges, feathering bearings, and lead/lag hinges. The combined mechanisms with pitch links, pitch horns, etc., produce a rather complicated looking rotor hub.

Blade Tip Vortices

One consequence of lift generation is that “tip vortices” form and trail from the tip of each rotating-blade, just as they would trail from the tips of a wing. The figure below shows an example of the physical nature of the vortical wake generated by a helicopter rotor in hovering flight compared to that of an airplane. In each case, the tip vortices are rendered visible by the natural condensation of water vapor in the air, which leave white wispy clouds in the flow behind the tips.

The vortical wake generated by a helicopter rotor compared to that of an airplane. The tip vortices are rendered visible by the natural condensation of water vapor in the air, forming long, thin white clouds.

For an airplane, the vortices trail downstream and are left behind in the wake, never to influence the wing again, at least directly. However, for a rotor, the vortices are convected downward below the rotor and form a series of interlocking, almost helical trajectories. Therefore, the rotor blades can encounter their own self-generated vortices as well as the vortices generated by other blades, which, overall, creates a much more complex flow environment at the rotor. For this latter reason, predicting the strengths and locations of the tip vortices plays an essential role in estimating blade airloads and rotor performance. The significant non-uniformity of the angles of attack that the blade sections experience as they sweep around the rotor disk during edgewise forward motion is one complication with the helicopter rotor that makes its aerodynamic analysis difficult.

Hovering Flight Analysis

Unlike airplanes, helicopters can hover, which is a flight condition where they are specifically designed to be operationally efficient. In hover, the primary purpose of the rotor is to provide a vertical lifting force in opposition to the weight of the helicopter. The generation of thrust requires torque (and power) to be applied through the rotor shaft. In hover or axial flight, the flow is a nominally axisymmetric streamtube that passes down through the rotor disk. This flow regime is the easiest to analyze in the first instance and, in principle, it should be the easiest to predict using mathematical models.

Although it must be remembered that the actual physical flow through the rotor will generate a complicated vortical wake structure, as previously discussed, the primary performance of the rotor can be analyzed by a more straightforward approach known as the momentum theory. This particular approach allows for predictions of the relationships between rotor thrust and power required to produce that thrust. Also, it exposes some of the other parameters that can be used determine rotor performance and efficiency, including disk loading, blade loading, power loading, and the figure of merit.

Time-Averaged Flow Field

Hover is a unique flight condition where the rotor has zero forward (edgewise) speed and zero vertical speed (no climb or descent). A set of velocity measurements in a diametric plane near a hovering rotor and its wake is shown in the figure below. The flow velocity smoothly increases as it is entrained into and through the rotor disk plane. There is no jump in velocity across the rotor disk, although because a thrust is produced on the rotor, there certainly must be a jump in time-averaged pressure over the disk.

Measurements of the velocity field in a diametric plane near and below a two-bladed rotor operating in hover.

A clear wake boundary or slipstream is apparent in these measurements, with the flow velocity outside this boundary being relatively calm or quiescent. Inside the wake boundary, the flow velocities are substantial and may be distributed non-uniformly across the streamtube and in the slipstream. Notice also the contraction in the diameter of the streamtube and wake boundary below the rotor corresponding to an increase in the slipstream velocity.

Flow Model

With the physical picture of the hovering rotor flow apparent, it is possible to develop a mathematical model. Consider the application of the three basic conservation laws (conservation of mass, momentum, and energy) to the rotor and its flow field. The conservation laws will be applied in a steady, incompressible, inviscid, axisymmetric one-dimensional integral formulation to a control volume surrounding the rotor and its wake.

This simplified approach permits the most basic analysis of the rotor performance (e.g., to determine the thrust produced and power required) but without considering the details of the flow environment or what is happening locally at each blade section. This approach, called the momentum theory, was first developed by William Rankine in 1865 for analysis of marine propellers and formally generalized by  Hermann Glauert in 1935 for application to helicopter rotors and autogiros. The assumptions are:

  1. One dimensional, steady, incompressible, inviscid flow.
  2. The rotor is an infinitesimally thin disk that acts as a pressure discontinuity within the moving fluid.
  3. The disk offers no resistance to fluid passing through it.
  4. The pressure and velocity is uniform over the disk.
  5. The flow is at ambient static pressure far upstream and downstream of the rotor.

Consider the figure shown below. Let cross-section 0 denote the plane far upstream of the rotor, where the air is still or quiescent. A denotes the rotor disk area. Cross sections 1 and 2 are the planes just above and below the rotor disk, respectively. The slipstream “far” wake is downstream of the rotor at \inftyThe flow through the rotor increases smoothly and continuously, although there must be a jump in static pressure over the disk to create thrust.

One-dimensional axisymmetric flow model used to analyze the performance of a hovering rotor.

A fundamental assumption in the momentum theory is that the rotor can be idealized as an infinitesimally thin actuator disk supporting a pressure difference; this concept is equivalent to an infinite number of blades of zero thickness. This actuator disk supports the thrust generated by rotating blades about the shaft and their action on the air. Work done by the rotor on the air leads to a gain in kinetic energy of the rotor slipstream, which is an unavoidable energy loss and a byproduct of thrust generation called induced power. By Newton’s third law, the force on the flow means an equal and opposite force is produced on the rotor, i.e., the rotor thrust, T.

Application of the Conservation Principles

At the plane of the rotor, assume that the velocity there, which is called the induced velocity, is v_i, In the far slipstream, the velocity will be increased over that at the plane of the rotor and this velocity is denoted by w.  The mass flow rate, \overbigdot{m}, must be constant within the boundaries of the rotor wake, i.e., inside the control volume. The only cross-section of the wake boundary that us uniquely defined  is at the rotor disk, so based on the assumed flow characteristics then

(3)   \begin{equation*} \overbigdot{m} = \varrho A_\infty w = \varrho A_2 v_i = \varrho A v_i \end{equation*}

The principle of conservation of momentum gives the relationship between the rotor thrust, T, and the net time rate-of-change of momentum of the air out of the control volume (Newton’s second law). The rotor thrust is equal and opposite to the force on the air. For an unconstrained flow, the net pressure force on the air inside the control volume is zero and so the effects of the external pressure can be neglected. Therefore, the rotor thrust can be written as

(4)   \begin{equation*} T = \overbigdot{m} w \end{equation*}

From the principle of conservation of energy, the work done on the rotor is equal to the gain in energy of the air per unit time. The work done per unit time, or the power consumed by the rotor, is Tv_i, so that

(5)   \begin{equation*} T v_i = \frac{1}{2} \overbigdot{m} w^2 \end{equation*}

From Eqs. 4 and 5 then it is apparent

(6)   \begin{equation*} v_i =  \frac{1}{2} w \quad \mbox{or~that} \quad w = 2 v_i \end{equation*}

This latter result, therefore, gives a simple relationship between the induced velocity in the plane of the rotor, v_i, and the velocity w in the slipstream. Notice also that based on ideal flow assumptions, the slipstream comprises an area that is exactly half of the rotor disk area.

Induced Velocity

The rotor thrust is related to the induced velocity at the rotor disk using

(7)   \begin{equation*} T = \overbigdot{m} w = \overbigdot{m} (2 v_i) = 2 (\varrho A v_i) v_i = 2 \varrho A v_i^2 \end{equation*}

Rearranging Eq. 7 to solve for the induced velocity gives

(8)   \begin{equation*} v_i = \sqrt{ \frac{T}{2 \varrho A} } = \sqrt{ \left( \frac{T}{A}\right) \frac{1}{2\varrho} } \equiv v_h \end{equation*}

Notice that v_h (\equiv v_i) is used for the induced velocity in hover because it becomes  a reference when the axial climb and descent and forward flight conditions are considered. Of significance here is that ratio T/A appears in Eq. {vieqn}, which is known as the disk loading and it an extremely important parameter in rotor analysis.

Power Required

The power required to hover, which is the time rate of work done by the rotor on the air, is given by

(9)   \begin{equation*} P = T v_i \equiv T v_h = T \sqrt{\frac{T}{2\varrho A}} = \frac{T^{3/2}}{\sqrt{2\varrho A} } \end{equation*}

This power value, called ideal power, is entirely induced in nature because the contribution of viscous effects has yet to be considered. In other words, this power value is the absolute lowest and hence the “ideal” amount required to generate a given rotor thrust.

Because w = 2 v_i, then it can also be written that

(10)   \begin{equation*} P = T v_h = 2 \overbigdot{m} v_h^2 = 2 (\varrho A v_h) v_h^2 = 2 \varrho A v_h^3 \end{equation*}

From this latter equation, it is noted that the power required to hover will increase with the cube of the induced velocity v_h at the disk. Therefore, to make a rotor hover generating a given amount of thrust, then the induced velocity v_h at the disk must be as low as possible. Obviously, if v_h is too low, then the rotor will generate no thrust because for a given mass flow rate then

(11)   \begin{equation*} T = 2 \overbigdot{m} v_h \end{equation*}

Therefore, to minimize the power required to produce thrust then, the goal is that v_h must be as low as possible, but the mass flow rate \overbigdot{m} through the disk must be as large as possible. This goal, consequently, requires a large rotor disk area to entrain the needed mass flow; large-diameter rotors are a fundamental design feature of all helicopters.

Pressure Variations

The pressure variation through the rotor flow field in the hover state can be found from the application of Bernoulli’s equation above and below the rotor disk. However, remember that there is a pressure jump across the disk as a result of energy added by the rotor, so Bernoulli’s equation cannot be applied across the disk.

Referring to the previous figure and applying Bernoulli’s equation up to the rotor disk between stations 0 and 1 produces

(12)   \begin{equation*} p_0 = p_{\infty} = p_1 + \frac{1}{2} \varrho v_i^2 \end{equation*}

and below the disk, between stations 2 and \infty, then

(13)   \begin{equation*} p_2 + \frac{1}{2} \varrho v_i^2 = p_\infty + \frac{1}{2} \varrho w^2 \end{equation*}

Because the jump in pressure \Delta p is assumed to be uniform across the disk, this pressure jump must be equal to the disk loading, T/A, that is

(14)   \begin{equation*} \Delta p = p_2 - p_1 = \frac{T}{A} \end{equation*}

Therefore, it can be written that

(15)   \begin{equation*} \frac{T}{A} = p_2 - p_1 = \left( p_\infty +\frac{1}{2} \varrho w^2 - \frac{1}{2} \varrho v_i^2 \right) - \left( p_\infty - \frac{1}{2} \varrho v_i^2 \right) = \frac{1}{2} \varrho w^2 \end{equation*}

from which it is seen that the rotor disk loading (a pressure) is equal to the dynamic pressure in the slipstream.

The pressure just above and below the disk can also be obtained in terms of disk loading. Just above the disk, the use of Bernoulli’s equation gives

(16)   \begin{equation*} p_1 = p_\infty - \frac{1}{2} \varrho v_i^2 = p_\infty - \frac{1}{2} \varrho \left( \frac{w}{2} \right) ^2 = p_\infty - \frac{1}{4} \left( \frac{T}{A} \right) \end{equation*}

and just below the disk, then

(17)   \begin{equation*} p_2 = p_0 + \frac{1}{2} \varrho w^2 - \frac{1}{2} \varrho \left(\frac{w}{2}\right)^2 = p_0 + \frac{3}{4} \left( \frac{T}{A} \right) \end{equation*}

Therefore, the conclusion to be drawn is that the static pressure is reduced by \frac{1}{4}(T/A) above the rotor disk and increased by \frac{3}{4}(T/A) below the disk.

Worked Example #1 – Distribution of Thrust

The simple momentum theory assumes that the jump in pressure across the actuator disk of a hovering rotor is constant and uniform. By considering an elemental annulus of the rotor disk, prove that this result must be consistent with a distribution of lift (or thrust force grading) across the rotor disk that varies linearly from a value of zero at the center (rotational axis) of the rotor to a maximum value around the edges of the rotor disk.

The area of the elemental annulus is

    \[ dA = 2\pi y dy \]

The rotor disk loading (a pressure) is

    \[ \frac{T}{A} = \Delta p = \mbox{constant} \]

where \Delta p is the pressure jump over the disk. Therefore, the thrust dT on the elemental annulus is

    \[ dT = \left( \Delta p \right)\, dA = \left( \Delta p \right) 2\pi y dy = 2\pi (\Delta p) \, y dy \]

Finally, the thrust per unit span or thrust distribution is

    \[ \frac{dT}{dy} =  2\pi (\Delta p) \, y \]

which is a linear distribution over the radial dimension of the rotor disk from the rotational axis to the edge of the disk.

Disk Loading & Power Loading

A parameter frequently used in helicopter analysis that appears in the preceding equations is the disk loading, T/A, denoted by DL. Because for a single-rotor helicopter in a hover, the rotor thrust, T, is equal to the helicopter’s weight, W, the disk loading is sometimes written as W/A or W/\pi R^2. However, to compute the disk loading for multi-rotor helicopters such as tandems and coaxials or tiltrotors, a first assumption is that each rotor carries an equal proportion of the vehicle’s weight.

Disk loading is measured in pounds per square foot (lb ft^{-2}) in USC or Newtons per square meter (N m^{-2}) in SI. In the SI system, the disk loading may also be quoted in terms of kilograms per square meter (kg m^{-2}). However, be aware that the direct use of the kilogram (kg) as a surrogate for a unit of force is strictly incorrect.

The power loading is defined as T/P, which is denoted by PL. Power loading is measured in pounds per horsepower (lb hp^{-1}) in USC or Newtons per kilowatt (N kW^{-1}) or kilograms per kilowatt (kg kW^{-1}) in the SI system. Remember that the induced (ideal) power required to hover is given by P = T v_h. The ideal power loading is inversely proportional to the induced velocity at the disk, i.e.,

(18)   \begin{equation*} v_h = \sqrt{ \frac{T}{2\varrho A}} = \sqrt{ \frac{DL}{2 \varrho}} = \frac{P}{T} = PL^{-1} \end{equation*}

According to the results in the figure below, the power loading, PL = T/P, decreases quickly with increasing disk loading, DL (= T/A); notice the logarithmic scales. Therefore, vertical lift aircraft with low effective disk loading will have relatively low power requirements per unit of thrust produced, i.e., they will have a high power loading. This outcome means that they will tend to be more efficient, i.e., the rotor will require less power and so consume less fuel to generate a given amount of thrust.

Hovering efficiency versus disk loading for a range of vertical lift aircraft. Traditional helicopters have the best efficiency compared to other VTOL aircraft.

Helicopters operate with low disk loadings in the region of 5 to 10 lb ft^{-2} or 24 to 48 kg m^{-2}, so they can provide a large amount of lift for relatively low power with power loadings of up to about 5 kg kW^{-1} (50 N kW^{-1} or 10 lb hp^{-1}). The results in the above figure show that the helicopter is a very efficient aircraft in hover compared to other VTOL aircraft. Tiltrotors, by virtue of their compromised design, have higher rotor disk loadings, so they are less efficient in hover than a helicopter of the same in-flight weight. Jet thrust concepts have very high effective disk loadings because of their high jet velocities.

Non-Dimensional Hovering Analysis

As for airfoils and wings, non-dimensional parameters are used in rotor analysis for generalizing aerodynamic performance. The non-dimensional value of the inflow, \lambda_i, called the induced inflow ratio, is written as

(19)   \begin{equation*} \lambda_i = \frac{v_i}{\Omega R} \end{equation*}

and in the hover case

(20)   \begin{equation*} \lambda_h = \frac{v_h}{\Omega R} \end{equation*}

Recall that the angular or rotational speed of the rotor is denoted by \Omega, and R is the rotor radius; the product is the tip speed, i.e., V_{\rm{tip}} = \Omega R.

For helicopter rotors, it is the convention to non-dimensionalize all velocities by the blade tip speed in hovering flight V_{\rm{tip}}, and the reference area is the rotor disk area, A. The rotor thrust coefficient is defined as

(21)   \begin{equation*} C_T = \frac{T}{\varrho A V_{\rm tip}^2} = \frac{T}{\varrho A \Omega^2 R^2} . \end{equation*}

Now it can be seen that the hover value of the inflow ratio, \lambda_h, is related to the thrust coefficient by

(22)   \begin{equation*} \lambda_h = \frac{v_i}{\Omega R} = \frac{1}{\Omega R} \sqrt{\frac{T}{2 \varrho A}} = \sqrt{ \frac{T}{2 \varrho A (\Omega R)^2}} = \sqrt{\frac{C_T}{2}} \end{equation*}

The corresponding rotor power coefficient is defined as

(23)   \begin{equation*} C_P = \frac{P}{\varrho A V_{\Omega R}^3} = \frac{P}{\varrho A \Omega^3 R^3}, \end{equation*}

Therefore, based on the momentum theory, the power coefficient for the hovering rotor becomes

(24)   \begin{equation*} C_P = \frac{T v_i}{\varrho A (\Omega R)^3} = \left( \frac{T}{\varrho A (\Omega R)^2} \right) \left( \frac{v_i}{\Omega R} \right) = C_T \lambda_i = \frac{C_T^{3/2}}{\sqrt{2}} \end{equation*}

Again, this result is calculated on the basis of uniform inflow over the rotor disk and no viscous losses, so it is called the ideal power.

The corresponding rotor shaft torque coefficient is defined as

(25)   \begin{equation*} C_Q = \frac{Q}{\varrho A V_{\Omega R}^2 R} = \frac{Q}{\varrho A \Omega^2 R^3} \end{equation*}

Notice that because power P is related to torque Q by P=\Omega Q, then numerically C_P has the same value as C_Q, although it would be incorrect to write that C_P = C_Q.

Measured Rotor Performance

In terms of coefficients, the ideal power to hover according to the simple momentum theory can be written as

(26)   \begin{equation*} C_P = \frac{C_T^{3/2}}{\sqrt{2}} \end{equation*}

The figure below shows a comparison of Eq. 26 with thrust and power coefficient measurements made for a hovering rotor. The form of presentation is called a power polar and is analogous to the drag polar used for airplane wings. Notice that the momentum theory underpredicts the actual power required, but the predicted trend that C_P \propto C_T^{3/2} is correct. These differences between the momentum theory and experiments occur because viscous effects (i.e., non-ideal effects) have not been included in the basic theory. However, this deficiency can be rectified using empirical corrections to the theory.

Power polar for a hovering rotor. The simple momentum theory, which is absent of effects with their origin in viscosity, substantially underpredicts the power required.

Non-Ideal Effects

Non-ideal but physical effects that are not included in the basic momentum theory include things such as non-uniform inflow, tip losses, wake swirl, less than ideal wake contraction, finite number of blades, and so on. One of the largest contributors to non-ideal effects is “tip loss,” which reflects the fact that a lifting surface cannot create a finite lift at its tips, and so the lift on the blade decreases rapidly as the tip is approached. In general, non-ideal effects can be split into lifting (induced) and non-lifting contributions.

Induced Effects

In the ideal rotor theory, then \kappa = 1. For an actual rotor, \kappa can be derived from rotor measurements or flight tests. For preliminary design, most helicopter manufacturers use their own measurements and experience to estimate values of \kappa. A typical average value is about 1.15 Values of \kappa can also be computed directly using more advanced blade element methods where the effects of the actual flight condition can be more accurately represented. This issue is particularly important for high-speed forward flight, where the increasing nonuniformity of the inflow from the reverse flow on the retreating blade must be accounted for.

Profile Drag Effects

Estimates for the profile power consumed by a rotor require a knowledge of the drag coefficients of the airfoils that make up the rotor blades. Thus drag coefficient will be a function of both the Reynolds number and Mach number, which vary along the span of the blade. However, a simple baseline result for the profile power can be obtained from an element-by-element summation of the sectional drag forces. i.e., the blade element method, the idea being shown in the figure below.

The principle of a blade element analysis of the rotor.

The power required to spin the blade in the absence of thrust (i.e., the profile power, P_0) can be obtained by radially integrating the sectional drag force along the length of the blade using

(27)   \begin{equation*} P_0 = \Omega N_b \int_0^R D y dy \end{equation*}

where N_b is the number of blades and D is the drag force per unit span at a section on the blade at a distance y from the rotational axis. The drag force on each blade element can be expressed conventionally as

(28)   \begin{equation*} D = \frac{1}{2} \varrho V^2 c C_{d}  = \frac{1}{2} \varrho (\Omega y)^2 c C_{d} \end{equation*}

where c is the blade chord, which is assumed constant in this case, i.e., the blade has a rectangular planform.

If the section profile drag coefficient, C_{d}, is also assumed to be constant, i.e., C_d = C_{d_{0}}, then the profile power integrates to

(29)   \begin{equation*} P_0 = \frac{1}{8} \varrho N_b \Omega^3 c C_{d_{0}} R^4 \end{equation*}

Converting this result to a power coefficient by dividing through by \varrho A (\Omega R)^3 gives

(30)   \begin{equation*} C_{P_{0}} = \frac{1}{8} \left( \frac{N_b c R}{A} \right) C_{d_{0}} = \frac{1}{8} \left( \frac{N_b c R}{\pi R^2} \right) C_{d_{0}} = \frac{1}{8} \left( \frac{N_b c}{\pi R} \right) C_{d_{0}} = \frac{1}{8} \sigma C_{d_{0}} \end{equation*}

The grouping

(31)   \begin{equation*} \frac{N_b c R}{A}  = \frac{N_b c R}{\pi R^2 }  = \frac{N_b c}{\pi R }   = \sigma \end{equation*}

is known as the rotor solidity. Typical values of \sigma for a helicopter rotor range between 0.05 and 0.12.

Modified Theory Versus Measurements

It is now possible to recalculate the rotor power requirements by using the modified  momentum theory such that

(32)   \begin{equation*} C_P = \frac{\kappa C_T^{3/2}}{\sqrt{2}} + \frac{\sigma C_{d_{0}}}{8} \end{equation*}

which is shown in the figure below, and which has been calculated by assuming \sigma = 0.1, \kappa = 1.15, and C_{d_{0}}=0.01.

Power polar for a hovering rotor using the modified momentum theory, which substantially improves predictions of the power required.

In the first case, to show the effect of the addition of profile power losses, it has been assumed that \kappa=1.0 (ideal induced losses), and in the second case, \kappa=1.15 (non-ideal losses). Notice the need to account for non-ideal induced losses and profile losses to give agreement with the measured data. The overall level of correlation thus obtained gives considerable confidence in the modified momentum theory approach for basic rotor performance studies, at least in hover.

Figure of Merit

There are several difficulties in defining an efficiency factor for a helicopter rotor because many parameters are involved, such as disk area, solidity, blade aspect ratio, airfoil characteristics, and tip speed. The power loading, PL, as discussed previously is an absolute measure of rotor efficiency because a helicopter of a given weight should be designed to hover with the minimum power requirements, i.e., the ratio T/P should be made as large as possible.

However, the power loading is a dimensional quantity and so a relative non-dimensional measure of hovering thrust efficiency is used, the figure of merit. This quantity is calculated using the simple momentum theory as a reference and is defined as the ratio of the ideal power required to hover to the actual power required, i.e.,

(33)   \begin{equation*} FM = \frac{\mbox{Ideal power required to hover}}{\mbox{Actual power required to hover}} < 1 \end{equation*}

The ideal power is given by the simple momentum result in Eq. 24. Therefore, for a real rotor the figure of merit will always be less than unity.

Using the modified form of the momentum theory with the non-ideal approximation for the power, the figure of merit can be written as

(34)   \begin{eqnarray*} FM =\frac{\mbox{Ideal power}}{\mbox{Induced power $+$ profile power}} = \frac{P_{\rm ideal}}{\kappa P_{\rm ideal} + P_0} = \frac{ \displaystyle{ \frac{C_T^{3/2}}{\sqrt{2}}}}{\displaystyle{ \frac{\kappa C_T^{3/2}}{\sqrt{2}}} + \displaystyle{ \frac{\sigma C_{d_{0}}}{8}} } \end{eqnarray*}

A representative plot of measured figure of merit versus rotor thrust is shown below. It will be apparent that the FM reaches a maximum and then remains constant or drops off slightly. This latter behavior is because of the higher profile drag coefficients (> C_{d_{0}}) obtained at higher rotor thrusts. For some rotors, especially those with less efficient airfoils, the curve can exhibit a peak in FM, followed by either a progressive or abrupt decrease thereafter. Therefore, the FM behavior in the high thrust range will, to some extent, be a function of the airfoils used on the blades and their stall type. In practice, maximum FM values between 0.65 and 0.75 represent a good hovering performance for a helicopter rotor.

Figure of merit predictions made with modified momentum theory to measured results for a hovering rotor.

The figure of merit for the best hovering efficiency can now be established, i.e., best power loading. The ratio of power required to hover to the thrust produced is

(35)   \begin{equation*} \frac{P}{T} = \frac{P}{W}= \frac{(\Omega R) C_P}{C_T} \end{equation*}

which can be written in terms of the modified momentum theory with the parameters \kappa and C_{d_{0}} as

(36)   \begin{equation*} \frac{P}{T} = \Omega R \frac{C_P}{C_T} = \Omega R  \left( \kappa \sqrt{\frac{C_T}{2}} + \frac{C_{P_{0}}}{C_T} \right) = \frac{\Omega R}{C_T} \left( \kappa \frac{C_T^{3/2}}{\sqrt{2}} + \frac{\sigma C_{d_{0}}}{8} \right) \end{equation*}

The operating C_T to give the best power loading can be obtained by differentiating Eq. 36 with respect to C_T, i.e.,

(37)   \begin{eqnarray*} \frac{d(P/T)}{d C_T} & = & -\Omega R C_T^{-2} \left( \frac{ \kappa C_T^{3/2}}{\sqrt{2}} + \frac{\sigma C_{d_{0}}}{8} \right) + \Omega R C_T^{-1} \left( \frac{3 \kappa C_T^{1/2}}{2 \sqrt{2}} \right) \nonumber \\ & = & -\frac{\Omega R}{C_T^{2} }\left( \frac{ \kappa C_T^{3/2}}{\sqrt{2}} + \frac{\sigma C_{d_{0}}}{8} \right) + \Omega R C_T^{-1/2} \left( \frac{3 \kappa}{2 \sqrt{2}} \right) \end{eqnarray*}

which must be zero for a minimum. Therefore,

(38)   \begin{equation*} \frac{3 \kappa}{2 \sqrt{2}} = \left( \frac{\kappa}{\sqrt{2}} + \frac{\sigma C_{d_{0}} C_T^{-3/2}}{8} \right) \end{equation*}

which on rearrangement gives

(39)   \begin{equation*} C_T = \left( \frac{\sqrt{2}}{4}\right)^{2/3}  \left( \frac{ \sigma C_{d_{0}}}{\kappa} \right)^{2/3} = \frac{1}{2} \left( \frac{ \sigma C_{d_{0}}}{\kappa} \right)^{2/3} \end{equation*}

Substituting the result that C_T = \frac{1}{2} ( \sigma C_{d_{0}}/\kappa )^{2/3} into the  figure of merit expression gives

(40)   \begin{equation*} FM = \frac{ \displaystyle{ \frac{1}{2 \sqrt{2}} } \left( \displaystyle{ \frac{\sigma C_{d_{0}}}{\kappa}} \right) }{ \displaystyle{ \frac{\kappa}{2 \sqrt{2}} } \left( \displaystyle{ \frac{\sigma C_{d_{0}}}{\kappa}} \right) + \displaystyle{ \frac{ \sqrt{2}}{8} \sigma C_{d_{0}} }} = \frac{\kappa^{-1}}{1 + \frac{1}{2}} = \frac{2}{3 \kappa} \end{equation*}

For an ideal rotor the best power loading is obtained at a figure of merit of 2/3. Using the modified momentum theory, this condition occurs at a figure of merit of 2/3\kappa.

For design purposes, solving for the rotor radius would determine its optimum value for a given gross weight of the helicopter, the rotor tip speed, and the operational density altitude. However, in most cases, the resulting radius is too high to be practical, and the rotor must be operated at a higher disk loading than the optimum. As shown in the figure below, at the most efficient operation of the rotor, its efficiency is relatively insensitive to the thrust in that the C_T/C_P curve is reasonably flat above a particular thrust coefficient. Therefore, there is some latitude in selecting the rotor radius, which may be constrained because of factors other than aerodynamics.

Finally, a word of caution about the figure of merit is appropriate. To be meaningful, the figure of merit must only be used as a gauge of rotor efficiency when two or more rotors are compared at the same disk loading, which can be seen if the figure of merit is written dimensionally as

(41)   \begin{equation*} FM = \frac{P_{\rm ideal}}{\kappa P_{\rm ideal} + P_0} = \frac{1}{\kappa + \displaystyle{ \frac{P_0}{P_{\rm ideal}} } } = \frac{1}{\kappa + \displaystyle{\frac{\sqrt{2\varrho}}{T} \frac{P_0}{\sqrt{DL}}}} \end{equation*}

Therefore, it would be considered inappropriate to compare the values of the figure of merit of two rotors with substantially different disk loadings because the rotor with the higher disk loading will generally always give the higher figure of merit, all other factors being equal.

Solidity & Blade Loading Coefficient

It will be seen from Eq. 34 that the solidity, \sigma, appears in the expression for figure of merit, FM. For a rotor with rectangular blades, the solidity represents the ratio of the lifting area of the blades to the area of the rotor, i.e.,

(42)   \begin{equation*} \sigma = \frac{\mbox{blade~area}}{\mbox{disk~area}} = \frac{A_b}{A} = \frac{N_b c R}{\pi R^2} = \frac{N_b c}{\pi R} \end{equation*}

As previously noted, for helicopter typical values of \sigma range from about 0.05 to 0.15.

If FM is plotted for rotors with different values of \sigma, the behavior is typified by the figure below. While the number of blades also affects rotor performance there are no known measurements of solidity effects independently of blade number. Results predicted by means of the modified momentum theory are also shown. From the measurements at zero thrust it was deduced that C_{d_{0}} = 0.011, and that \kappa was about 1.25.

Measured and predicted figure of merit versus thrust coefficient for a hovering rotor with different values of solidity.

It will be noted that higher values of FM are obtained with the lowest possible solidity at the same design C_T, i.e., same aircraft gross weight or disk loading. This result is hardly an unexpected from Eq. 34, all other terms such as \kappa being assumed constant, and means that the viscous drag on the rotor is being minimized by reducing the net blade area. However, the minimization of \sigma must be done with caution because reducing blade area must always result in a higher angle of attack of the blade sections (and higher lift coefficients) to obtain the same values of C_T. Therefore, the lowest allowable value of \sigma must ultimately be limited by the onset of blade stall.

This latter effect is shown by the results for the lowest solidity of 0.042, where a progressive departure occurs from the theoretical predictions for C_T > 0.004. For a full-scale rotor this would occur at higher values of C_T because of the higher values of maximum lift found at the higher Reynolds numbers on the blades. Therefore, an alternative presentation is to plot the figure of merit versus blade-loading coefficient, C_T/\sigma, as shown in the figure below. In this case, C_T/\sigma can be written as

(43)   \begin{equation*} \frac{C_T}{\sigma} = \frac{T}{\varrho A (\Omega R^2)} \left( \frac{A}{A_{b}} \right) = \frac{T}{\varrho A_{b} (\Omega R)^2 } \end{equation*}

where A_{b} is the area of the blades.

Measured and predicted figure of merit versus blade-loading coefficient for a hovering rotor with different solidities.

Notice that reducing the value of \sigma results in higher values of C_T/\sigma for the same value of C_T. Although the rotor operates at higher values of FM with an increased blade loading coefficient, the maximum value is limited by the onset of blade stall. Typically, for a contemporary helicopter rotor, the maximum realizable value of the blade loading coefficient without stall is about 0.12–0.14. However, the influence of Reynolds number on blade stall must also be considered, especially with subscale rotors.

The maximum attainable value of C_T/\sigma will also depend on the distribution of local lift coefficients along the blade, which in turn depends on both the blade twist and its planform shape. The local lift coefficients can be related to the blade loading using the blade element theory, and so the blade twist and blade planform can be designed to delay the effects of stall to higher values of C_T/\sigma. A rotor that uses airfoils with higher values of the maximum-lift coefficient can also be designed to have lower solidity. This approach has the benefits of a lower blade and hub weight, both of which are significant contributors to total helicopter weight.

Ground Effect

Just like airplanes, helicopter performance is affected by the presence of the ground or any other boundary that may alter or constrain the flow into the rotor or constrain the development of the wake, as shown in the figure below. Because the ground must be a streamline to the flow, the rotor slipstream tends to rapidly expand as it approaches the ground. This behavior alters the slipstream velocity, the induced velocity in the plane of the rotor, and, therefore, the rotor thrust and power. Similar effects are obtained both in hover and forward flight, but the effects are most substantial in the hovering state.

The effects of the ground will distort the rotor wake and change the rotor performance. The effects are strongest in hover and diminish as the helicopter flies forward.

A representative set of power polars for a rotor hovering near the ground is shown in the figure below. Notice that the results suggest significant effects on hovering performance for heights less than one rotor diameter. When the hovering rotor is operating in ground effect, then the rotor thrust is increased for a given power. Alternatively, this effect can be viewed as a reduction in power for a given thrust (weight). Remember that a straight line drawn from point (0,0) to any point on any polar curve gives the ratio of C_T/C_P so is a measure of the epower loading or efficiency. Notice that for the lowest rotor heights off the ground the efficiency is the highest.

Power polars for rotors hovering in ground effect. The effects on rotor performance diminish quickly for rotor heights greater than one diameter above the ground.

Worked Example #2 – Estimating Power Requirements for Flight

In 1907, Paul Cornu built a primitive twin-rotor helicopter. Each rotor of his machine was approximately 19.7 ft in diameter and it had a net gross weight (with pilot) of about 575 lb. Use momentum theory to verify the power requirements for flight, free of the ground and out of ground effect.

Assuming each rotor lifted half of the total aircraft weight, then the momentum theory gives a result for net minimum possible power (or ideal power) required to drive both rotors using

    \[ P_{\rm ideal} = 2 \left( \frac{(W/2)^{3/2}}{\sqrt{2\varrho A}} \right) \]

where the total take-off weight W = 575 lb and where each rotor had a swept disk area, A=304 ft^2. Assuming sea level air density, this gives the ideal shaft power (in horsepower) required to drive both rotors of Cornu’s machine as

    \[ P_{\rm ideal} = \frac{2}{550} \left( \frac{(575/2)^{3/2}}{\sqrt{2 \times 0.002378 \times 304}} \right) = 14.7~\mbox{hp} \]

Therefore, an installed power of at least 14.7 hp would be required for free flight, but only if the rotors were aerodynamically 100% efficient and there were no transmission losses. Realistically, with the primitive types of rotors used by Cornu, it could be expected the aerodynamic efficiency of the rotors are no better than 50% (a figure of merit of 0.5) giving a power required of about 30 hp.

Cornu also used an inefficient belt and pulley system to drive the rotors from an engine that produced only 24 hp. In his logbooks, Cornu constantly talks about the challenges of slipping belts. Therefore, taking into account the aerodynamic efficiency of the rotors and with a conservative estimate of belt slippage and transmission losses, for Cornu to hover his machine free of the ground the installed power required would need to have been about 40 hp. Therefore, the conclusion is that using an engine with a power output of only 24 hp it is highly unlikely that Paul Cornu’s machine ever flew in sustained flight free of the ground.


Worked Example #3 – Hovering Power Required

A tiltrotor has a gross weight of 45,000 lb (20,400 kg). The rotor diameter is 38 ft (11.58 m). On the basis of the momentum theory, estimate the power required for the aircraft to hover at sea level on a standard day out of ground effect where the density of air is 0.002378 slugs ft^{-3} or 1.225 kg m^{-3}. Assume that the figure of merit of the rotors is 0.75 and transmission losses amount to 5%.

A tiltrotor has two rotors, which are each assumed to carry half of the total aircraft weight, that is, T = 2,500 lb. For each of the rotors, the disk area is, A = \pi (38/2)^2 = 1134.12 ft^2. The induced velocity in the plane of the rotor is

    \[ v_h = \sqrt{ \frac{T}{2 \varrho A} } = \sqrt{ \frac{22500}{2 \times 0.002378 \times 1134.12} } = 64.56~\mbox{ ft s$^{-1}$} \]

The ideal power per rotor will be

    \[ T v_h = 22,500 \times 64.56 = 1,452,600}~\mbox{ft-lb s$^{-1}$} \]

This result is converted into horsepower (hp) by dividing by 550 to give 2,641 hp per rotor. Remember that the figure of merit accounts for the aerodynamic efficiency of the rotors. Therefore, the actual power required per rotor to overcome induced and profile losses will be 2,641/0.75 = 3,521.5 hp, followed by multiplying the result by two to account for both rotors, that is, 2 \times 3,521.5 =7,043 hp. Transmission losses account for another 5%, so that the total power required to hover is 1.05 \times 7,043 =7,395 hp.

The problem can also be worked in SI units. In this case, T = 10,200 \times 9.81 = 100,062 N. The disk area is, A = \pi (11.58/2)^2 = 105.32 m^2. The induced velocity in the plane of the rotor is

    \[ v_i = \sqrt{ \frac{T}{2 \varrho A} }  = \sqrt{ \frac{100062}{2 \times 1.225 \times 105.35} } = 19.69~\mbox{ m s$^{-1}$} \]

The ideal power per rotor will be

    \[ T v_h = 100,062 \times 19.69 = 1,970.2}~\mbox{kW} \]

The actual power required per rotor to overcome induced and profile losses will be 1,970.2/0.75 = 2,626.9 kW followed by multiplying the result by two to account for both rotors, that is, 5,253.8 kW. Transmission losses mean that the total power required to hover will be 5,515.7 kW.


Worked Example #4 – Induced Power Factor & Profile Power

A student makes measurements of rotor performance at a fixed rotor speed for a series of blade pitch angles. The rotor has a solidity of 0.1. The values of thrust coefficient, C_T, that were measured were 6.0000E-06, 0.001049, 0.002375, 0.004075 and 0.005582, and the corresponding values of power coefficient, C_P, were 0.000196, 0.000225, 0.000281, 0.000404 and 0.000554, respectively. The student wants to estimate the induced power factor, the zero thrust (profile) power, and the mean section drag coefficient for this rotor.

The simple momentum theory gives the ideal power as

    \[ C_{P_{\rm ideal}} = \frac{C_T^{3/2}}{\sqrt{2}} \]

and the modified semi-empirical theory is

    \[ C_{P_{\rm}} = \frac{\kappa \, C_T^{3/2}}{\sqrt{2}} + C_{P_{0}} \]

The student wants to find values of \kappa and C_{P_{0}} so we can write

    \[ C_{P_{\rm}} = \kappa \, C_{P_{\rm ideal}} + C_{P_{0}} \]

and so to find these values we can plot C_{P_{\rm}} versus C_{P_{\rm ideal}}, which should be close to a straight-line.
The best straight-line fit (least-squares) gives the slope \kappa and the intercept on the y-axis is C_{P_{0}}. In this case the value of \kappa is 1.206 and C_{P_{0}} is 0.000192. It is then possible to estimate the average drag coefficient of the airfoils that comprise the rotor using

    \[ C_{P_{0}} = \frac{\sigma C_{d_{0}}}{8} \]


    \[ C_{d_{0}} = \frac{8 C_{P_{0}}}{\sigma} \]

If \sigma = 0.1 as stated, then C_{d_{0}} = 0.0154, which seems fairly reasonable.

Axial Climbing & Descending Flight

Adequate climbing flight performance is an essential operational consideration for a helicopter, and sufficient power reserves must be available to ensure adequate climbing performance over a wide range of flight weights and operational density altitudes. It seems obvious that it takes more power to increase altitude than to lose altitude. Estimates of the power required to climb and descend can also be established from a momentum theory analysis.

Climbing Flight

The three conservation laws are applied to a control volume surrounding the climbing rotor and its flow field, as shown in the figure below. As before, consider the problem to be one-dimensional, and the flow properties will be assumed to vary only in the vertical direction over cross-sectional planes parallel to the disk, and at each cross-section, the flow properties are distributed uniformly.

Flow model for momentum theory analysis of a rotor in axial climbing flight.

In contrast to the hover case where the climb velocity is identically zero, the relative velocity far upstream relative to the rotor will now be V_c. At the plane of the rotor, the velocity will now be V_c+v_i, and the slipstream velocity is now V_c+w.

The mass flow rate \overbigdot{m} is constant within the slipstream boundary and so can be defined at the rotor, i.e..

(44)   \begin{equation*} \overbigdot{m} = \varrho A_\infty (V_c + w) = \varrho A (V_c + v_i) . \end{equation*}

The thrust on the rotor in this case will be

(45)   \begin{equation*} T = \overbigdot{m} (V_c + w) - \overbigdot{m} V_c = \overbigdot{m} w \end{equation*}

Notice that this is the same equation for the rotor thrust in the hover case, i.e., Eq. 4.

Because the work done by the climbing rotor is T(V_c + v_i), then

(46)   \begin{eqnarray*} T (V_c + v_i)  = \frac{1}{2} \overbigdot{m} (V_c + w)^2 - \frac{1}{2} \overbigdot{m} V_c^2 = \frac{1}{2} \overbigdot{m} w (2V_c + w) \end{eqnarray*}

From Eqs. 45 and 46 it is readily apparent that w = 2 v_i.

The relationship between the rotor thrust and the induced velocity at the rotor disk in hover is

(47)   \begin{equation*} v_h \equiv v_i = \sqrt{ \frac{T}{2 \varrho A} } \end{equation*}

and for the climbing rotor using Eq. 45 then

(48)   \begin{equation*} T = \overbigdot{m} w = \varrho A (V_c + v_i) w = 2 \varrho A (V_c + v_i) v_i \end{equation*}

so that

(49)   \begin{equation*} \frac{T}{2\varrho A} = v_h^2 = (V_c + v_i) v_i =  V_c v_i + v_i^2 \end{equation*}

which is a quadratic equation in v_i. Dividing through by v_h^2 to make it non-dimensional gives

(50)   \begin{equation*} \left( \frac{v_i}{v_h} \right)^2 + \frac{V_c}{v_h} \left( \frac{v_i}{v_h} \right) - 1 = 0 \end{equation*}

which is a quadratic equation in v_i/v_h.  This equation has the solution

(51)   \begin{equation*} \frac{v_i}{v_h} = -\left(\frac{V_c}{2v_h} \right) \pm \sqrt{ \left(m \frac{V_c}{2 v_h}\right)^2 +1 } \end{equation*}

Although there are two possible solutions (one positive and one negative), v_i/v_h must always be positive in the climb so as not to violate the assumed flow model. The valid solution is

(52)   \begin{equation*} \frac{v_i}{v_h} = -\left(\frac{V_c}{2v_h} \right) + \sqrt{ \left( \frac{V_c}{2 v_h}\right)^2 +1 } \end{equation*}

Descending Flight

The climb flow model cannot be used in a descent (where V_c \ll 0) because now V_c is directed upward and so the slipstream will be above the rotor. This will be the case whenever |V_c| is more than twice the average induced velocity at the disk. For cases where the descent velocity is in the range -2v_h \le V_c \le 0, the velocity at any plane through the rotor slipstream can be either upward or downward. Under these circumstances, a definitive control volume surrounding the rotor and its wake cannot be established.

The assumed flow model and control volume surrounding the descending rotor is shown in the figure below. To proceed, the assumption must be made that |V_c|> 2v_h so that a well-defined slipstream will always exist above the rotor and encompassing the limits of the rotor disk. Far upstream (well below) the rotor, the magnitude of the velocity is the descent velocity, which is equal to |V_c|. Notice that to avoid any ambiguity, it will be assumed that the velocity is measured as positive when directed in a downward direction.  At the plane of the rotor, the velocity is |V_c|-v_i. In the far wake (above the rotor), the velocity is |V_c|-w.

Flow model used for momentum theory analysis of a rotor in a vertical (axial) descent.

The mass flow rate, \overbigdot{m}, through the rotor disk is

(53)   \begin{equation*} \overbigdot{m} = \varrho A_\infty (V_c + w) = \varrho A (V_c + v_i) \end{equation*}

The thrust in this case can be expressed as

(54)   \begin{equation*} T = ( -\overbigdot{m}) (V_c + w) - (-\overbigdot{m}) V_c = -\overbigdot{m} w \end{equation*}

Notice that T is not negative because \overbigdot{m} is negative by means of the assumed sign convention.

The work done by the rotor is

(55)   \begin{equation*} T (v_i+V_c)  = \frac{1}{2} \overbigdot{m} V_c^2 - \frac{1}{2} \overbigdot{m} (V_c + w)^2 = -\frac{1}{2} \overbigdot{m} w (2V_c + w) \end{equation*}

which is a negative quantity. Therefore, the rotor must be extracting power from the airstream and this operating condition is known as the windmill state. More usually it is referred to as the windmill brake state because the rotor in this condition decreases or “brakes” the velocity of the flow.

Using Eqs. 54 and 55 it is seen, again, that w = 2 v_i. Note, however, that the net velocity in the slipstream is {\it less} than |V_c|, and so from continuity considerations the wake boundary expands above the descending rotor disk. For the descending rotor then

(56)   \begin{equation*} T = -\overbigdot{m} w = -\varrho A (V_c + v_i) w = -2 \varrho A (V_c + v_i) v_i \end{equation*}

so that

(57)   \begin{equation*} \frac{T}{2\varrho A} = v_h^2 = -(V_c + v_i) v_i = - V_c v_i -v_i^2 \end{equation*}

Dividing through by v_h^2 gives

(58)   \begin{equation*} \left( \frac{v_i}{v_h} \right)^2 + \frac{V_c}{v_h} \left( \frac{v_i}{v_h} \right) + 1 = 0 \end{equation*}

which is a quadratic equation in v_i/v_h. This equation has the solution

(59)   \begin{equation*} \frac{v_i}{v_h} = -\left(\frac{V_c}{2v_h} \right) \pm \sqrt{ \left( \frac{V_c}{2 v_h}\right)^2 -1 } \end{equation*}

Again, like the climb case, there are two possible solutions for v_i/v_h in descent. The only valid solution is

(60)   \begin{equation*} \frac{v_i}{v_h} = -\left(\frac{V_c}{2v_h} \right) - \sqrt{ \left( \frac{V_c}{2 v_h}\right)^2 -1 } \end{equation*}

which is applicable for V_c/v_h\le -2.

Induced Velocity Curves

Results from the foregoing analysis are shown below. It is apparent that as the climb velocity increases the induced velocity at the rotor decreases. This is called the normal working state of the rotor, with hover being the lower limit. The branch of the induced velocity curve denoted by the broken line gives a solution to Eq. 52 for negative values of V_c, i.e., a descent. However, as the rotor begins to descend there can be two possible flow directions, which violates the assumed flow model and so this solution is physically invalid. This condition is called the Vortex Ring State (VRS) and can only be described on the basis of measurements.

Induced velocity variation as a function of climb and descent velocity based on momentum theory (complete induced velocity curve).

Power Required Curves

Because both climb and descent change the induced velocity at the rotor, the induced power will also be affected. The power ratio can be written as

(61)   \begin{equation*} \frac{P}{P_h} = \frac{V_c + v_i}{v_h} = \frac{V_c}{v_h} + \frac{v_i}{v_h} \end{equation*}

where the first term is the useful work to change the potential energy of the rotor (helicopter) and the second term is the work done on the air by the rotor, i.e., the irrecoverable induced losses.

Using  Eq. 52 and substituting and rearranging gives the power ratio for a climb as

(62)   \begin{equation*} \frac{P}{P_h} = \frac{V_c}{2v_h} + \sqrt{ \left( \frac{V_c}{2 v_h}\right)^2 +1 } \mbox{~~which is applicable for~$\displaystyle{\frac{V_c}{v_h}} \ge 0$} \end{equation*}

In a descent, Eq. 60 is applicable, and substituting this into Eq. 61 and rearranging gives the power ratio as

(63)   \begin{equation*} \frac{P}{P_h} = \frac{V_c}{2v_h} - \sqrt{ \left( \frac{V_c}{2 v_h}\right)^2 - 1 } \mbox{~~which is applicable for~$\displaystyle{\frac{V_c}{v_h}} \le -2$} \end{equation*}

The figure below shows the total rotor power ratio, P/P_h, plotted versus the climb ratio, V_c/v_h. Notice that the power required to climb is always greater than the power required to hover. However, as the climb velocity increases the induced power becomes a progressively smaller percentage of the total power required to climb. It is also significant to notice that in a descent, at least above a certain rate,  the rotor extracts power from the air and uses less power than required to hover, i.e., the rotor now operates like a windmill. However, a helicopter rotor will never operate in this condition.

Total power required as a function of climb and descent velocity (universal power curve).

Vortex Ring State

In the region -2\le V_c/v_h \le 0, called the vortex ring state (VRS), the momentum theory is invalid because the flow can take on two possible directions and a well-defined slipstream ceases to exist, as shown in the figure below. In the VRS the rotor can experience highly unsteady flow with regions of concurrent upward and downward velocities, and the flow can periodically break away from the rotor disk. This means that a control volume cannot be defined that encompasses only the physical limits of the rotor disk.  From a piloting perspective, VRS is not a sustainable flight condition. Because of the flow unsteadiness at the rotor in VRS, it can lead to a loss of rotor control. If the VRS occurs on the tail rotor, such as during sideways flight or hovering in a crosswind, then directional (yaw) control may be seriously impaired.

The four operating states of a rotor in axial flight. The most complicated is the vortex ring state of VRS, which cannot be analyzed by means of the momentum theory.

The induced velocity curve in the VRS can still be defined empirically, albeit only approximately, on the basis of experiments with rotors.  Even then, measurements of rotor thrust and power are difficult to make.The average induced velocity is then obtained indirectly from the measured rotor power and thrust using the assumed form

(64)   \begin{equation*} P_{\rm meas} = T(V_c + \overline{v}_i) + P_0 \end{equation*}

where P_0 is the profile power and where \overline{v}_i is recognized as only an averaged value of the induced velocity through the disk. Using the result that P_h=T \overline{v}_h then

(65)   \begin{equation*} \frac{V_c + \overline{v}_i}{\overline{v}_h} = \frac{P_{\rm meas} - P_0}{P_h} = \frac{P_{\rm meas} - P_0}{T \sqrt{T/2\varrho A}} = \frac{(P_{\rm meas} - P_0)\sqrt{2\varrho A}}{W^{3/2}} \end{equation*}

Therefore, in addition to the measured rotor power P_{\rm meas}, to obtain an estimate for the averaged induced velocity ratio it is necessary to know the rotor profile power. As shown previously by means of Eq. 30, one simple estimate for the profile power coefficient of a rotor with rectangular blades is C_{P_{0}}= \sigma C_{d_{0}}/8. Because of the high levels of turbulence near the rotor in the VRS, the derived measurements of the average induced velocity contain a relatively large amount of scatter.

These measurements can then be used to find a “best-fit”  approximation for v_i at any rate of descent.  One approximation is

(66)   \begin{equation*} \frac{v_i}{v_h} = \left\{\begin{array}{ll} 1 - \displaystyle{\frac{V_c}{v_h}} & \mbox{~~for~} -1.5 \le \displaystyle{\frac{V_c}{v_h}} \le 0 \\[24pt] 7 + 3 \displaystyle{\frac{V_c}{v_h}} & \mbox{~~for~} -2 \le\displaystyle{\frac{V_c}{v_h}} \le -1.5\end{array} \right. \end{equation*}


The principle of autorotation can be seen in nature in the flight of sycamore or maple seeds, which spin rapidly as they slowly descend and are often carried on the wind for a considerable distance. In practice an autorotation is a maneuver that can be used to recover the helicopter to the ground in the event of an engine failure, transmission problems, or loss of the tail rotor. It requires that the pilot let the helicopter descend at a sufficiently high but controlled rate, where the energy to drive the rotor can be obtained by giving up potential energy (altitude) for energy taken from the relative upward flow through the rotor, thereby averting a ballistic fall.

Notice that from the power curve shown previously that there is actually a value of V_c/v_h for which zero net power is required for the rotor [i.e., P = T(V_c + v_i) = 0 or P/P_h  =0]. This condition is called ideal autorotation for vertical flight. For a given thrust it is a self-sustained operating state where the energy to drive the rotor comes from potential energy converted to kinetic energy from the relative descent velocity (which is upward relative to the rotor). On the basis of assuming the validity of Eq. 66, it will be apparent that the power curve crosses the ideal autorotation line V_c + v_i = 0 at

(67)   \begin{equation*} \frac{V_c}{v_h} = -\left( \frac{7 \kappa}{1 + 3 \kappa} \right) \end{equation*}

which gives V_c/v_h=-1.75 for an ideal rotor (\kappa=1). In practice, a real (actual) autorotation in axial flight occurs at a slightly higher rate than this, because in addition to induced losses at the rotor, there are also a proportion of profile losses to overcome. In a real autorotation then

(68)   \begin{equation*} P = \kappa T \left( V_c + v_i \right) + P_0 = 0 \end{equation*}

Therefore, in a stable autorotation an energy balance must exist where the decrease in potential energy of the rotor T V_c just balances the sum of the induced (\kappa Tv_i) {\it and} profile (P_0) losses of the rotor. Using Eq. 68, this condition is achieved in vertical descent when

(69)   \begin{equation*} \frac{V_c + v_i}{v_h} = -\frac{P_0}{\kappa T v_h} = - \frac{P_0 \sqrt{2 \varrho A}}{\kappa T^{3/2}} = - \frac{P_0 \sqrt{2 \varrho A}}{\kappa W^{3/2}} \end{equation*}

which depends primarily on the disk loading. Also, using the definition of figure of merit (and assuming the induced and profile losses do not vary substantially from the hover values), then

(70)   \begin{equation*} \frac{P_0 \sqrt{2 \varrho A}}{T^{3/2}} = \left( \frac{1}{FM} - \kappa \right) \end{equation*}

Using Eq. 66 for the induced velocity with Eq. 70 gives the real autorotation condition

(71)   \begin{equation*} \frac{V_c}{v_h} = -\frac{FM^{-1} - \kappa}{1 + 3 \kappa} - \frac{7 \kappa}{1 + 3 \kappa} \end{equation*}

The first term on the right-hand side of Eq. 71 will vary in magnitude from -0.04 to -0.09, depending on the rotor efficiency. Compared to the second term, the extra rate of descent required to overcome profile losses is relatively small. Therefore, on the basis of the foregoing, it is apparent that a real vertical autorotation will occur for values of V_c/v_h between -1.85 and -1.9.

It is found that with a helicopter, autorotations must be performed at relatively high rates of descent. Using the result that v_h \approx 14.49 \sqrt{T/A}, where T/A is in lb/ft^2, gives V_c \approx -26.81 \sqrt{T/A} ft s^{-1} for autorotation at sea level conditions, which for a representative disk loading of 10 lb ft^{-2} leads to a vertical rate of descent of about 5,000 ft min^{-1}. However, with some forward speed, the power required at the rotor is considerably lower than in the hover case.

Worked Example #5 – Climb Power Required

Given a helicopter of weight, W = 6,000 lb, calculate the power required in hover and 600 ft/min axial rate-of-climb. The radius of the main rotor is 20 ft and the rotor has a figure of merit of 0.75. Assume sea level conditions. Discuss the factors that will determine the maximum vertical climb rate of a helicopter.

The power required in an axial climb can be estimated from the momentum theory result that

    \[ \frac{P}{P_h} = \frac{V_c}{2v_h} + \sqrt{ \left( \frac{V_c}{2 v_h}\right)^2 +1 } \approx \frac{V_c}{2v_h} + 1 \mbox{~~for low to moderate climb rates.} \label{powerclimb3} \]

In hover, the induced velocity, v_h, is

    \[ v_h = \sqrt{ \frac{T}{2 \varrho A} } = \sqrt{ \frac{ 6,000}{2 \times 0.00238 \times \pi \times (20.0)^2 }} = 31.67~\mbox{ft/s} \]

where at sea level conditions \varrho = 0.00238 slugs/ft^3. The power required to hover will be

    \[ P_h = (1/0.75) \frac{T^{3/2}}{\sqrt{ 2 \varrho A}} = \frac{ 1.333 \times (6,000)^{3/2}}{\sqrt{2 \times 0.00238 \times \pi \times (20.0)^2 }} \times \frac{1}{550} = 460.66~\mbox{hp} \]

The power required to climb at 600 ft/min (10 ft/s) will be

    \[ P \approx P_h \left( \frac{V_c}{2 v_h} + 1 \right) = 460.66 \left( \frac{10.0}{2 \times 31.67} + 1 \right) = 533.49~\mbox{hp} \]

The maximum rate of climb of the helicopter will be determined by the excess power available over and above that required to hover at the same weight and density altitude. The vertical drag produced on the airframe may also be a factor in the climb condition, albeit at high rates of climb.

Forward Flight Analysis

Under forward flight conditions the rotor moves through the air with an edgewise component of velocity that is parallel to the plane of the rotor disk. Under these conditions the axisymmetry of the flow through the rotor is lost. Now the flow field in which the rotor operates is considerably more complex than that of a fixed-wing and gives rise to a number of aerodynamic problems that ultimately limit the rotor performance.

Flow Environment

The overall aerodynamic complexity of the helicopter in forward flight can be appreciated from the schematic below. One issue is that blade tips on the advancing side of the rotor disk can start to penetrate into supercritical and transonic flow regimes, with the associated formation of compressibility zones and, ultimately, strong shock waves. In addition to the occurrence of wave drag and the possibilities of shock induced flow separation, both phenomena require much more power to drive the rotor. The periodic formation of shock waves is also a source of obtrusive noise.

The increased power demands placed on the rotor system, when compressibility effects manifest, will eventually limit forward flight speeds. Although compressibility effects on contemporary rotors can be relieved to some extent by the use of swept tip blades and thin “transonic” airfoils, the problems of increased power requirements and noise are only delayed to moderately higher forward flight speeds and are not eliminated.

Momentum Theory Analysis in Forward Flight

Despite the inherently more complicated nature of the rotor flow in forward flight, the simple momentum theory can be extended to forward flight on the basis of certain assumptions. Because helicopter rotors are required to produce both a lifting force (to overcome the weight of the helicopter) and a propulsive force (to propel the helicopter forward), the rotor disk must be tilted forward at an angle of attack relative to the oncoming flow. The following treatment of rotor performance in forward flight was first derived by Glauert, where the analysis is performed with respect to an axis aligned with the rotor disk, as shown in the figure below.

Glauert’s flow model for the momentum analysis of a rotor in forward flight.

Mass Flow Rate

In this case, Glauert defines the mass flow rate, \overbigdot{m}, through the actuator disk as

(72)   \begin{equation*} \overbigdot{m} = \varrho A U \end{equation*}

where U is the resultant velocity at the disk as given by

(73)   \begin{equation*} U = \sqrt{(V_{\infty}\cos\alpha)^2 + (V_{\infty}\sin\alpha + v_i)^2} = \sqrt{ V_{\infty}^2 + 2V_{\infty}v_i \sin\alpha + v_i^2 } \end{equation*}


The application of the conservation of momentum in a direction normal to the disk gives

(74)   \begin{equation*} T = \overbigdot{m} ( w + V_{\infty}\sin\alpha) - \overbigdot{m} V_{\infty}\sin\alpha = \overbigdot{m} w \end{equation*}

Power Required

By the application of conservation of energy, then

(75)   \begin{equation*} P  =  ( v_i + V_{\infty}\sin\alpha ) = \frac{1}{2} \overbigdot{m} (V_{\infty}\sin\alpha + w)^2 - \frac{1}{2} \overbigdot{m} V_{\infty}^2 \sin^2\alpha \end{equation*}

which gives

(76)   \begin{equation*} P  =  \frac{1}{2} \overbigdot{m} \left( 2V_{\infty}w\sin\alpha + w^2 \right) \end{equation*}

Using Eqs. 74 and 76 then

(77)   \begin{equation*} 2wv_i + 2V_{\infty}w\sin\alpha = 2V_{\infty}w\sin\alpha + w^2 \end{equation*}

giving w = 2v_i, which is the same relationship as for the axial flight cases. Therefore,

(78)   \begin{equation*} T = 2 \overbigdot{m} v_i = 2 \varrho A U v_i = 2 \varrho A v_i \sqrt{V_{\infty}^2 + 2V_{\infty}v_i \sin\alpha + v_i^2} \end{equation*}

Notice that for hovering flight, V_{\infty}=0, so that Eq. 78 reduces to the result for hover (Eq. 7), i.e.,

(79)   \begin{equation*} T = 2\varrho A v_i^2 = 2\varrho A v_h^2 \end{equation*}

As forward flight speed increases such that V_{\infty} \gg v_i, then Eq. 78 reduces to

(80)   \begin{equation*} T = 2 \varrho A v_i V_{\infty} \end{equation*}

which is called Glauert’s “high-speed” approximation.

The rotor thrust is given by

(81)   \begin{equation*} T = 2 \overbigdot{m} v_i = 2(\varrho A U) v_i \end{equation*}

which can be written as

(82)   \begin{equation*} T = 2 \varrho A v_i \sqrt{(V_{\infty}\cos\alpha)^2 + (V_{\infty}\sin\alpha + v_i)^2} \end{equation*}

Recall from Eq. 79 that for hovering flight v_h^2 = T/2\varrho A, so the induced velocity v_i in forward flight becomes

(83)   \begin{equation*} v_i = \frac{v_h^2}{\sqrt{(V_{\infty}\cos\alpha)^2 + (V_{\infty}\sin\alpha + v_i)^2}} \end{equation*}

The idea of a tip speed ratio or advance ratio, \mu, can now be introduced. By using the velocity parallel to the plane of the rotor, then it is defined as

(84)   \begin{equation*} \mu = V_{\infty} \cos\alpha/\Omega \end{equation*}

In most cases, the value of \alpha is sufficiently small so that

(85)   \begin{equation*} \mu = V_{\infty} /\Omega \end{equation*}


The inflow ratio is \lambda=(V_{\infty} \sin\alpha + v_i)/\Omega R so

(86)   \begin{equation*} \lambda = \frac{V_{\infty} \sin\alpha}{\Omega R} + \frac{v_i}{\Omega R} = \mu \tan\alpha + \lambda_i \end{equation*}

Also, Eq. 83 becomes

(87)   \begin{equation*} \lambda_i = \frac{\lambda_h^2}{\sqrt{\mu^2 + \lambda^2}} \end{equation*}

But, it is also known from the hover case that \lambda_h = \sqrt{C_T/2}, therefore,

(88)   \begin{equation*} \lambda_i = \frac{C_T}{2\sqrt{\mu^2+\lambda^2}} \end{equation*}

Finally, the solution for the inflow ratio, \lambda, is

(89)   \begin{equation*} \lambda = \mu\tan\alpha + \frac{C_T}{2 \sqrt{\mu^2+\lambda^2}} \end{equation*}

which is a form of transcendental equation in that the unknown value, \lambda, appears on both sides of the equation.

While analytic solutions to Eq. 89 can be found under certain assumptions is preferable to use a numerical method to solve for \lambda. The simplest approach is a fixed-point iteration. The algorithm consists of a loop to iteratively compute new estimates of \lambda until a termination criterion has been met. To this end, Eq. 89 can be written as the iteration equation

(90)   \begin{equation*} \lambda_{n+1} = \mu \tan \alpha + \frac{C_T}{2 \sqrt{\mu^2+\lambda_n^2}} \end{equation*}

where n is the iteration number. The starting value for \lambda_0 is usually the hover value (i.e., \lambda_0 = \lambda_h=\sqrt{C_T/2}). The error estimator is

(91)   \begin{equation*} \epsilon = \bigg\| \frac{\lambda_{n+1} - \lambda_n}{\lambda_{n+1}} \bigg\| \end{equation*}

and the convergence criteria is when \epsilon < 0.0005. One normally finds that between 10 and 15 iterations are required for convergence.

Results for the inflow ratio \lambda/\lambda_h computed are shown in the figure below for several different values of \alpha (both positive and negative) and over a range of values of \mu/\lambda_h typical of a helicopter, i.e.,

(92)   \begin{equation*} \lambda = \lambda_i + \mu \tan \alpha \end{equation*}


(93)   \begin{equation*} \frac{\lambda}{\lambda_h} = \frac{\lambda_i}{\lambda_h} + \frac{\mu}{\lambda_h} \tan \alpha \end{equation*}

The induced part of the total inflow \lamda_i decreases with increasing advance ratio but the total inflow increases at higher becomes dominated by the \mu \tan \alpha term at higher advance ratios.

Power Requirements in Forward Flight

The rotor power in forward flight is given by

(94)   \begin{equation*} P  =  T (V_{\infty}\sin\alpha + v_i) = T V_{\infty}\sin\alpha + T v_i \end{equation*}

The first term on the right-hand side of the above equation is the power required to propel the rotor forward and also to climb. The second term is the induced power. As for the axial flight case, the rotor power in forward flight can be referenced to the hover result and so

(95)   \begin{equation*} \frac{P}{P_h} = \frac{P}{T v_h} = \frac{T (V_{\infty}\sin\alpha + v_i)}{T v_h} = \frac{ V_{\infty}\sin\alpha + v_i}{v_h} = \frac{\lambda}{\lambda_h} \end{equation*}

Recall that

(96)   \begin{equation*} \lambda = \mu\tan\alpha + \frac{C_T}{2 \sqrt{\mu^2+\lambda^2}} = \mu\tan\alpha + \frac{\lambda_h^2}{\sqrt{\mu^2+\lambda^2}} \end{equation*}


(97)   \begin{equation*} \frac{P}{P_h} = \frac{\lambda}{\lambda_h} = \frac{\mu}{\lambda_h} \tan\alpha + \frac{\lambda_h}{\sqrt{\mu^2+\lambda^2}} \end{equation*}

The first term on the right-hand side of the above equation is the extra power to meet propulsion and climb requirements, whereas the second term is the induced power.

The inflow and power required for flight depend on the disk angle of attack, which must always be tilted forward slightly for propulsion, which in turn requires a knowledge of the helicopter’s drag, D. Assuming straight-and-level flight, the disk angle of attack, \alpha, can be calculated from a simple force equilibrium, as shown in the figure below.

A simple force equilibrium “trim” analysis of a helicopter in forward flight.

For vertical equilibrium T\cos\alpha = W, and for horizontal equilibrium T\sin\alpha=D\cos \alpha \approx D. Therefore, the disk angle of attack can be found from

(98)   \begin{equation*} \tan \alpha = \frac{D}{W} = \frac{D}{L} \approx \frac{D}{T} \end{equation*}

Therefore, the power equation in straight-and-level flight can be written as

(99)   \begin{equation*} \frac{P}{P_h} = \frac{\mu}{\lambda_h} \left( \frac{D}{T} \right) + \frac{\lambda_h}{\sqrt{\mu^2+\lambda^2}} \end{equation*}

The determination of the drag D on the helicopter, however, requires a knowledge of both the drag on the rotor and the drag on the airframe, the latter of which is called parasitic drag. The rotor drag must be estimated using blade element theory.

Forward Flight Performance

For a helicopter in level forward flight, the total power required at the rotor, P, can be expressed by

(100)   \begin{equation*} P = P_i + P_0 + P_p \end{equation*}

where P_i is the induced power, P_0 is the profile power required to overcome viscous losses at the rotor, and P_p is the parasitic power required to overcome the drag of the helicopter. Each contributing part can now be analyzed and the effects summed together.

Induced Power

The induced power of the rotor, P_i, is given by

(101)   \begin{equation*} P_i = \kappa T v_i \end{equation*}

or in coefficient form

(102)   \begin{equation*} C_{P_{i}}= \kappa C_T \lambda_i \end{equation*}

where \kappa is the now familiar empirical correction to account for a multitude of  non-ideal effects. The value of \kappa cannot necessarily be assumed independent of advance ratio, but the use of a mean value between 1.15 and 1.25 is usually sufficiently accurate for preliminary predictions of power requirements.

Substituting the value of \lambda_i in forward flight gives

(103)   \begin{equation*} C_{P_{i}} = \frac{ \kappa C_T^2}{2 \sqrt{\lambda^2 + \mu^2 }} \approx \frac{\kappa C_T^2}{2 \mu} \mbox{ for larger $\mu$} \end{equation*}

Notice that is the forward velocity is sufficiently high, say \mu > 0.1, then the induced velocity can be approximated by Glauert’s “high-speed”asymptotic , i.e.,

(104)   \begin{equation*} C_{P_{i}} \approx \frac{\kappa C_T^2}{2 \mu} \end{equation*}

Profile Power

The profile power coefficient can be approximated as

(105)   \begin{equation*} C_{P_{0}} = \frac{\sigma C_{d_{0}}}{8} ( 1 + K \mu^2) \end{equation*}

which comes from blade element theory and where the numerical value of K varies depending on the various assumptions and/or approximations that are made in the blade element integration.

In practice, a value of K \!=\! 4.65 is often used for helicopter performance predictions for \mu < 0.5. At higher advance ratios, experimental evidence suggests that profile power grows more quickly than given by Eq. 105, as result of radial and reverse flow, as well as compressibility effects on the rotor.

Parasitic Power

The parasitic power, P_p, is a power loss as a result of drag on the airframe, rotor hub, and so on. Because helicopter airframes are much less aerodynamic than their fixed-wing counterparts, often with regions of large-scale flow separation, this source of drag can be very significant.

The parasitic power contribution can be written as

(106)   \begin{equation*} P_p = \left( \frac{1}{2} \varrho V_{\infty}^2 S_{\rm{ref}} \, C_{D_{f}} \right) V_{\infty} \end{equation*}

where S_{\rm{ref}} is some reference area and C_{D_{f}} is the drag coefficient based on this reference area. In nondimensional form, this becomes

(107)   \begin{equation*} C_{P_{p}} = \frac{1}{2} \left( \frac{S_{\rm{ref}}}{A} \right) \mu^3C_{D_{f}} = \frac{1}{2} \left( \frac{f}{A} \right) \mu^3 \end{equation*}

where A is the rotor disk area and f (=C_{D_{f}} S_{\rm ref}) is known as the equivalent wetted area or equivalent flat-plate area. This area parameter accounts for the drag of the hub, fuselage, landing gear, and so on, in aggregate.

The concept of equivalent wetted area comes from noting that while the drag coefficient can be written in the conventional way as

(108)   \begin{equation*} C_{D_{f}} = \frac{D_f}{\frac{1}{2} \varrho V_{\infty}^2 S_{\rm{ref}}} \end{equation*}

where S_{\rm{ref}} is a reference area, the definition of S_{\rm{ref}} may not be unique. Thus an equivalent wetted area is used, which is defined as

(109)   \begin{equation*} f = \frac{D_f}{\frac{1}{2} \varrho V_{\infty}^2 } \end{equation*}

Such an approach avoids any confusion that may arise through the definition of S_{\rm{ref}}.

It is found that values of f range from about 10 ft^2 (0.93 m^2) on smaller helicopters to as much as 50 ft^2 (4.65 m^2) on large utility helicopter designs.

Tail Rotor Power

The power required by the tail rotor typically varies between 3 and 5% of the main rotor power in normal flight, and up to 20% of the main rotor power at the extremes of the flight envelope. It is calculated in a similar way to the main rotor power, with the thrust required being set equal to the value necessary to balance the main rotor torque reaction on the fuselage. The use of vertical tail surfaces to produce a side force in forward flight can help to reduce the power fraction required for the tail rotor, albeit at the expense of some increase in parasitic and induced drag.

The interference between the main rotor and the tail rotor, and between the tail rotor and the vertical fin, is usually neglected in preliminary analysis. However, these effects may be accounted for by an increase in the induced power factor, \kappa, to take into account the generally higher nonuniform inflow at the tail rotor location. Although the tail rotor power consumption is relatively low, interference effects may increase the power required by up to 20%, depending on the tail rotor and fin configuration. Because of the relatively low amount of power consumed by the tail rotor, for first estimates of performance the power required can be expressed as a fraction of the total main rotor power, with a good estimate being 5%.

Total Power

In light of the forgoing, the total power coefficient for the helicopter in forward flight can be written in the form

(110)   \begin{equation*} C_P \equiv C_Q = \frac{\kappa C_T^2}{2 \sqrt{\lambda^2 + \mu^2}} + \frac{\sigma C_{d_{0}}}{8} ( 1 + K \mu^2 ) + \frac{1}{2} \left( \frac{f}{A} \right) \mu^3 + C_{P_{\hbox{\scriptsize\it TR}}} \end{equation*}

The tail rotor power must always be added to obtain a proper estimate of total helicopter power requirements. For larger values of \mu, then \lambda \ll \mu, so that Glauert’s formula allows Eq. 110 to be simplified to

(111)   \begin{equation*} C_P \equiv C_Q = \frac{\kappa C_T^2}{2 \mu} + \frac{\sigma C_{d_{0}}}{8} ( 1 + K \mu^2 ) + \frac{1}{2} \left( \frac{f}{A} \right) \mu^3 + C_{P_{\hbox{\scriptsize\it TR}}} \end{equation*}

Comparison with In-Flight Measurements

Representative results of net power required for an exemplar helicopter in straight-and-level flight is shown in the figure below. A gross weight of 16,000 lb (7,256 kg) and an operating altitude of 5,200 ft (1,585 m) has been assumed. The rotor disk angle of attack was calculated at each airspeed to satisfy the horizontal force equilibrium, which, although not a complete trim calculation, provides reasonably acceptable results. The predicted components of the total rotor power are also shown, including that of the tail rotor. The equivalent flat-plate area, f, of the helicopter is 23.0 ft^2 (2.137 m^2). For both the main and tail rotors, it is assumed that \kappa = 1.15 and C_{d_{0}} = 0.008. The distance between the main and tail rotor shafts, x_{\hbox{\scriptsize\it TR}}, is 32.5 ft (9.9 m).

Predicted power curve for a helicopter versus flight test measurements. The breakdown of the constituent parts of the power is also shown.

Notice that the induced and propulsive part of the power initially decreases with increasing airspeed but increases again as the disk is progressively tilted forward to meet greater propulsion requirements. It is insufficient to assume induced losses are only a result of lift generation, so induced losses decrease rapidly with airspeed to a point and then start to increase again from losses associated with propulsive forces.

The power required for forward flight increases very quickly at higher airspeeds because the parasitic losses are proportional to \mu^3. The rate of power growth is even higher when reverse flow and compressibility losses on the rotor are considered. However, the airframe drag makes a major contribution to the total power required in high-speed flight. In the practice of design, much can be done to expand the flight envelope by striving for a more streamlined airframe.

Reverse Flow Region

In forward flight, there is a region of the rotor disk on the retreating side called the “Reverse Flow Region.” In this region, the flow over the rotor blade section comes from tail to nose rather than from nose to tail, as shown in the figure below. Because the flow in this region is “reversed” with the sharp trailing edge pointed into the airflow, the lift is negative and the contribution of the profile drag on the blade section to the overall rotor torque and power required will be somewhat higher. The effects can usually be neglected at lower flight airpeeds.

In the reverse flow region, the flow at the blade approaches from the trailing edge toward the leading edge.

The geometry of this reverse flow region can be calculated using Eq. 2 with V \le 0, i.e.,

(112)   \begin{equation*} V (y, V_{\infty}) =\Omega y + V_{\infty} \sin \psi \le 0 \end{equation*}

This means that

(113)   \begin{equation*} \Omega y \le  V_{\infty} \sin \psi \end{equation*}

and for the boundary of the reverse flow region then

(114)   \begin{equation*} y =\frac{V_{\infty}}{\Omega} \sin \psi \quad \mbox{or} \quad r = \mu \sin \psi \end{equation*}

with r = y/R, which is the equation of a circle in polar coordinates with a center located at (r, \psi) = (\mu/2), 3 \pi/2).

Effects of Weight

Representative results showing the effect of the helicopter’s weight on the power required are shown below for the exemplar helicopter at mean sea-level (MSL) conditions. Notice that it requires more power for flight at higher flight weights. In this case, the power available at MSL is 2,800 hp and for a turboshaft engine this stays relatively constant with airspeed.

Predictions of power required in forward flight at different flight weights.

The airspeed obtained at intersection of the power required curve with the power available or torque available gives the maximum level flight speed. Most helicopters powered by turboshaft engines may be torque limited, which is a gearbox and transmission strength limitation, rather than being limited by the shaft power from the engine. However, the maximum airspeed may also be limited by the onset of rotor stall and compressibility effects before this point is reached.

Effects of Density Altitude

An important operational consideration for any aircraft is the effect of altitude on performance. As shown in the figure below, increasing density altitude increases the power required in hover and at lower airspeeds. At higher airspeeds and higher altitudes, the result of lower air density results in a lower power requirement because of the reduction of parasitic drag.

Predictions of power required in forward flight at different density altitudes.

However, a higher density altitude will also affect the engine power available. At 9,000 ft the power available is about 25% less than that available at sea-level conditions, resulting in a large decrease in the excess power available at any airspeed relative to that at sea-level conditions. Again, remember that helicopters powered by turboshaft engines may be torque limited, which is a gearbox and transmission strength limitation, rather than being limited by the shaft power from the engine.

Climb Performance

The general power equation can be used to estimate the climb velocity, V_c, that is possible at any given airspeed based on the excess power available, i.e.,

(115)   \begin{equation*} W \, V_c = P_{\rm avail} - \left( P_0 + P_p + \displaystyle{ \frac{\kappa T^2}{2 \varrho A V_{\infty}}} \end{equation*}

It is realistic to assume that for low rates of climb or descent the rotor induced power, P_i, the profile power, P_0, and the airframe drag, D, remain nominally constant so

(116)   \begin{equation*} V_c = \frac{P_\rm{avail} - P_{\rm{level}}}{T} = \frac{\Delta P}{T} = \frac{\Delta P}{W} \end{equation*}

Notice that P_{\rm{level}} is simply the net power required to maintain level flight conditions at the same forward speed. If the installed power available is P_{\rm avail} (which may vary with flight condition) then it will be seen that the power available to climb varies with forward flight speed. The climb velocity can then be obtained from

(117)   \begin{equation*} V_c = \frac{P_{\rm{avail}} -(P_i + P_0 + P_p )}{T} = \frac{\Delta P}{W} \end{equation*}

where \Delta P is the excess power available at that combination of airspeed and altitude.

Example of the rate of climb versus airspeed for the exemplar helicopter.

Calculations of the maximum rate of climb as a function of flight speed and density altitude are shown in in the figure below for the exemplar helicopter. Notice that climb performance is substantially affected by density altitude These curves will mimic the excess power available because the climb (or descent) velocity is determined simply by the excess (or decrease) in power required, \Delta P, relative to steady level flight conditions. Notice that the rate of climb improves markedly when translating from hover into forward flight. Pilots often call the tendency of the helicopter to climb  when accelerating from hover by the name “translational lift.” However, the term is a misnomer because the helicopter climbs by virtue of the excess power available and not extra rotor lift.

Worked Example #6 – Power Required in Forward Flight

A helicopter is operating in level forward flight at a true airspeed of 210 ft/s under the following conditions: shaft power supplied = 655 hp, W = 6,000 lb, \varrho = 0.00200 slugs/ft^{3}. The rotor parameters are R = 19 ft, \sigma = 0.08, \Omega R = 700 ft/s, \kappa = 1.15, and C_{d_{0}} = 0.01. Neglect the effects of reverse flow, stall, and compressibility. (i) How much power is required to overcome induced losses? (ii) How much power is required to overcome profile losses? (iii) What is the equivalent flat-plate area, f? (iv) If the installed power is 800 hp, estimate the maximum rate of climb possible at this airspeed.

(i) The induced power of the rotor, P_i, can be approximated by

    \[ P_i = \kappa T v_i \]

If the forward velocity is above \mu>0.1 (\mu = 210.0/700.0 = 0.3 in this case) then the induced velocity can be approximated by Glauert’s “high-speed” formula. Therefore, the induced power required does not need a solution to the inflow equation and can be determined from

    \[ P_i = \kappa T \left(\frac{T}{2 \varrho A V_{\infty}} \right) = \frac{\kappa W^2}{2 \varrho A V_{\infty}} \]

Substituting the appropriate parameters gives the induced power at this operating condition as

    \[ P_i = \frac{ 1.15 \times (6,000)^2}{2 \times 0.002 \times \pi \times (19)^2 \times 210.0} \times \frac{1}{550} = 79.0 \mbox{hp} \]

(ii) The profile power coefficient can be estimated from

    \[ C_{P_{0}} = \frac{\sigma C_{d_{0}}}{8} ( 1 + K \mu^2) \]

where K can be assumed to be 4.7; different values may be assumed for K. Therefore, at this operating condition the profile power is

    \begin{eqnarray*} P_0 & = & \frac{1}{8} \varrho A (\Omega R)^3 \sigma C_{d_{0}} \left( 1 + 4.7 \mu^2 \right) \nonumber \\ & = & \left( \frac{1}{8} \right) \times 0.002 \times \pi \times 19^2 \times 700^3 \times 0.08 \times 0.01 \times \nonumber \\ & & \hspace*{30mm}\left( 1 + 4.7 \times 0.3^2 \right) \times \frac{1}{550} \nonumber \\  & = & 201.3~\mbox{hp} \end{eqnarray*}

(iii) The total power in forward flight is the sum of induced, profile, and parasitic power, i.e.,

    \[ P = P_i + P_0 + P_p \]

Knowing the induced, parasitic and total shaft power supplied means that the parasitic power will be P_p = 655 – 79.0 – 201.3 = 374.7 hp. The parasitic power coefficient can be written as

    \[ C_{p_{p}} = \frac{1}{2} \left( \frac{f}{A} \right) \mu^3 \]

where f is the equivalent parasitic drag area. In dimensional units the parasitic power is

    \[ P_p = \frac{1}{2}  \varrho f  V_{\infty}^3 = 0.5 \times 0.002 \, f \, 210^3 = 374.7 \times 550 \]

Solving for f gives

    \[ f = \frac{2 \times 374.7 \times 550}{0.002 \times 210^3} = 22.25~\mbox{ft$^2$} \]

(iv) The maximum rate-of-climb, V_c, is determined by the excess power available over and above that required from straight-and-level flight at the same airspeed. In this case

    \[ V_c = \frac{\Delta P}{W} = \frac{ (800 - 655 ) \times 550}{6,000} = 13.26~\mbox{ft/s} \]

Conventionally, in aviation this result would be expressed in terms of feet-per-minute, which gives V_c = 13.26 \times 60 =798 ft/min.

Fuel Flows

For many aircraft performance problems, such as the calculation of range and endurance, a knowledge of the fuel burn of the engine(s) is required. For the helicopter, from the power required curves the fuel consumption can be estimated for any given type of engine, e.g., piston engine or turboshaft engine. Engine performance characteristics are usually expressed in terms of a brake specific fuel consumption or BSFC (in units of lb hp^{-1} hr^{-1} or kg kW^{-1}hr^{-1}) versus shaft power (in units of hp or bhp or kW). These curves are a function of atmospheric conditions, so a series of curves are required for different altitudes and operating temperatures. Fuel flow curves must be derived versus indicated airspeed and gross weight.

For a normally-aspirated (non-supercharged) piston (reciprocating) engine, the power curves vary almost linearly with density ratio, \sigma. One common approximation is

(118)   \begin{equation*} P_{\rm alt} = P_{\rm MSL} \left( 1.133 \sigma - 0.133 \right) \end{equation*}

where P_{\rm alt} is the power available at altitude and P_{\rm MSL} is the power available at mean sea level conditions. The value of the density ratio \sigma can be found from the ISA model depending on the pressure altitude and outside air temperature.

For a turboshaft engine a good approximation is

(119)   \begin{equation*} P_{\rm alt} = P_{\rm MSL} \left( \frac{\delta}{\theta} \right) \end{equation*}

where \delta is the pressure ratio at that altitude and \theta is the corresponding absolute temperature ratio, which again can be found from the ISA model.

Because helicopters will operate with the engines operating at close to their rated power for much of the flight, a first level approximation is to assume that the BSFC remains independent of power output. In this regard the fuel flow rate \overbigdot{W_f} is given by

(120)   \begin{equation*} \overbigdot{W_f} \frac{W_f}{dt} = {\rm BSFC} \, P_{\rm alt} \end{equation*}

where P_{\rm alt} is the total power required for the helicopter at altitude. Generally, fuel flow curves versus airspeed (at a given density altitude) are fairly flat. Therefore, for most helicopters the fuel flow curves will mimic the shapes of the power required curves. In this regard, a representative fuel flow curve for a helicopter is shown in the figure below. Notice that the best flight endurance will occur at the airspeed for the lowest fuel burn rate. The best range will occur at the airspeed for the lowest fuel burn per unit distance or airspeed, which is obtained at the tangent point as shown.

Range & Endurance

The fuel flow curves provide the information needed to determine range/payload and endurance/payload charts. These charts provide critical information on the effects of aircraft range and endurance, especially when trading off payload for fuel, and will be used by pilots for flight planning. Helicopters have relatively low useful load capability compared to airplanes, so the trade of payload with fuel is usually severe. When calculating flight endurance and/or range then specific mission profile for any aircraft must be defined, although most flight plans will involve a flight from point A to point B. Engine characteristics must be taken into account to determine both the maximum endurance and maximum range, i.e., its BSFC characteristics.

The fuel burn rate, W_F, with respect to distance, R, will be

(121)   \begin{equation*} \frac{d W_F}{d R} = \frac{P \times (\rm BSFC)}{V_{\infty}} \end{equation*}

where BSFC is the specific fuel consumption of the engine(s). The power required, P, varies with gross weight and density altitude, and as already shown the BSFC itself depends on power and density altitude. Because the weight decreases as fuel is burned, Eq.~121 must be integrated numerically to find the range. Fuel burned during takeoff, climb, and descent is factored into the calculation, along with a mandated fuel reserve in minutes of flying time.

Because the fuel weight on a helicopter is normally a small fraction of the total gross weight, then the the fuel burn rate can be evaluated fairly accurately at the point in the cruise where the aircraft weight is equal to the initial gross weight (gross takeoff weight W_{\rm GTOW}) is less half the initial fuel weight, where W_F is the initial fuel weight. In this case the range, R, of the helicopter is given by

(122)   \begin{equation*} R = W_F \left[ \frac{V_{\infty}}{P \times \rm {BSFC}} \right]_{W=W'} \end{equation*}

less an allowance for the other contingency factors described previously.

By a similar process, the estimated endurance, E, will be given by

(123)   \begin{equation*} E = W_F \left[ \frac{1}{P \times {\rm BSFC} } \right]_{W=W'} \end{equation*}

Generally, it is sufficiently accurate to estimate endurance by dividing the useable fuel on board by the average fuel flow rate.

Controlling the Helicopter

One of the greatest mysteries to most people about the helicopter is how it is controlled during its flight. In the case of an airplane it is much more intuitive because control is obtained by using easily identified aerodynamic surfaces, which are deflected to give the appropriate forces and moments on the airplane during its flight. In the case of a helicopter, control is obtained by using forces and moments produced by the rotor, which are obtained by tilting the orientation of the rotor disk. The question now is how this is done.

Main Rotor

First, it can be shown that the dynamic natural frequency of a rotor blade in its flapping motion (up and down) about the flapping hinge is equal to the rotational frequency, \Omega. While this result is strictly valid for a blade with a hinge at the rotational axis, it is close enough for any practical rotor and does not change the physics of the following discussion.

The critical issue is understanding that if any first-order dynamic system is excited by a forcing function at its natural frequency, then it will respond at the natural frequency but with a 90^{\circ} phase lag between the forcing and the response. For example, in the case of a rotor blade, if its lift (the forcing) is increased, then the blade response (the flapping angle) does not change instantaneously but will increase over time and only reach its maximum flapping displacement 90^{\circ} later. This 90^{\circ} lag between changes in blade pitch (lift) and the flapping response (displacement) is the key to understanding how to control the helicopter.

To change the blade lift, some form of mechanism is required to change the blade pitch angles. Most often thus is done using a swashplate, an example used on an actual helicopter being shown in the photograph below. A swashplate consists of two plates, the upper one rotating with the rotor and the lower one fixed and connected to the pilot’s controls. The upper swashplate is connected to the blades using pitch links, so the movement of the swashplate (up and down and tilting) then translates into displacements of the pitch links, producing changes in the blade pitch angles. A set of bearings between the two disks allows the upper disk to rotate with the rotor while the lower is non-rotating. Both disks can be slid up and down the shaft in response to collective pitch inputs, and the swashplate can also be tilted to an arbitrary orientation in response to cyclic pitch inputs, both originating from the pilot’s controls.

Photograph of a swashplate, which is used to impress a one-per-revolution change in blade pitch (and hence blade lift) to cause the blades to flap about the hinges and so tilt the orientation of the rotor disk and corresponding thrust vector.

The inputs produced by the swashplate consist of the average or collective pitch \theta_0 and the lateral cyclic, \theta_{1c}, and the longitudinal cyclic, \theta_{1s}, i.e.,

(124)   \begin{equation*} \theta(\psi)  = \theta_0 + \theta_{1c} \cos \psi + \theta_{1s} \sin \psi \end{equation*}

The collective pitch, \theta_0, controls the average pitch and so the total rotor thrust. The cyclic pitch controls the orientation or tilt of the rotor disk and so the direction of the rotor thrust vector. The blade pitch motion itself is induced through a pitch or feathering bearing. A pitch horn is attached to the blade outboard of the pitch bearing. A pitch link is attached to the pitch horn and the upper (rotating) part of the swashplate so that as the upper plate rotates, the vertical displacement of the pitch link produces blade pitch motion.

The novel part of the helicopter swashplate is the ability to tilt it to an arbitrary orientation, which requires a gimbal or spherical bearing between the swashplate and the rotor shaft. This mechanism then allows a first harmonic blade-pitch input with any phase angle. The upshot is that the rotor disk plane can be tilted to any orientation in space, the basic idea being shown in the schematic below.

The swashplate allows a first harmonic blade-pitch input with any phase angle, and so the rotor disk plane can be tilted to any orientation.

The two figures below explain in more detail the relationship behavior between blade pitch (and lift) and blade flapping displacements. It is unnecessary to impose pitch changes at other that once-per-revolution for the purposes of control. Again, the blade pitch (or feathering) motion imposed by the swashplate is described by

(125)   \begin{equation*} \theta(\psi)  = \theta_0 + \theta_{1c} \cos \psi + \theta_{1s} \sin \psi \end{equation*}

remembering that the imposed (controlled) value of \theta_0 modulates the value of the rotor thrust.

The first harmonic blade of the blade flapping response is given by

(126)   \begin{equation*} \beta(\psi)  = \beta_0 + \beta_{1c} \cos \psi + \beta_{1s} \sin \psi \end{equation*}

The mean value, \beta_0, is called the coning angle, which results from a balance of the moment of the blade lift and the centrifugal force about the flapping hinge.

For \theta_{1c} and \theta_{1s} inputs then the blade flapping response will be

(127)   \begin{eqnarray*} \beta ( \psi ) & = & \beta_0 + \theta_{1c} \sin \psi - \theta_{1s} \cos \psi \\ & = & \beta_0 + \theta_{1c} \cos \bigg( \psi - \frac{\pi}{2} \bigg) + \theta_{1s} \sin \left(\psi - \frac{\pi}{2} \right) \end{eqnarray*}

Notice that because of the dynamic behavior of the blade, the flapping response lags the blade pitch (aerodynamic) inputs by \pi/2 or 90^{\circ}.

Therefore, if swashplate produces a once-per-revolution aerodynamic forcing, it will causes the rotor blades to flap and once-per-revolution and the rotor disk plane will then precess to a new orientation in space, thereby tilting the orientation of the thrust vector. The system is really fundamentally very simple; an aerodynamic forcing is applied at (or close) to the natural frequency of the flapping blade and the blades respond so that a unit of cyclic pitch input results in (almost) a unit of flapping response. Strictly speaking this behavior is for a rotor with a flapping hinge at the rotational axis, but even with a hinge offset the underlying physics and rotor response is essentially similar.

The figure below illustrates more clearly what happens with a \theta_{1s} blade pitch input (longitudinal cyclic) from the swashplate, which means that the blade lift will be phased around the rotor disk such that it will reach a maximum at \psi = 90^{\circ} and a minimum at \psi = 270^{\circ}. Because of the 90^{\circ} phase lag between the generation of lift and the blade flapping response, in this case the blades will flap up and reach a maximum displacement at \psi = 180^{\circ} and a minimum displacement at \psi = 0^{\circ}, the net effect in this case being that the rotor disk plane will tilt backwards.

The use of cyclic pitch to produce a longitudinal flapping displacement, thereby tilting the rotor disk backward in this case.

The second figure, as shown below, illustrates what happens with a \theta_{1c} blade pitch input (lateral cyclic) from the swashplate, which means that the blade lift will be phased around the rotor disk such that it will reach a maximum at \psi = 0^{\circ} and a minimum at \psi = 180^{\circ}. Again, because of the 90^{\circ} phase lag between the generation of lift and the blade flapping response, the blades in this case will flap up and reach a maximum displacement at \psi = 90^{\circ} and a minimum displacement at \psi = 270^{\circ}, the net effect is that the rotor disk plane will tilt to the left (to port) when viewed from behind. The mystery of how to control the helicopter is now solved!

The use of cyclic pitch to produce a lateral flapping displacement, thereby tilting the rotor disk to port in this case.

Tail Rotor

A primary purpose of the tail rotor is to provide a sideward force on the airframe in a direction and of sufficient magnitude to counter the main rotor torque reaction.  If the distance from the main rotor shaft to the tail rotor shaft is x_{\hbox{\scriptsize\it TR}}, the tail rotor thrust required will be

(128)   \begin{equation*} T_{\hbox{\scriptsize\it TR}} = \frac{(P_i + P_0 + P_p)}{\Omega~x_{\hbox{\scriptsize\it TR}}} \end{equation*}

where \Omega is the angular velocity of the main rotor. This latter equation assumes that there is no interference or off-loading of the tail rotor by the fin in hover or forward flight.

The second purpose of the tail rotor is to provide directional (yaw) control, which is done by modulating the tail rotor’s thrust; the thrust is controlled by the pilot’s feet by pushing on a set of floor mounted pedals much like the rudder pedals on an airplane. For example, for a rotor turning in the conventional direction (counterclockwise when viewed from above) pushing on the left pedal increases tail rotor thrust (positive to starboard) and the helicopter will yaw nose left about its center of gravity.

As for the main rotor, the power required to drive the tail rotor depends on the disk loading. Whereas a larger diameter may be preferable for low induced power requirements, this is outweighed by several factors. First, a larger diameter usually means a heavier design and this is undesirable because of adverse effects on the helicopter’s center of gravity location. Second, to meet certification requirements it is usually desirable that the tail rotor disk loading and induced velocities be high enough so that sideward flight without the tail rotor entering into the vortex ring state (VRS). Both of these constraints dictate the use of a relatively small tail rotor with a high disk loading.

Tail rotors may be either of the pusher or tractor variety, as shown in the figure below, and can be located either on the left- or right-hand side of a vertical fin. All designs suffer from interference effects between the rotor and the fin, more or less, these effects being a function of the tail rotor size or disk area, fin area, and spacing of the rotor plane from the fin.


Types of tail rotors include the pusher and tractor designs.

In contrast, the tractor design has a vertical fin inside the high-energy region from the tail rotor wake. While this “blockage” effect tends to increase the tail rotor thrust, there is also a significant force applied to the vertical tail that is in the opposite direction to the anti-torque thrust requirement. It is found, however, that the net effect is a decrease in thrust compared to what would be obtained if the rotor was operating in isolation. In both cases, the interference effects become more significant with larger fins and/or smaller rotors. As a result, most modern helicopters use a pusher tail rotor design because this configuration tends to have a higher overall anti-torque-producing efficiency.

The tail rotor must also provide the specified yaw acceleration in the maximum specified crosswind conditions, taking into consideration possible losses in efficiency because of aerodynamic interference effects between the tail rotor and the vertical fin. Furthermore, when the main rotor thrust or power is increased, for example, to climb, the reaction torque on the fuselage is increased. This effect means that the tail rotor thrust must also increase to balance this torque reaction. Therefore, when the pilot increases the collective pitch to climb, foot pressure must be applied to the appropriate pedal to keep the nose pointed straight in the desired direction of flight.

Tail rotors typically have two or four blades, with no particular aerodynamic advantage of one number over the other. Only collective pitch is required because there is no need to control the orientation of the tail rotor disk plane. Tail rotor blades may have some built-in twist to help minimize induced power requirements. Although some blade designs may use cambered airfoil
sections, the tail rotor blades on many helicopters use symmetric airfoils because of their excellent overall performance and low pitching moments. Generally, tail rotors are designed to operate at tip speeds that are comparable to those of the main rotor. Lower tip speeds are desirable to
minimize noise. However, for a given thrust, tail rotors operating at lower tip speeds require higher solidity to prevent blade stall. A lower tip speed also increases the torque requirement. Both of these factors will increase the weight of the drive system.

A secondary effect of the anti-torque side force is the tendency for the helicopter to drift sideways. This effect is corrected by the main rotor, which is tilted slightly to the left (using cyclic pitch inputs) so that a component of the main rotor thrust produces an equal and opposite side force. This behavior is the reason why it will be noticed that a helicopter will tend to hover with one wheel (or skid) lower than the other. On larger helicopters, the main rotor shaft is physically tilted slightly (as part of the design, thereby introducing a pretilt) so that the pilot does not require as much cyclic pitch input to counter the tail rotor side force. The tail rotor thrust and the component of the main rotor side force act together, producing a couple and, thereby, a rolling moment about the center of gravity (c.g.). To reduce this moment, the tail rotor is located vertically up on the tail structure so that the line of action of its thrust vector is close to the helicopter’s center of gravity.

Common amongst all tail rotors is the lack of any cyclic pitch; only collective pitch is used because control of the tail rotor disk orientation is not required. Nevertheless, the tail rotor must be provided with flapping hinges so that the blades may be allowed to respond to the changing aerodynamic environment. Lead/lag hinges are not usually used to save weight and reduce mechanical complexity. Instead, pitch/flap coupling is built into the tail rotor design, which can be discerned from the photograph below. This approach provides a means of allowing the blades to pitch cyclically in such a way as to minimize blade flapping produced by the changing aerodynamic loads.

Pitch/flap coupling can be introduced into the tail rotor design such that blade flapping acts to change the blade pitch and so minimize the flapping. In this case, the coupling is introduced by locating the pitch horn 45^{\circ} in front of the blade feathering axis.

Summary & Closure

This lesson provides an overview of the factors affecting the performance of a helicopter. The momentum method was used to evaluate rotor performance in various conditions including hover, climb, descent, and forward flight. The concept of disk loading was shown to be a crucial factor in rotor performance, with low disk loading being important for good hovering efficiency. Power loading and figure of merit were also introduced as design and comparison tools. The basic momentum theory was modified to account for non-ideal effects and provide better agreement with experimental measurements. The momentum theory was also applied to forward flight, where numerical solutions are required to solve for the inflow through the rotor disk. Basic performance characteristics of helicopters in forward flight were also discussed.

5-Question Self-Assessment Quickquiz

For Further Thought or Discussion

  • The history of airplanes shows that growth (in terms of number of prototypes being developed) was rapid and monotonic. However, while the growth of rotorcraft (helicopters) started earlier it progressed more in a series of discrete events that might be called “fits and starts.” Discuss this point further assuming associations with technology developments.
  • Give a brief overview of the main technological problems that were encountered when trying to build and fly a successful helicopter prior to 1900. In your discussion, consider issues such as overall levels of aeronautical knowledge, stability and control requirements, the availability of suitable engines, construction materials, and anything else you feel relevant.
  • Engines are often viewed as the one of the most important enabling technologies for powered flight, especially for a helicopter. However, it could be argued that it was actually the realization of a readily available supply of suitably formulated and inexpensive fuel for a piston engine (i.e., gasoline) that was really the key enabling technology. Discuss this viewpoint.
  • Most modern helicopters are of the single main rotor and tail rotor (conventional) configuration.  For the same overall aircraft gross weight, what might be the relative advantages of a using a tandem rotor helicopter over a conventional helicopter? Also, compare the potential relative merits of a coaxial rotor and a side-by-side rotor configuration over a tandem rotor design.
  • Discuss the physical and other features that distinguish a helicopter from an autogiro. Discuss also the reasons as to why the autogiro was quickly eclipsed by the success of the helicopter when it appeared in the mid to late 1930s. Do you think there are prospects for the autogiro or gyroplane in the future?
  • Although tiltrotor concepts date back more than sixty years, it is only recently that civilian tiltrotor aircraft have completed much of the flight-testing toward civil certification, i.e., the AW-609. Discuss the technical, economic and any other reasons for this long gestation period.  Consider also the prospects that you may fly on a tilt-rotor operated by an airline within the next decade.
  • Use conservation of mass and momentum but use Bernoulli’s equation instead of the general energy equation to prove that the induced velocity w in the fully contracted wake (slipstream) of a climbing rotor is twice the induced velocity in the rotor plane.

Other Useful Online Resources

For additional resources on helicopters and other rotorcraft, follow up on some of these online resources:

  • Early film. How it Works: Sikorsky H-19 Helicopter.
  • History Of Helicopters – Military Helicopter Invention Documentary.
  • Great film on the Birth of the Bell Helicopter.
  • A video showing the detail of the swashplate.
  • Video on the top 7 U.S. military helicopters of all time!
  • Your first helicopter lesson in a Robinson R-44!
  • The massive Sikorsky CH-53K helicopter in action!
  • Bell 206 JetRanger helicopter review + how to fly a helicopter.
  • MD 520N NOTAR helicopter review and flight test.


  1. The author is indebted to one of his teachers, Professor Alfred Gessow, who was a pioneer in the field of helicopter aerodynamics and co-author of the book "Aerodynamics of the Helicopter".