Introduction

The modern propeller is a remarkable accomplishment of aeronautical engineering. Aviation history shows that propellers have undergone continuous innovations and technological advancements for over a century, in many ways setting the pace for developing new aircraft. The earliest propeller concepts can be traced back to Leonardo da Vinci in the 1480s, who sketched the “aerial screw,” as shown in the figure below. This concept was an aerodynamic device derived from an Archimedes water screw, which some may consider as a precursor to the modern helicopter rotor. Although never built in its intended human-carrying scale,^[1] Da Vinci’s design laid down the concept of using rotating wings to create an aerodynamic force parallel to the axis of rotation, a force that is called thrust.

In the 19th century, George Cayley contributed significantly to understanding the principles of flight, experimenting with gliders, and recognizing the need for a propulsion mechanism to overcome drag if sustained flight was ever to be achieved. At least one of his airplane concepts and an airship, the latter shown in the illustration below, indicated that the means of thrust and propulsion was by using “airscrews” or propellers. The power source was not specified, although the only option at the time would have been a steam engine.^[2] The use of “screw” propellers for propulsion was nothing new, having been used in marine applications since the 1830s. Still, the propeller blade designs that were suitable for good efficiency in the air were as yet unknown. In the 1870s, Thomas Moy built the Aerial Steamer, an early powered aircraft with a primitive propeller driven by a steam engine. Like the Cayley concept, however, although it did not achieve sustained flight, it was a step forward in critical thinking about how an airplane might be propelled.

In 1903, the Wright brothers made the first successful, powered flight with their Wright Flyer, equipped with twin wooden propellers designed and built in their workshop in Dayton, Ohio. The Wrights discovered that there was no theory or other analysis for describing or predicting propeller performance, and the state of the art at the time was simply trial and error. They then set out to study the aerodynamics of propellers in their own wind tunnel, showing that high aspect ratio blades with a nose-down twist (called washout) along their length gave them the best propulsive efficiency. By 1908, they had improved their propeller designs for the Wright Flyer III, enabling flights of much longer duration and range. Their “raked” propeller had a swept leading edge and more blade area toward the tip, as shown in the figure below, which gave further efficiency improvements, allowing more of the limited engine power to be converted into thrust. The broader propeller tips were also structurally stiffer, preventing them from buzzing and fluttering, a shortcoming of early propeller designs.

During WWI, propeller technology advanced rapidly by the demand for airplanes with greater performance. Initially, fixed-pitch wooden propellers made from laminated spruce were standard, as shown in the photograph below. However, as engine power and aircraft speeds increased, the limitations of wooden propellers quickly became evident, including structural cracking and delamination. They were also very vulnerable to damage in air-to-air combat, with a single bullet through a wooden blade often causing its catastrophic failure.

While wood was easy to work with and propellers could be carved to almost any shape, the quest for more durable propellers led to metal blades, initially made of steel. Steel offered much greater strength and resistance to damage, although it introduced other challenges, such as significantly increased weight and the need for more complex manufacturing processes. One of the earliest airplanes using steel propellers was the Handley Page O/400 bomber, which significantly improved its performance and durability in the demanding conditions of long-range bombing missions under enemy fire.

With the formation of dedicated research institutions like the NACA and the Royal Aircraft Establishment (RAE), the aerodynamic principles governing propeller performance began to be systematically quantified. From the 1920s to the 1950s, NACA conducted wind tunnel tests and collated detailed data on propeller performance. Engineers experimented with various blade planforms, including tapered and elliptical shapes, as well as specific tip shapes and airfoil sections that were better suited for propellers. The RAE also researched propellers extensively, focusing on theoretical aerodynamic analyses, mainly by Hermann Glauert^[3], supported also by wind tunnel measurements. These propeller developments laid the groundwork for more sophisticated airplane designs, contributing to the rapid evolution of aviation technology.

Materials like duralumin, an aluminum alloy invented in the mid-1920s, were used to make propellers lighter and stronger. Engineers soon began developing variable-pitch aluminum propellers, allowing pilots to adjust the blade pitch angles for optimal performance during flight, significantly improving both the propeller efficiency and the airplane performance over a broader range of airspeeds. In the 1930s, the Hamilton Standard propeller company introduced the first controllable pitch propeller. On takeoff, the pilot could use a lower blade pitch for best thrusting performance and then shift to a higher blade pitch for efficient cruising flight. This innovation soon led to the constant-speed propeller, which automatically adjusted the blade pitch during flight using a governor^[4] to maintain a constant rotational speed and continuously maximize the propeller’s propulsive efficiency.

During WWII, propeller-driven aircraft technology reached new heights, with significant advancements in propeller design, aerodynamics, engine efficiency, structural reliability, and overall weight reduction.. The introduction of multi-bladed propellers was another crucial development. Initially, two-blade and three-blade propellers were standard, but engineers soon adopted four-blade and five-blade designs to absorb the power from increasingly powerful piston engines. Additional blades also provided smoother operation, reducing vibrations from the engine and propeller, and improving the airplane’s overall performance. As engine power levels increased even further after the advent of supercharging and turbocharging, counter-rotating tandem propellers were used to prevent the size of a single propeller from becoming prohibitively large.

Iconic WWII aircraft like the British Supermarine Spitfire and the American North American P-51 Mustang exemplified propeller and engine technology advancements. The Spitfire, with its powerful Rolls-Royce Merlin engine and Rotol advanced four-bladed aluminum propeller design, as shown in the photograph below, achieved exceptional performance in terms of airspeed and altitude of operation. Later models of the Spitfires were equipped with a counter-rotating tandem propeller powered by a version of the Rolls-Royce Griffon engine, representing the pinnacle of piston engine/propeller performance. The P-51 Mustang, when fitted with a four-bladed Hamilton Standard propeller, became renowned for its long range, high speed, and effectiveness as an escort fighter.

These technological improvements with propellers were also applied to bombers and transport aircraft, such as the Boeing B-17 Flying Fortress, the Consolidated B-24 Liberator, the Avro Lancaster, and the Douglas C-47. Propellers played a crucial role in improving the speed, altitude, and range of these aircraft, ultimately contributing to the Allied victory in WWII. The increased efficiency and reliability of these propeller-driven aircraft allowed for more effective long-range bombing campaigns, strategic airlifts, and supply missions. Additionally, advancements in aerodynamics and engine integration enabled these aircraft to carry heavier payloads while maintaining maneuverability and endurance in combat operations.

The transition to using gas turbine (turbojet) engines after WWII soon surpassed the use of propellers in most military and commercial aviation. Nevertheless, propeller technology continued to evolve for specific applications, including fighter airplanes suitable for high-speed flight. However, the most significant interest in new propeller development was for the turboprop engine, which used a gas turbine engine to drive a propeller, and offered high propulsive efficiency at lower speeds and altitudes than pure jet engines. This made turboprop “jet” airliners ideal for shorter routes where fuel efficiency and short takeoff and landing capabilities, such as at regional airports, were critical.

The Lockheed L-188 Electra, which entered service in 1958, demonstrated the enduring significance of propeller-driven airplanes; one of its four-bladed propellers is shown in the photograph below. Airplanes like the Lockheed C-130 Hercules, the modern Airbus A400M Atlas, and the C-130J Super Hercules have since leveraged advances in turboprop technology for their improvements in efficiency, versatility, and reliability. Today, the turboprop is still a preferred propulsion system for many airplanes and can be found on high-performance single-engine airplanes such as the Pilatus PC-12 and Cessna Caravan 208.

While the basic engineering principles used for aircraft propeller design have remained the same for decades, numerous detailed improvements have led to substantial gains in propulsive efficiency and operational reliability. Propeller construction methods have continued to advance using carbon and graphite composites, which provide lighter and more aerodynamically efficient blades with lower noise. Engineers continue to use techniques such as computational fluid dynamics (CFD) and computer-aided design (CAD) to optimize propeller performance even further, tailoring them to specific airplanes and flight conditions.

Most modern propellers have swept leading edges and thin airfoil sections, an example of which is shown in the photograph below. Scimitar propellers, which have highly swept blades made primarily of composite materials, were developed in the 1980s. Like the swept wings used for high-speed aircraft, these propeller blades delay the onset of drag rise and the creation of wave drag, allowing them to remain efficient propulsors at higher flight speeds and flight Mach numbers. They also reduce noise, a significant concern for all modern aircraft because of the increasingly stringent noise standards set by the ICAO. This type of swept-blade propeller design is used in many modern turboprop airplanes, such as the ATR 72, the Bombardier Dash 8, and the Pilatus PC-12.

Ongoing developments in propeller designs suited for electrically powered aircraft focus on improving efficiency and reducing noise, particularly for urban air mobility (UAM) and uncrewed aerial vehicles (UAVs). Engineers are exploring innovative blade geometries and lightweight materials to create quiet, compact propeller systems that meet the unique demands of drones, electric airplanes, and eVTOL concepts. The goals include optimizing flight performance using multi-rotor systems and variable-pitch propellers while leveraging improvements in battery technology and manufacturing techniques. While the tipping point in favor of electric propulsion is likely decades away, technology developments are progressing rapidly.

Propeller Fundamentals

Propellers convert rotational motion into thrust by creating aerodynamic lift forces on their blades, which act as rotating wings. As the blade moves through the air, the airfoil shapes create an aerodynamic pressure difference between the upper and lower surfaces, resulting in blade lift. The resolved forward component of the lift vectors is known as thrust. The engine supplies a torque to the propeller shaft to overcome the drag forces on the blades and maintain their rotation. Efficient propeller design maximizes this thrust by optimizing blade shape, angles of attack, and rotational speed, ensuring that as much of the engine’s power as possible is effectively converted into propelling an aircraft.

Blade Twist

A notable design feature of propeller blades is that they are significantly twisted along their span, i.e., they have a form of washout. Each blade is an airfoil, similar in cross-section to an airplane wing, and the twist ensures that each part of the blade meets the flow at an optimal angle of attack, maximizing efficiency and thrust production across the entire blade span. As shown in the figure below, by using a velocity triangle, the local pitch angle must change from a relatively high value at the root (next to the hub or boss) and progressively decrease in value from section to section when moving out to the blade tip.

The primary purpose of using blade twist is to get the local aerodynamic angles of attack, ,  at each section along the span of the blade to be low enough such that it operates close to the conditions where the section is most aerodynamically efficient for lift production, i.e., at their best lift-to-drag ratios. The upshot then will be the most thrust for the lowest amount of torque and power supplied to the propeller shaft. Most propeller blades operate with their local angles of attack between 7 and 12 degrees.

For example, if was to be assumed constant for all of the airfoils along the length of a propeller blade, the blade twist must be adjusted along its span to compensate for the radially varying local flow velocities. At a radial position from the rotational axis, the rotational velocity is , and the airspeed is . From the velocity triangle, the inflow angle is

(1)  

using the small angle approximation, . For a propeller operating at a sufficiently high airspeed, the induced velocity is small enough to be neglected. Therefore, because , then

(2)  

This expression shows that must follow a hyperbolic variation proportional to . The twist angle is larger near the root and decreases toward the tip, thereby ensuring a constant angle of attack along the blade span.  However, because of the variations in Mach number, Reynolds number, and section thickness-to-chord ratio, the best angle of attack is not necessarily a constant. The net twist varies for different propellers, but washout angles may be as large as 40 degrees over the blade span; of course, these twist angles are much larger than those used on any fixed wing.

Fixed-Pitch Versis Variable Pitch

Fixed-pitch propellers have a very limited operating envelope; they are either suitable for takeoff and climb and poor for cruise, or alternatively good for cruise and have a poorer takeoff and climb performance. The solution to this dilemma is to change the average blade pitch angles^[5] to allow the efficient operation of the propeller to be optimized as a function of the flight condition, the idea being shown in the figure below.

To maximize thrust and efficiency, a fine (low) blade pitch will be used for takeoff and landing, and a coarser (higher) blade pitch will be used for cruise flight. Regulating the pitch angles of the blades in this manner allows the efficient operation of the propeller to be optimized as a function of the flight condition. Therefore, an aircraft with a variable pitch or constant speed propeller can maintain good levels of propulsive efficiency over a broader range of airspeeds. In cruise flight, a speed governor automatically adjusts the blade pitch to keep the propeller at a constant rotational speed, offloading much workload in controlling the peopeller from the pilot.

The highest thrust from the propeller will be limited by the onset of stall, just as a regular wing is limited in its angle of attack capability by stall. However, because the local incident Mach number varies from the root to the tip of the propeller, and the stall angle of attack of an airfoil decreases with increasing Mach number, the allowable angle of attack before stall will decrease from root to tip. The onset of stall will eventually occur if the blade pitch is increased too much, which will be accompanied by a loss of thrust, an increase in power, and a marked drop in propulsive efficiency.

If the engine stops in flight, the propeller blades need to be “feathered” into the flow to reduce the otherwise high drag of the stationary blades, i.e., the leading edges of the blades point directly into the airflow. In this regard, strong springs are used on a constant-speed propeller so that the blades are automatically forced into the fully feathered position if the propeller stops. The drag of an unfeathered stationary propeller is considerable, so an airplane with a fixed-pitch propeller will inevitably have a poor glide ratio. The propeller blades may sometimes be set to a negative pitch to produce negative thrust and act as an aerodynamic brake during landing. However, this latter feature is usually only included in high-performance turboprop airplanes.

Types of Propellers

Many propellers are currently used, from those that use just two blades to those with four or more blades, some with fixed pitch, and others with variable pitch, a selection being shown in the photographs below. In addition, some propellers may have swept blades, a design feature used to reduce compressibility drag loss at the tips of the propellers when the aircraft operates at higher flight speeds, just like what is done with airplane wings. As engine power increases, more blades (or more blade area) are needed to deliver the power to the air. In some cases, to prevent the propeller diameter from becoming too large, counter-rotating propellers may be used to absorb high amounts of available power from the engine shaft.

Propellers with larger numbers of blades also tend to be relatively more efficient (when compared based on the same thrust and total blade area), which is because they act increasingly like an actuator disk.^[6] However, the propeller’s net efficiency depends on other factors, including its operating speed (rpm), tip Mach number, blade section Reynolds numbers, diameter, and blade pitch. Today, there is an increasing emphasis on obtaining lower noise from propellers, which has driven engineers of modern designs (even on general aviation airplanes) to use more blades and reduced diameters with lower tip Mach numbers.

Momentum Theory of the Single Propeller

The Rankine propeller theory,^[7] also known as the Rankine-Froude theory,^[8] provides a fundamental framework for understanding the performance of propellers. This theory models the propeller as an idealized actuator disk that imparts momentum to the air, creating a difference in pressure and velocity across the disk. The physical problem is that the propeller does work on the air as it passes through the propeller disk, i.e., the propeller applies a force to the air in the downstream direction and changes the momentum and kinetic energy of the air. As a result, the force on the propeller, which is produced because of a pressure difference between one side of the propeller disk and the other, is in the opposite direction to the force on the fluid, i.e., the thrust is directed upstream.

Flow Model

The flow model for the single propeller problem is shown in the figure below. The upstream and downstream sections labeled 1 and 2 bound the control volume. Remember that when the integral form of the conservation equations is used, then points out of the control volume by convention. The upstream or free-stream (undisturbed) velocity is , and the pressure there is . It will be assumed it is uniform, which is a reasonable assumption unless the propeller is affected by a wing or another part of the airframe. It will also be assumed that a one-dimensional, steady, incompressible flow applies throughout so that the flow velocities only change with downstream distance.

The flow is entrained and accelerated into the propeller, so at the plane of the propeller, the flow velocity is plus an increment , i.e., the velocity there is . The static pressure will change there, too. The propeller works on the air to increase its momentum and kinetic energy, so the velocity is downstream. In the slipstream, the static pressure will also recover to ambient conditions, i.e., the pressure downstream will slowly return to .

Application of the Conservation Laws

The conservation of mass requires constant mass flow through the control volume. The areas of the upstream and downstream parts of the streamtube (stations 1 and 2) are unknown, but the area of the propeller disk, , is known ( where is the diameter of the propeller), so the mass flow rate through the propeller (and hence through the boundaries of the control volume) is

(3)  

The conservation of momentum requires that the net change in momentum of the fluid (as applied by the propeller) is equal to the force on the fluid, i.e.,

(4)  

where the direction of is downstream, and the thrust on the propeller, , is in the opposite direction (pointing left in the figure) so that

(5)  

The conservation of energy states that work done on the propeller (to move it forward) plus the work done on the air (to create the aerodynamic force) must be equal to the gain in kinetic energy of the slipstream as it passes through the propeller, i.e.,

(6)  

so that the power required for the propeller to produce the thrust is

(7)  

Notice that the term on the left-hand side of the above equation can be written as the sum of the term, which is the work done to move the propeller forward (this is the useful work). The term is the work done on the air or the induced power loss; the induced power is an irrecoverable power loss, which constitutes a loss in the propeller’s efficiency in producing useful work.

Induced Velocity

The induced velocity in the plane of the propeller can be found by substituting Eq. 5 into Eq. 7, which gives

(8)  

or simplifying gives

(9)  

This gives the relationship between and , i.e.,

(10)  

For the next step, take Eq. 5 and substitute in Eq. 3 (mass flow rate) and the connection between and () to get

(11)  

The induced velocity in the plane of the propeller can now be solved for in terms of thrust, i.e.,

(12)  

and expanding out gives

(13)  

The ratio is called the propeller disk loading. It can be seen that the latter equation is quadratic in , which can be solved to get

(14)  

for which there must be two roots, i.e., from the term. Only one can be a physical root (the other one violates the assumed flow model where the flow is assumed to be through the propeller from left to right), which is

(15)  

or non-dimensionally as

(16)  

Therefore, it will be apparent that for any given propeller producing a thrust , its induced velocity will decrease rapidly with increasing airspeeds, i.e., . In the design of a propeller, the induced losses can often be ignored.

Power Required

The corresponding power required to drive the propeller and produce a thrust, , becomes

(17)  

Notice that the useful power for propulsion is , and the second term is the irrecoverable induced loss. Rearranging the preceding equation gives

(18)  

This power equation can also be written as non-dimensionally as

(19)  

Notice that in the limiting case when the airspeed, , becomes high, then , meaning that the induced losses tend to zero. Therefore, a greater fraction of the power delivered to the propeller goes into useful work to propel the airplane forward. However, in practice, some additional profile (non-lifting) power, , will be required to overcome the viscous (shear) drag of the blades so that the total power required, , can be written as

(20)  

Propulsive Efficiency

The propulsive efficiency of the propeller can also be derived from the preceding analysis. The useful power for propulsion is so the efficiency of the propeller, , can be written as

(21)  

which says that the propeller becomes more efficient at higher airspeeds where the induced velocity becomes a smaller fraction of the airspeed. This result also confirms that the efficiency of the propeller at higher airspeeds as will be dictated by the profile losses on the blades, i.e.,

(22)  

The profile power losses, denoted by , depend on the blade sections and the net blade area, which is measured by the propeller’s solidity or activity factor.

Wake Swirl Effects

Incorporating the swirl in the momentum theory involves adding the in-plane tangential velocity component to the otherwise purely axial flow, i.e., a rotational flow component parallel to the rotor’s plane of rotation. Rankine’s actuator disk theory considers only the axial momentum balance, assuming the flow is steady, inviscid, and incompressible, although Hermann Gluert further considered this term. When swirl is included, the average angular velocity component of the downstream flow must now be considered because it affects the power requirements and propulsive efficiency. While the effects are best considered within the framework of the blade element theory, an approximation can be derived from the momentum theory.

Propeller swirl refers to the rotational velocity of the air in the downstream wake behind the plane of a propeller, which, in effect, creates a spiral or corkscrew flow, as shown in the figure below. This effect occurs because of viscosity, where the propeller blades drag along a certain amount of fluid as they rotate. The upshot imparts a rotational velocity component to the fluid, , as a consequence of generating thrust. The higher the thrust and torque, the more significant his effect. Swirl causes efficiency losses because some kinetic energy is lost to rotational fluid motion over and above that from the thrust generation.

Recall that the thrust generated by a propeller with a pure axial velocity component, , is given by

(23)  

where is the mass flow rate through the propeller. The power required to generate this thrust, considering the presence of an average swirl velocity, , is

(24)  

where the effective velocity, , is given by

(25)  

Therefore, the power required is

(26)  

which is greater than the power required for a pure translational change in momentum through the propeller.

In the wake of a propeller, the tangential velocity component, , can be derived from the angular momentum imparted by the propeller blades. The torque, , generated by the propeller can be calculated using the relationship between torque and power, i.e.,

(27)  

where is the power absorbed by the propeller and is the angular velocity of the propeller. The time rate of change of angular momentum, which is a torque, is given by

(28)  

where is the angular momentum. By using the relationship between torque and angular momentum, the effective radius at which the torque acts can be considered as the mean aerodynamic radius of the propeller, which can be approximated by

(29)  

where is the propeller’s radius (= ). Finally, the average tangential velocity, , is

(30)  

In practice, the effect of wake swirl on propeller performance is minimal, perhaps reducing the overall propulsive efficiency by 1% to 2%. However, for large propellers driven by engines delivering large amounts of torque, the effects may be as high as 5%. Coaxial counter-rotating propellers, ducted propellers, and swirl recovery vanes have been used to reduce wake swirl and improve the net propulsive efficiency.

Propeller Coefficients

The thrust coefficient for a propeller is a non-dimensional thrust and is defined as

(31)  

where is the thrust generated by the propeller, is the propeller’s diameter, and is the number of revolutions of the propeller per second. If the rotational angular velocity of the propeller is , then . The rotational speed is often measured by revolutions per minute or rpm, so rpm equals .

For a propeller, a non-dimensional airspeed called a tip speed ratio or advance ratio, given the symbol , is defined by

(32)  

Finally, the power coefficient for a propeller is defined as

(33)  

where would be the brake power, i.e., the power delivered to the propeller through the driving shaft. In terms of the torque, , then .

Notice that propeller thrust coefficient, advance ratio, and power coefficient definitions differ from those used for helicopter rotors. It is essential not to confuse them. The differences are clarified in the table below. Notice that = 2.

Rotating-wing system	Thrust coefficient,	Power coefficient,	Advance ratio
Helicopter rotor
Propeller

The momentum theory results for a propeller can be written in coefficient form. For the induced velocity, then

(34)  

and for the induced power, then

(35)  

The corresponding propulsive efficiency is then

(36)  

Momentum Theory of the Tandem Propeller

Tandem coaxial propellers are arranged on the same axis but rotate in opposite directions, allowing for a more compact design than separate, non-coaxial propellers. Additionally, the opposing rotation of the propellers cancels out torque reaction effects, which can be very significant when large amounts of power and torque are being transmitted from the engine. In a tandem propeller system, two propellers are aligned along the same axis but operate at different stations, effectively working on the same slipstream. The second propeller acts on the accelerated flow produced by the first propeller, leading to further acceleration and additional thrust generation.

There are two conditions of interest:

1. The rear propeller is sufficiently separated to work in the front propeller’s slipstream and induced velocity field.
2. Both propellers are located so close that they share the same induced velocity.

Separated Tandem Propellers

The mass flow rate through both propellers must be conserved for the tandem system, as shown in the figure below. If the first propeller increases the velocity from to , and the second propeller further increases it to , the mass flow rates through each propeller are

(37)  

The thrust produced by each propeller can then be determined, i.e.,

(38)  

The total thrust generated by the tandem system is then the sum of the thrusts produced by each propeller, i.e.,

(39)  

The total power required is the sum of the power required by each propeller, i.e.,

(40)  

The addition of the profile power is necessary to account for the drag associated with the propeller blades. It affects the overall efficiency of both single and tandem propeller systems. The efficiency of each propeller can be expressed as

(41)  

For a tandem configuration with two propellers, the efficiency of each propeller is

(42)  

Therefore, the overall efficiency of the tandem system can then be expressed as

(43)  

Including profile power affects the efficiency calculation because both thrust production and the power required to overcome drag must be balanced. If the profile power becomes significant relative to the thrust generated, it will lower the overall efficiency, particularly at higher speeds where profile losses will dominate. In this regard, it will be noted that

(44)  

which is the same result as found for the single propeller, i.e., its net propulsive efficiency mostly depends on the profile power required.

Tandem Propellers with Shared Induced Velocity

Now consider two tandem propellers positioned closely enough to share the same induced velocity increment , as shown in the figure below. The flow velocity at the propeller planes is . The analysis simplifies under the assumption that the induced velocity is the same for both propellers. The mass flow rate through the tandem propeller system is

(45)  

where is the area of each propeller disk, which is assumed to be the same for both propellers.

The thrust generated by each propeller can be expressed as

(46)  

Because both propellers share the same induced velocity , the total thrust is

(47)  

The power required for each propeller is given by

(48)  

and

(49)  

Therefore, the total power required is

(50)  

The efficiency of the tandem propeller system can be analyzed by considering the ratio of useful work to the total power required. The useful work is related to the thrust produced and the forward velocity . The efficiency of each propeller can be expressed as

(51)  

Incorporating the profile power associated with the drag of the propeller blades, the overall efficiency of the tandem system can be expressed as

(52)  

which again comes to

(53)  

Comparing Efficiencies

A common question is how a tandem propeller design compares to a single propeller’s efficiency. The answer, at least partly, lies in the basis for comparison, e.g., constant net thrust, continuous power, or constant solidity (blade area). Nothing suggests that one system is better than another, at least within the assumptions and limitations of the momentum theory. It is the profile power requirements of the blades that dictate the net propulsive efficiency of the propeller at higher airspeeds. Any swirl recovery effects are a small fraction of the net efficiency.

When comparing efficiencies, consideration must be given to the added complexity of managing the propellers’ power and the propulsive system’s overall weight efficiency. Depending on the specific design and operating conditions, the overall propulsive efficiency of the tandem system could either outperform or underperform compared to a well-optimized single propeller system. The primary advantage of a tandem propeller is that it will have a smaller diameter to absorb the same amount of power as a single large propeller. In all cases, the emphasis is on the design and optimization of the propeller configuration to give a required level of performance, including constraints such as size, weight, costs, etc.

General Propeller Performance

A better way to predict a propeller’s performance is by using the blade element theory, which can also help design the shapes of propeller blades to give the best efficiency. Developing a computer program to calculate the performance of a propeller using the blade element method is a relatively straightforward task. The principle used here is calculating the angle of attack and the corresponding lift and drag at each propeller blade section, as shown in the figure below. All blades are twisted along their span from root to tip (which is done to give good aerodynamic efficiency), so the local value of blade pitch differs from point to point along the span.

Notice that the relative flow angles and, hence, the angle of attack of the blade element are obtained by using the vector addition of the relative velocity components, i.e., the vector sum of the rotational velocity, , of the blade element as it rotates about the shaft and the free-stream velocity, , where the span position from the rotational axis is denoted by . This combination yields what is usually called the velocity triangle, which helps visualize and calculate the relative flow angles.

In the propeller theory, it is usually assumed that the induced velocity, , is small compared to the free-stream (flight) velocity, i.e., , and can be neglected. However, at lower airspeeds, a good approximation for the average induced velocity over the propeller disk can be obtained using Eq. 15, i.e.,

(54)  

Because is a function of thrust, , and thrust is a function of , including the induced inflow requires an iterative approach.

Theoretical Development

The local pitch of the blade is as it varies along the span, so the angle of attack of any blade section is

(55)  

The lift coefficient on the blade element then follows as the product of the angle of attack of the blade and the local lift-curve slope of the airfoil section, i.e.,

(56)  

where is measured from the zero-lift angle and recognizing that the lift-curve slope will depend somewhat on the shape of the blade section used, as well as the local incident Mach number at the section and to some extent Reynolds number too. The lift on a blade element will be

(57)  

where is the local chord of the blade ( is the area of the blade element), and the resultant local velocity at the blade element, , is

(58)  

When the lift on all the sections of the propeller blade is obtained, then the net thrust of the propeller can be obtained by resolution of the local lift vectors in the direction of the thrust component followed by spanwise integration, i.e.,

(59)  

where is the number of propeller blades, and is the propeller’s radius. The component of the drag as it affects the thrust can be ignored. Also, it is reasonable to assume that there is axisymmetry in the problem. Hence, the angles of attack at any given blade station are the same for any rotational angular position of the propeller blades. The integration process is usually performed numerically.

By analogous arguments, the power required to rotate the propeller will be

(60)  

where is the local profile drag that again depends on the airfoil shape used on the blade and the incident Mach number. Notice the inclusion of the moment arm to get to the torque (torque is a moment, so the product of a force times a distance or “arm”), and power is just the product of torque and angular velocity. In this case, however, the component of the lift (which is the induced drag) will contribute to the net drag of the section and, hence, the torque and power required to rotate the blades.

If the section drag coefficient is known, then

(61)  

However, the challenge with a propeller is to properly represent the section because it depends on the Mach number, in this case, the helical Mach number, i.e.,

(62)  

where is the local speed of sound, i.e., ). Of course, suppose the helical Mach number becomes too large and exceeds the drag divergence Mach number of the airfoil sections. In that case, the propulsive efficiency of the propeller will decrease rapidly.

Numerical Implementation

The propeller blade can be discretized into segments, as shown in the figure below. To give any reasonable definition of the spanwise aerodynamic loads, should be between 30 and 100. Each segment is then small enough to assume constant flow properties. Let denote the spanwise position of the -th segment, where , and let be the spanwise width of each segment.

For each segment, the resultant local velocity, , is computed using

(63)  

and so the inflow angle is

(64)  

The local angle of attack is then obtained using

(65)  

The local lift coefficient, , is

(66)  

If the drag coefficient is known or assumed, e.g., , it can be used directly; otherwise, the normal process is to interpolate from airfoil data tables (i.e., a table look-up approach) based on the local helical Mach number . Reynolds number effects can also be included if such data are available.

Computing the lift force, , and the drag force, , on each segment gives

(67)  

Finally, the thrust and power contributions from all segments are then summed up using

(68)  

which, for simplicity of illustration here, uses the trapezoidal rule of integration.

Propeller Performance Charts

Propeller performance curves are presented in the form of thrust coefficient , power coefficient , and propulsive efficiency , respectively, analogous to how airfoil and wing aerodynamic coefficients are used.^[10] All propellers have their characteristics quantified in this manner, and there is a separate set of curves for each reference blade pitch angle. By convention, the reference pitch angle is not the angle of attack of the blade sections of the propeller but the reference pitch at the 75% blade span (i.e., at ), which is a geometric quantity and can be measured.

Helical Pitch and Slip

Often the pitch of a propeller is measured in units of length, which refers to the helical pitch that the reference blade section traces out during one revolution, i.e., like a screw thread. Hence, the old name of a propeller, which is known as an airscrew. The helical pitch is the actual distance the propeller advances along its axis in one complete revolution, considering the effects of fluid resistance. If the propeller were moving through a perfectly rigid medium (like a screw in a solid material), it would advance precisely the distance of its geometric pitch per revolution, as shown in the figure below. However, the propeller covers a shorter distance because air has viscosity and can be deformed.

The helix angle, , describes the inclination of the path traced by a point on the propeller blade relative to the plane of rotation. A point on the propeller blade at radius moves in a circular path with a circumferential distance per revolution given by

(69)  

Simultaneously, the propeller advances axially by , which is given by

(70)  

where is the airspeed and is the rotational speed in revolutions per second. The helix angle is the angle between the helical path and the plane of rotation, given by

(71)  

Substituting the values gives

(72)  

For a reference radius and diameter , the standard form of the helix angle equation is

(73)  

where is the propeller’s diameter.

The propeller slip describes the difference between the geometric pitch (theoretical advance per revolution) and the actual advance in air. This effect occurs because air is not a solid medium, and some of the propeller’s effort is lost to the deformation of the air. The propeller slip ratio, , is given by

(74)  

where is the geometric pitch, i.e., the distance the propeller would ideally move in one revolution, and is the actual pitch, i.e., the actual distance the propeller moves forward in the air per revolution.

A high slip ratio indicates that the propeller is inefficient in converting rotational motion into forward thrust, whereas a low slip ratio (near zero) means the propeller is operating efficiently. Ideally, the actual advance per revolution would equal the geometric pitch. In practice, there is always some loss of efficiency. The slip ratio is just one way of measuring propeller efficiency, but it has no particular aerodynamic significance and is not the same as its propulsive efficiency.

Propeller Dimensions

Propeller dimensions are typically given in inches because of historical conventions in U.S. aviation, where USC units remain the industry standard for aircraft manufacturing and maintenance. This ensures consistency with aircraft manuals, FAA regulations, and existing designs, making it easier for pilots, mechanics, and manufacturers to communicate and compare specifications. Diameter and pitch are expressed in inches. For example, the Sensenich 69CK propeller has a 69-inch diameter and is available in a helical pitch range from 42 to 58 inches, so 69 x 42, 69 x 44, etc. While metric conversions are used internationally, general aviation favors inches for compatibility and ease of use.

Advance Ratio

The propeller advance ratio, , is a fundamental aerodynamic parameter that helps determine the efficiency and effectiveness of a propeller in producing thrust under different operating conditions. It connects forward motion, rotational speed, and propeller design, serving as a crucial non-dimensional parameter in propeller and rotor analysis. The advance ratio, , is defined as

(75)  

where is the airspeed (in units of m/s or ft/s), is the rotational speed of the propeller (revolutions per second, fps), and = diameter of the propeller (in units of m or ft). Physically, the advance ratio can be interpreted as the distance traveled by the propeller per revolution in terms of diameters, i.e., a non-dimensional distance or advance.

Thrust Coefficient

Recall that the thrust coefficient for a propeller is defined as

(76)  

where is the thrust generated by the propeller, is the propeller’s diameter, and is the number of revolutions of the propeller per second. If the rotational angular velocity of the propeller is , then . Normally, the results for and other propeller characteristics are plotted as a function of the advance ratio, .

Representative versus results for a propeller are shown in the figure below, with a separate curve for each reference blade pitch. Notice that results from the blade element method discussed previously (i.e., the “theory”) agree well with the measurements.

Power Coefficient

Recall that the corresponding power coefficient for the propeller is defined as

(77)  

where would be the brake power, i.e., the power delivered to the propeller through the driving shaft. Normally, the torque, , would be measured, so then . The corresponding versus results for a propeller are shown in the figure below. Again, notice that the theory is in good agreement with the measurements.

Propulsive Efficiency

Propeller efficiency measures how effectively the propeller converts available power into useful thrust. It is defined as the ratio of useful power output (thrust power) to the total power input, i.e., engine power at the shaft or “brake” power. Propellers are typically designed for optimal performance at specific operating points. The propulsive efficiency of a propeller is defined as

(78)  

which is just a non-dimensional statement that the propeller’s efficiency is the ratio of the useful power to the input power. Notice that the symbols or are used to denote propeller efficiency, being preferred in most contexts to avoid any possible confusion with the efficiency of the propulsion system as a whole.

Using the previous definitions of and then

(79)  

The corresponding representative versus results for a propeller are shown in the figure below, again with one curve for each reference pitch angle.

Notice that for a given propeller operated with any given blade pitch and rotational speed , its propulsive efficiency increases with increasing forward airspeed to reach a maximum and then diminishes rapidly. Consequently, a propeller of a given (fixed) blade pitch cannot operate with high propulsive efficiency over a wide range of values of (or airspeed for a given rotational speed).

This latter outcome occurs because propeller blades are wings (rotating wings). All wings can only operate aerodynamically over relatively small ranges of the angle of attack, i.e., local blade section angles of attack between 2 to 14 degrees, depending on the local Mach number and Reynolds number. The local sectional angles of attack on the propeller depend not only on the rotational speed of the propeller and airspeed or advance ratio but also on how the propeller is twisted along its span. So, as airspeed changes, (propeller speed in terms of revolutions per second) is assumed constant. Then, the blade pitch must be increased to progressively maintain the angle of attack on the propeller. By gradually increasing the blade pitch, the best efficiency can be obtained over a much wider range of airspeeds, which is precisely the purpose of a continuously variable pitch or “constant speed” propeller.

Further Discussion of Propeller Performance

With this understanding of the flow at the blade section and the creation of thrust from the propeller, the various curves of , , and , as shown previously, can now be explained in greater detail. At low values of , the corresponding angles of attack of the blade sections are relatively high. So, the blade sections produce relatively high lift but are close to the point of stall. The propeller still produces thrust but requires high power and is inefficient. As airspeed and increase, the blade sections operate at lower angles of attack and closer to their best section lift-to-drag ratios. As a result, the thrust is maintained, but the drag on the blades decreases, so propulsive efficiency increases markedly.

The lowest angles of attack will produce little lift on the blades or thrust on the propeller. However, there is a range of airspeeds (assuming blade pitch does not change) for which good propulsive efficiency is obtained. Therefore, the best aerodynamic efficiencies will only be obtained when all blade sections (or most of them) operate at or near the angles of attack for their best lift-to-drag ratio, which is usually between 2 to 8 degrees, depending on the incident Mach number.

There is eventually a point at higher airspeeds (or high ) where the blade sections encounter diminished angles of attack and higher helical Mach numbers, simultaneously decreasing thrust and efficiency unless the blade pitch increases further. Eventually, the blade pitch cannot be mechanically increased to improve efficiency, so the efficiency drops off.

The results also explain the differences in propulsive efficiency of fixed-pitch versus constant-speed propellers, as shown in the figure below. Notice that if the propeller pitch is fixed, its propulsive efficiency increases slowly with airspeed, reaching a maximum and then decreasing rapidly. The relatively low efficiency of a fixed-pitch propeller at low airspeeds means that the aircraft’s takeoff and climb performance will be relatively poor. Further increases in airspeed beyond the airspeed for peak efficiency will cause propeller efficiency to decrease precipitously. This behavior decreases the power delivered to the airstream as useful work, effectively setting an upper barrier to the airspeed achievable by the airplane.

The preceding situation is very different for a variable pitch or constant speed propeller, which can be set into a fine pitch for takeoff, giving good propulsive efficiency and low airspeed and giving the airplane markedly better takeoff and climb performance. As airspeed builds, the blade pitch can be increased to maintain a constant rpm schedule so the propeller efficiency can closely follow the envelope of peak efficiency. This reason is why airplanes with constant-speed propellers have much better overall flight performance and can cruise at much higher airspeeds, as shown in the figure below. A constant-speed propeller also maintains a steady load on the engine, which is essential for long engine life.

Worked Example #1 – Calculating propeller performance

Refer to the propeller charts below, which are a standard performance presentation for all propellers. Assume that an actual propeller has a diameter of 7 ft and is a constant-speed propeller with a rotational speed of 2,000 rpm. The propeller operates at an equivalent of 8,000 ft ISA density altitude.

For each blade pitch angle measured at 75% radius and at the point of maximum propulsive efficiency in each case, estimate the following:
(a) What are the advance ratio values and corresponding airspeed values?
(b) What are the propeller thrust coefficient values and the propeller’s corresponding thrust?
(c) What are the values of the propeller power coefficient and the corresponding shaft torque and power required to rotate the propeller?

Show solution/hide solution.

(a) At the peak efficiency, the values of the advance ratio can be read off the first chart. We can easily do this to two decimal places; the chart can be digitized for better accuracy. We are also given information about the specific propeller, which is relatively small and would likely be for a general aviation aircraft, so in each case, we can calculate the corresponding airspeed for a given value of , i.e.,

   

so

   

It is best to use a table to show the results, i.e.,

Blade pitch ()			(ft/s)
15	0.82	0.65	151.7
20	0.85	0.82	191.3
25	0.87	1.04	242.7
30	0.87	1.25	292.7
35	0.86	1.45	338.3
40	0.86	1.70	398.7
45	0.84	1.95	455.0

(b) The propeller thrust coefficient can be read off the second chart for each value of the advance ratio, as was identified in the previous part. The thrust coefficient for a propeller is defined as

   

so the corresponding thrust (in units of force) from the propeller is

   

We are told that the propeller operates at the equivalent of 8,000 ft ISA density altitude, so according to the ISA equations, the density at this altitude is 0.001869 slugs ft. Inserting the information gives

   

Again, it is best to use a table to show the results, i.e.,

Blade pitch ()
15	0.65	0.025	124.7
20	0.82	0.038	189.5
25	1.04	0.040	199.4
30	1.25	0.047	234.4
35	1.45	0.052	259.3
40	1.70	0.060	299.2
45	1.95	0.072	359.0

(c) The propeller power coefficient can be read off the third chart for each value of the advance ratio identified in the previous part. The power coefficient for a propeller is defined as

   

so the corresponding power needed to drive the propeller is

   

Inserting the known information gives

   

where we have converted to horsepower (hp) by dividing the result in ft-lb s by 550. Again, it is best to use a table to show the results, i.e.,

Blade pitch ()			(hp)
15	0.65	0.022	46.5
20	0.82	0.035	74.0
25	1.04	0.048	101.3
30	1.25	0.065	137.5
35	1.45	0.09	190.3
40	1.70	0.12	253.8
45	1.95	0.17	359.7

Helical Tip Mach Number

As airspeed and the values of increase, a concern is that the tip speed of the propeller can approach supersonic conditions. If this condition occurs, the propeller will lose propulsive efficiency and also produce more noise. This reduction in efficiency means that more power is required to produce the same amount of thrust, which can adversely affect fuel consumption and overall flight performance. Supersonic airflow generates shock waves, resulting in a dramatic increase in noise levels from a propeller. A propeller’s noise is a significant concern in aviation, impacting passenger comfort, community acceptance, and regulatory compliance.

Let be the helical tip velocity based on the vector sum of the rotational and airspeed components at the blade elements. Therefore, the rotational tip speed of a propeller characterized by radius or diameter (= 2) is

   

and so the helical tip speed of the propeller is

   

where is the forward airspeed. Therefore, the helical Mach number will be

   

where is the speed of sound at the atmospheric conditions at which the propeller operates.

The measurements shown in the figure below, which is for a high-speed NACA propeller with thin blades, eventually reach supersonic helical tip Mach numbers. In this case, the measurements were made at constant blade pitch for variations in rotational speed. Notice that rapid losses in propulsive efficiency (from a peak of about 0.95) are produced at higher airspeeds if the rotational tip speeds are kept at high values. Increasing compressibility losses begin to appear when helical tip Mach numbers of about 0.8 are reached. While every propeller will be different, exceeding some limiting critical tip Mach number at the blade tip, which may be around 0.7 for a standard blade and 0.8 for a thin airfoil section and swept leading edge at the tip, can be expected to decrease propulsive efficiency, increase noise, or, more usually, both.

Propellers produce spiraling sound waves emanating from each of the blades, their noise being the most intense in the propeller’s plane of rotation. As supersonic tip Mach numbers are approached, the waves coalesce and become a shock wave, so the amplitude, impulsiveness, and harshness of the noise increase markedly. These intertwining spiral waves are the propeller equivalent of the Mach cone produced by a body following a rectilinear trajectory, e.g., the sonic boom. The schlieren flow visualization image below shows the phenomenon for a tip Mach number of 0.87. The tip vortex and the turbulent wake behind the blade are also visible, both of which will contribute to the “high-speed” noise produced by the propeller.

The fundamental frequency of propeller noise occurs at the product of the rotational frequency and the number of blades, i.e., at . However, higher frequencies are also produced by other sources, such as the turbulent wake. These frequencies tend to be in the range where the human ear is most sensitive to. Measurements of propeller noise suggest that the noise in terms of SPL (Sound Pressure Level) for a given thrust is linearly proportional to the tip Mach number, i.e.,

(80)  

which seems to hold even for .

The main aerodynamic affecting propeller performance is shock-induced flow separation rather than the presence of shock waves per se. Therefore, this operating condition limits the propeller’s useful operating envelope, and at the tip must be kept below a certain threshold, depending on the blade design. Today, thin, swept propeller blades are common, which will operate at the lowest possible values of tip speed, ,  to meet the propulsion and noise requirements.

Example #2 – Calculating a propeller’s helical tip Mach number

Refer back to Worked Example #1. What are the propeller’s helical tip speed values and helical Mach number? Comment on your results. Note: The helical tip speed is the vector sum of the rotational speed at the tip of the propeller and the free stream (airspeed).

Show solution/hide solution.

Let be the helical tip velocity based on the vector sum of the rotational and airspeed components. The rotational tip speed of a propeller of diameter is

   

where and so the helical tip speed of the propeller is

   

where is the forward airspeed. The helical Mach number will be

   

where is the local speed of sound at the conditions at which the propeller operates. At 8,000 ft ISA density altitude, 1085.3 ft/s.As a final table to show the results, the helical tip speed and Mach number are:

Blade pitch ()		(ft/s)	(ft/s)
15	0.65	151.7	748.6	0.69
20	0.82	191.3	757.6	0.70
25	1.04	242.7	772.2	0.71
30	1.25	292.7	788.9	0.73
35	1.45	338.3	807.3	0.74
40	1.70	398.7	833.5	0.77
45	1.95	455.0	862.8	0.79

We notice that for an airspeed above 400 ft/s, the propeller blade tips would likely begin to operate near or just beyond the critical Mach number (i.e., the onset of transonic flow), which for the thin tips of propeller blades is about 0.8. Under these conditions, the propeller will likely lose propulsive efficiency.

Speed Power Coefficient (Weick) Method

The speed power coefficient, , is a non-dimensional propeller parameter first used by Fred Weick at NACA that can be used to help characterize the performance of a propeller that is optimally matched to its aerodynamic and operational conditions. The Weick method provides a means to normalize propeller performance data across different flight speeds, power requirements, and propeller diameters, facilitating the selection of an optimal propeller size and pitch for a given application.

Weick^[11]noted that while the , , and coefficients help compare the thrust, torque, and power of different propellers, “they are not particularly helpful, however, in selecting or designing propellers to fit certain specific requirements for driving some form of airplane.” Furthermore, Weick noted that “the parameter is not a fair basis upon which to compare the efficiencies of various propellers that might fit the same aircraft at the same values of and but have different diameters and, therefore, other values of .”

Propeller testing often involves using a series of geometrically similar designs, with variations in pitch applied in uniform increments, as previously shown and discussed. For example, the NACA tested propellers in their 20-foot wind tunnel using a rig that allowed continuous adjustments in pitch over an extensive range of angles. For a fixed-pitch propeller of a given blade design, the range of pitch angles may be more limited, perhaps only three or four. From these tests, an appropriate propeller can be selected to meet specific operating conditions (, , , and ) required to propel a given aircraft. To make this selection process possible, using a coefficient that encapsulates these critical parameters without dependence on the propeller’s diameter or physical dimensions is advantageous. Therefore, a coefficient based solely on (, , and ) provides a consistent and equitable metric for evaluating the efficiency of different propellers intended for the same aircraft.

Weick showed that new propeller coefficient involving , , and without may be obtained from and using

(81)  

where is the air density, is the true airspeed of the propeller through the air, is the shaft power available to the propeller, and is the propeller’s rotational speed in revolutions per second. Notice that the values of must be calculated using consistent engineering units. is called a speed-power coefficient because it contains the rotational and forward speeds and the power absorbed.^[12] This form, however, covers an extensive range of values, making its use impractical. the fifth root of the expression, where

(82)  

A fixed-pitch propeller’s diameter and pitch selection are based on identifying the optimal balance between efficiency and operational constraints in a specific design condition for the aircraft. Because a fixed-pitch propeller operates most efficiently at a single point in its performance envelope, the coefficient can help determine the best-matched propeller for the desired airspeed and power available.

The selection process involves computing the values of as a function of the advance ratio . The assumption, of course, is that these data (or data for a similar candidate propeller) are available. The coefficient can be evaluated using

(83)  

An example is shown in the figure below, the data values being obtained from wind tunnel tests as previously discussed, gives all the values of as a function of along with the envelope of peak propulsive efficiency for different blade pitch angles.

Once the values are determined, the corresponding advance ratio =  can be obtained from the efficiency curves, the idea being to work along the curves of peak propulsive efficiency in terms of . The resulting curve, which is marked as the “Curve of best efficiency” in the figure above, can then be used to guide the determination of an appropriate propeller diameter and pitch. This approach provides a basis to ensure that the propeller will operate within its optimal efficiency range at the given airspeed and propeller rotational speed.

For a fixed-pitch propeller, the blade angle or pitch selection is always a compromise. It is necessary to choose a blade angle that shows peak efficiency at a somewhat smaller value of than the one calculated for the flight condition. Because the pitch of a fixed-pitch propeller cannot be adjusted in flight, careful selection is necessary to prevent performance degradation at off-design conditions, e.g., at lower or higher airspeeds. The corresponding value of can then be determined. The needed propeller diameter is then determined using

(84)  

Notice that the use of the coefficient in propeller selection provides one systematic method to ensure that the chosen propeller maximizes its thrust and efficiency while remaining within the other constraints of the aircraft’s design.

Low Reynolds Number Propellers

Low Reynolds number propellers are designed to perform efficiently in flow regimes characterized by Reynolds numbers based on blade sectional chord at 75% radius, typically below 10⁵. Such operating conditions are generally found on small airplanes, UAVs, and drones. In this regime, the blade sectional drag values are higher than at chord Reynolds numbers above 10⁶, which significantly impacts the thrust and lowers the overall propulsive efficiency of the propeller.

For example, the results shown in the figure below suggest that the turbulent flat-plate solution is a good approximation to the viscous (shear) sectional drag on airfoils over a fairly wide range of chord Reynolds numbers for above . In this case, it will be sufficient to assume that the zero-lift drag coefficient is given by

(85)  

where is the reference chord Reynolds number for which a reference value of drag is known for a given airfoil section; not all airfoils will have known (measured) drag coefficients at all Reynolds numbers, but this approach allows the drag coefficient for other Reynolds numbers to be estimated with good confidence.

However, it will be seen that the aerodynamic behavior at lower chord Reynolds numbers leads to higher drag coefficients, perhaps by nearly an order of magnitude. Changing the scaling coefficient from -0.2 to -0.4 in Eq. 85 gives a good approximation (line fit) based on the results shown over a lower range of if a reference drag coefficient is known. However, the overall sectional drag behavior of airfoils in this regime tends to be less predictable, especially at other than shallow angles of attack, because the boundary layers are thicker, often with laminar separation bubbles, and the onset of flow separation and stall occurs more readily.

Therefore, the impact of these airfoil section effects on propeller performance at low chord Reynolds numbers is profound. For example, the figure below shows the efficiencies of a two-bladed propeller at “full-scale” and at low Reynolds numbers. Notice the marked drop in the maximum efficiency at lower Reynolds numbers and the range over which it can operate efficiently. Most of these effects on propulsive efficiency are primarily related to the higher sectional drag coefficient values at these low Reynolds numbers.^[13] However, there is also a reduction in maximum sectional lift values, which limits thrust production. These considerations are essential for propeller applications to most UAVs and drones, which can expect maximum propulsion efficiencies of between 50% and 60% rather than 80% or higher for “full-scale” propellers.

Unique low Reynolds number airfoil designs are typically employed to minimize sectional drag on smaller propellers, much in the manner done for fixed wings, but these must be designed carefully. A larger solidity (blade area) also assists in generating the necessary thrust under these conditions, perhaps at the expense of some other losses in propulsive efficiency. Designing low Reynolds number airfoils suitable for propellers often involves using advanced computational fluid dynamics (CFD) simulations to analyze and optimize performance. These simulations help engineers understand the flow behavior around the propeller blades and improve design parameters. Wind tunnel testing is then essential for validating the design assumptions and performance predictions.

Selecting Propellers for a Drone

Selecting the correct propeller(s) for a UAV or drone involves several interrelated factors, such as the drone’s size, weight, its aerodynamic characteristics, motor characteristics and performance, and the desired flight performance of the drone such as airspeed, range, endurance, etc. A VTOL-capable drone may use a separate set of propellers for lifting, and another for propulsion, i.e., a hybrid concept, where the lifting propellers are switched off in forward flight, and the lift to overcome the drone’s weight is generated by a conventional fixed wing. In some cases, the propellers may be used for both lifting and propulsion, e.g., a tiltrotor or tilting wing concept, which will require further compromises in selecting the best propeller. The assumption is that such propellers are of fixed pitch only, so the selected propeller(s) will inevitably be a compromise.

Basic Selection Process

The following are the basic steps to select the appropriate propeller(s). Inevitably, without access to a test rig and a wind tunnel, the process will involve trial and error and good judgment. Fortunately, at this smaller scale, propellers are inexpensive, so the final decisions may be made based on flight testing using a selection of different propellers until the best level of performance is obtained.

1. To determine the propeller requirements, consider the drone’s weight, aerodynamic efficiency (e.g., lift-to-drag ratio), estimated motor specifications, battery type, voltage, and desired flight characteristics. Heavier drones will need larger diameter and higher-pitch propellers to generate sufficient thrust to overcome their drag. A VTOL drone will need enough thrust from the lifting propellers to overcome the drone’s weight.
2. Calculate the thrust and power requirements. A thrust-to-weight ratio of about 2:1 is necessary for good hovering, forward flight, and maneuvering performance for a quadcopter drone. For example, if a drone has a mass of 1 kg, the design thrust must be capable of carrying 2 kg, e.g., 500 g per motor/propeller for a quadcopter. The power requirements can be estimated using the momentum theory, starting from an initial estimate of the needed propeller diameter.
3. Match the initial size and pitch of the propeller to the motor’s recommended range, which includes considering the motor’s power or “KV rating.” Different propellers can dramatically affect the drone’s speed, maneuverability, and other flight characteristics.
4. A propulsive propeller will require significantly less thrust than the drone’s weight, depending on the drone’s lift-to-drag ratio. The goal is to obtain the highest level of propulsive efficiency at the propeller’s “” value for the combination of thrust required, propeller rpm, and airspeed. To this end, propeller charts., i.e., versus , versus and versus should be consulted. The power speed coefficient, , should be evaluated.
5. Review the small-scale propeller measurements made by Professor Selig and his students at the University of Illinois. They are available in chart form and as data files. They cover a comprehensive range of propellers that could be suitable for drones. Measurements are available in the static thrust condition (i.e., hover) and forward flight over a range of values. Remember that small propellers have relatively low thrust-producing efficiencies of about 50%, impacting flight duration for a given battery charge.
6. Consult the motor specifications. The motor’s datasheet may recommend a propeller size (diameter) and pitch. Manufacturers often provide a range of suitable propellers for their motors. It is usually wise to stay within the recommended ranges, although other propellers may also be appropriate.
7. Decide on the propeller geometry. Propellers are specified in terms of diameter x pitch. For example, a propeller marked as 20 x 14 means it has a diameter of 20 inches and a helical pitch distance of 14 inches; the propeller could also be specified as cm units. Propellers come in quantum sizes such as 12 inches, 14 inches, etc. Lifting (hovering) propellers will have relatively low pitches, whereas propulsive propellers will have higher pitches. The faster the drone must fly, the higher the propeller pitch needed for good propulsive efficiency.
8. Decide on the propeller material. Plastic propellers are lightweight and inexpensive but less durable. Carbon fiber propellers give better aerodynamic performance but are significantly more expensive. It is often a good choice to buy and test three inexpensive propellers at first, perhaps with higher and lower pitches than the first choice, before settling on a carbon fiber propeller.

Hovering Propeller

Propellers generally have low static thrust or hovering efficiency compared to helicopter rotors, in part because of their typically higher disk loadings and non-optimal blade designs. From momentum theory, the induced velocity at the propeller’s disk is

(86)  

where is the thrust required. The ideal induced power required is

(87)  

Expressing this result in terms of the propeller thrust coefficient, i.e., , gives

(88)  

Therefore, the ideal induced power coefficient using the propeller definition is

(89)  

Accounting for induced losses with the induced power factor , the actual induced power coefficient is

(90)  

For the profile power, the coefficient remains the same as the helicopter case, i.e.,

(91)  

where is the solidity of the propeller.

Finally, the total power coefficient for hovering flight using the modified momentum theory is

(92)  

Alternatively, the power coefficient can be written in terms of the figure of merit, , as

(93)  

As shown by the power polar in the figure below, with suitable values for and or , the modified momentum theory can give good predictions of hovering performance. The low Reynolds number combined with the typical (excessive) spanwise blade twist and non-optimal chord distribution used on these propellers generally gives relatively poor hovering performance. In this case, = 1.35 and = 0.2 ( 0.5) gives good agreement with these measurements. Notice that increasing propeller pitch produces more thrust for the same rpm; the resulting measurements then move up the curve to higher values of and . Combined with the effects of rpm are the effects of the Reynolds number, which slightly increases thrust and reduces power at the higher values of rpm.

Propulsion Propeller

Selecting a propeller for a drone to meet its propulsion requirements must be performed more rigorously and requires access to propeller charts, i.e., thrust coefficient versus advance ratio , power coefficient versus , and efficiency versus . Such charts may come from wind tunnel tests or calculations using a propeller model (code). Propulsive propellers for drones, with their high rpm values and lower airspeeds (compared to a conventional airplane), operate at relatively low values, typically less than unity.

The figure below shows wind tunnel measurements for a small-scale propeller with three different pitch values that are being considered to meet propulsive requirements. At low values of , the propeller blades may experience stall because of excessive pitch relative to the incoming airflow. As airspeed builds and increases, the blade angles of attack decrease, and the propeller starts to produce useful thrust, also with increasing propulsive efficiency. At some point, the propeller reaches its maximum efficiency. Beyond this, for a given rpm, the blade section angles of attack continue to decrease, reducing the lift produced by the blades and the net thrust from the propeller. Eventually, the thrust reduces to zero, and at sufficiently high for a given blade pitch, the propeller starts to generate negative thrust, acting as a drag element.

The idea in selecting a fixed-pitch propeller is to operate it at or near its best propulsive efficiency. The advantage of non-dimensional coefficients, including the power speed coefficient, is that the results can scale to different propeller diameters, at least within some range where the Reynolds numbers do not change much, maybe up to 20%. Inevitably, oversized propellers will have to operate at values of that are too low to be efficient, and smaller propellers will have to operate at much higher values of (to give the required thrust). Higher blade pitch values will increase the value of , at which the maximum propulsive efficiency is obtained. The process is inevitably iterative, based on the power required for flight (and hence the drag of the drone), the final propeller pitch being a compromise but is nonetheless a point design, hopefully with some good operating margins in terms of its efficiency over a range of airspeeds.

Summary & Closure

Propellers are an effective and robust means of generating thrust, so they are widely used in aviation across various propulsion systems. Fixed-pitch propellers, while straightforward and lightweight, have inherent limitations in their efficiency across different flight conditions. Variable-pitch and constant-speed propellers expand the operational range, allowing for better efficiency at various speeds and altitudes, though they come with increased mechanical complexity, weight, and cost. Whether paired with piston engines, turboshaft engines like turboprops, or even electric motors, correctly matching the propeller to its specific powerplant is critical to achieving optimal performance.

A key trend in the design of modern airplanes is the shift toward smaller-diameter propellers with more blades. This configuration helps reduce blade tip speeds, minimize noise, and improve efficiency. Advances in composite materials, such as Kevlar and carbon fiber, have further transformed propeller design. These materials enable thinner, swept blades that enhance aerodynamic efficiency at higher tip Mach numbers and flight speeds while reducing weight, vibration, and acoustic footprint. Despite these innovations, wooden propellers remain in use for low-performance airplanes because of their cost-effectiveness and ease of manufacturing.

For UAVs and drones, propeller selection plays an even more crucial role in overall efficiency, endurance, and mission effectiveness. Fixed-pitch propellers are typically preferred because of their simplicity and reliability, but they require careful optimization. The relationship between blade pitch, advance ratio, and rotational speed must be fine-tuned to ensure efficient thrust production across various flight conditions. Non-dimensional coefficients, such as the power-speed coefficient, assist in scaling performance across different propeller sizes, but trade-offs are inevitable. The iterative nature of propeller design means that achieving the right balance between thrust production and power consumption is essential. Ultimately, the goal is to develop propellers that maintain high efficiency across various operating conditions while meeting noise, weight, and performance constraints. As UAV and drone technology advances, so will propeller innovations, pushing the boundaries of aerodynamic efficiency, power optimization, and operational versatility.