Aircraft Propellers

J. Gordon Leishman

42 Aircraft Propellers

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 leading the development of new aircraft. Even earlier 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 was an aerodynamic device derived from the concept of an Archimedes water screw, which some consider a precursor to the modern helicopter rotor. Though never built to its intended human-carrying scale,^[1] Da Vinci’s design laid down the concept of using rotating wings to create an aerodynamic force, which is referred to as thrust.

The “aerial screw” can be found in Paris Manuscript B, folio 83v of Leonardo da Vinci’s unpublished and fragmented notebooks.

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 if sustained flight was ever to be achieved. At least one of his airplane concepts and an airship, the latter being shown in the illustration below, indicate the means of 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, but the designs suitable for good efficiency in 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 thinking about how an airplane might be propelled.

A Cayley airship concept used twin “airscrews” as a means of propulsion.

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. 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 large aspect ratio blades with nose-down twist (washout) along their length gave them good propulsive efficiency. By 1908, they had improved their propeller designs for the Wright Flyer III, enabling longer flights. Their “raked” propeller had 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.

Early airplane propellers were relatively primitive and developed by trial and error.

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 cracking and delamination. They were also very vulnerable to damage from bullets in air-to-air combat, with a single bullet through a wooden blade often causing its catastrophic failure.

An early wooden propeller used on a WWI vintage Sopwith Camel.

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.

With the establishment of dedicated research institutions like the NACA and the Royal Aircraft Establishment (RAE), the aerodynamic principles governing propeller performance began to be systematically quantified. Throughout the 1920s and 1930s, 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. The RAE also researched propellers extensively, focusing on theoretical aerodynamic analyses (mainly by Glauert^[3]) supported 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, made propellers lighter and stronger. Engineers soon began developing variable-pitch aluminum propellers, allowing pilots to adjust the blade angle for optimal performance during flight, significantly improving efficiency and control. 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 performance and shift to a higher pitch for efficient cruising flight. This innovation led to the constant-speed propeller, which automatically adjusted blade pitch using a governor or constant-speed unit^[4]to maintain a constant rotational speed and so continuously maximize the propulsive efficiency of the propeller

Hamilton Standard Propeller, controllable-pitch, two-blade, metal, Ruth Nichols (A19400027000). Studio photograph in support of "Pioneers of Flight" exhibit — Hamilton Standard controllable-pitch, two-blade, metal propeller circa 1932.

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

Iconic WWII aircraft like the versatile British Supermarine Spitfire and the multi-role American North American P-51 Mustang exemplified propeller and engine technology advancements. The Spitfire, with its powerful Rolls-Royce Merlin engine and Rotol advanced 4-bladed metal 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. The P-51, 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 and the Consolidated B-24 Liberator.

The Rotol 4-blade propeller was introduced in later Spitfire variants to improve performance by increasing thrust and efficiency.

The transition to using jet engines after WWII soon surpassed the use of propellers in military and commercial aviation. Nevertheless, propeller technology continued evolving for specific applications, including high-speed flight. The most significant interest in propeller development was for the turboprop, which uses a gas turbine to drive a propeller and offers 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 at regional airports were critical.

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

A 4-bladed propeller powered by a turboshaft engine provided a blend of power and propulsive efficiency for civil and military airplanes.

While the basic engineering principles used for aircraft propeller design have remained the same over a century, 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. Engineers continue to use techniques such as computational fluid dynamics (CFD) and computer-aided design (CAD) to optimize propeller performance 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 of composite materials, were developed in the 1980s. Like the swept wings used for high-speed aircraft, these types of propeller blades delay the onset of drag rise and the creation of wave drag, thereby allowing them to remain efficient propulsors at higher flight speeds. They also reduce noise, which is a significant concern for aircraft. This type of swept-blade propeller design is used in many modern turboprop aircraft, such as the ATR 72, the Bombardier Dash 8, and the Pilatus PC-12.

A modern constant speed (variable pitch) propeller designed for a high-performance twin-engine turboprop airplane.

Ongoing developments in propeller design for electrically powered aircraft focus on enhancing efficiency and noise reduction, 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.

Learning Objectives

Have an appreciation of the historical developments of propeller technology from the days of the Wright brothers to the present.
Understand the basic operational principles of a propeller, including the methods used to calculate its thrust, the power required, and its efficiency.
Be familiar with the momentum theory and blade element theory for calculating the performance of a propeller.
Know how to read and interpret a propeller performance chart.
Understand the operational advantages of variable-pitch propellers compared to fixed-pitch designs.

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, their airfoil shapes create aence between their upper and lower surfaces, resulting in blade lift and pressure differ drag. 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 rotation. Efficient propeller design maximizes this thrust by optimizing blade shape, angle of attack, and rotational speed, ensuring that the engine’s power is effectively converted into propelling an aircraft.

The propeller’s rotating blades are each designed to produce a lift force, so the propeller has a net thrust. A torque (and so power) is needed to do this, which is delivered through the propeller shaft.

A notable 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 is twisted along its length. This twist ensures that each part of the blade meets the fluid at an optimal angle of attack, maximizing efficiency and thrust across the entire blade length. As shown in the figure below, the local pitch angle changes from a relatively high value at the root (next to the hub) and progressively decreases in value from section to section when moving out to the blade tip. The net twist varies for different propellers, but washout angles may be as large as 40 degrees over the blade span; of course, these angles are much larger than those used on a wing. The primary purpose of using blade twist is to get the local angles of attack at each section along the span of the blade to be low enough such that it operates close to the aerodynamic conditions where the section is most aerodynamically efficient.

A propeller blade is inevitably twisted along its length, the twist being used to optimize the angles of attack for the best aerodynamic efficiency.

Fixed-pitch propellers have a very limited operating envelope; they are either good for takeoff and climb and poor for cruise, or good for a cruise and poor for takeoff and climb. The solution to this dilemma is to be able to change the average pitch angles of the blades^[5] allows the efficient operation of the propeller to be optimized as a function of the flight condition, as shown in the figure below. A fine (low) blade pitch is used for takeoff and landing, and a coarse (high) blade pitch is 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.

These velocity triangles show how a variable blade pitch allows for control of the thrust and overall performance of the propeller as airspeed builds, whereas a fixed-pitch propeller will show diminished thrust.

The highest section angles of attack attainable on the blades 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 and an increase in power.

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. 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 un-feathered stationary propeller is considerable, so an airplane with a fixed-pitch propeller may 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.

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. 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.

There are many types of propellers, including those with different numbers of blades. Counter-rotating coaxial propellers may be used for high-power applications.

Propellers with larger numbers of blades also tend to be relatively more efficient (when compared based on the same thrust and total blade area). However, the propeller’s net efficiency depends on other factors, including its operating rpm, tip Mach number, 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 using more blades and reduced diameters with lower tip Mach numbers.

Momentum Theory of the Single Propeller

The Rankine propeller theory,^[6] also known as the Rankine-Froude theory, 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 $T$ 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 $d\vec{S}$ points out of the control volume by convention. The upstream or free-stream (undisturbed) velocity is $V_{\infty}$ , and the pressure there is $p_{\infty}$ . It will be assumed it is uniform (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.

Control volume used for the momentum theory analysis of a single thrusting propeller.

The flow is entrained and accelerated into the propeller, so at the plane of the propeller, the flow velocity is $V_{\infty}$ plus an increment ${v}$ , i.e., the velocity there is $V_{\infty} + v$ . The static pressure will change there, too. The propeller works on the air to increase its momentum and kinetic energy, so the velocity is $V_{\infty} + w$ downstream. In the slipstream, the static pressure will also recover to ambient conditions, i.e., the pressure downstream will be $p_{\infty}$ .

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, $A$ , is known ( $A = \pi D^2/4$ where $D$ is the diameter of the propeller, so the mass flow rate through the propeller (and hence through the boundaries of the control volume) is

(1) $\begin{equation*} \overbigdot{m} = \varrho_{\infty} A (V_{\infty} + v ) \end{equation*}$

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.,

(2) $\begin{equation*} F_{\rm fluid} = -\overbigdot{m} V_{\infty} + \overbigdot{m} (V_{\infty} + w) = -\overbigdot{m} V_{\infty} + \overbigdot{m} V_{\infty} + \overbigdot{m} w = \overbigdot{m} w \end{equation*}$

where the direction of $F_{\rm fluid}$ is downstream, and the thrust on the propeller, $T$ , is in the opposite direction (pointing left in the figure) so that

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

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.,

(4) $\begin{eqnarray*} T (V_{\infty} + v ) & = & \frac{1}{2} \overbigdot{m} (V_{\infty} + w)^2 - \frac{1}{2} \overbigdot{m} V_{\infty}^2 \\ & = & \frac{1}{2} \overbigdot{m} (V_{\infty}^2 + 2 V_{\infty} w + w^2) - \frac{1}{2} \overbigdot{m} V_{\infty}^2 \nonumber \\[6pt] & = & \frac{1}{2} \overbigdot{m} ( 2 V_{\infty} w + w^2) \end{eqnarray*}$

so that the power required $P$ for the propeller to produce the thrust $T$ is

(5) $\begin{equation*} P = T (V_{\infty} + v ) = \frac{1}{2} \overbigdot{m} ( 2 V_{\infty} w + w^2) \end{equation*}$

Notice that the $T (V_{\infty} + v)$ term on the left-hand side of the above equation can be written as the sum of the $T \, V_{\infty}$ term, which is the work done to move the propeller forward (this is the useful work). The $T v$ term is the work done on the air or the induced power loss; the induced power is an irrecoverable power loss and so constitutes a loss in efficiency of the propeller in producing useful work.

Induced Velocity

The induced velocity in the plane of the propeller can be found by substituting Eq. 3 into Eq. 5, which gives

(6) $\begin{equation*} (V_{\infty} + v ) w = \frac{1}{2} ( 2 V_{\infty} w + w^2) \end{equation*}$

or simplifying gives

(7) $\begin{equation*} 2 V_{\infty}w + 2 v w = 2 V_{\infty} w + w^2 \end{equation*}$

This gives the relationship between ${v}$ and ${w}$ , i.e.,

(8) $\begin{equation*} w = 2 v \end{equation*}$

For the next step, take Eq. 3 and substitute in Eq. 1 (mass flow rate) and the connection between ${v}$ and ${w}$ ( $w = 2 v$ ) to get

(9) $\begin{equation*} T = \varrho_{\infty} A (V_{\infty} + v ) w = 2 \varrho_{\infty} A (V_{\infty} + v ) v \end{equation*}$

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

(10) $\begin{equation*} (V_{\infty} + v ) v = \frac{T}{2 \varrho_{\infty} A} \end{equation*}$

and expanding out gives

(11) $\begin{equation*} v^2 + V_{\infty} v - \frac{T}{2 \varrho_{\infty} A} = 0 \end{equation*}$

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

(12) $\begin{equation*} v = \frac{1}{2} \bigg( -V_{\infty} \pm \sqrt{V_{\infty}^2 + \frac{2T}{\varrho_{\infty} A}} \bigg) \end{equation*}$

for which there must be two roots, i.e., from the $\pm$ 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

(13) $\begin{equation*} v = \frac{1}{2} \bigg( -V_{\infty} + \sqrt{V_{\infty}^2 + \frac{2T}{\varrho_{\infty} A}} \bigg) \end{equation*}$

or non-dimensionally as

(14) $\begin{equation*} \dfrac{v}{V_{\infty} } = \frac{1}{2} \bigg( \sqrt{1 + \frac{2T}{\varrho_{\infty} A V_{\infty}^2}} - 1 \bigg) \end{equation*}$

Therefore, it will be apparent that for any given propeller producing a thrust $T$ , then its induced velocity $v$ will decrease rapidly with increasing airspeeds, i.e.,

(15) $\begin{equation*} v \rightarrow 0 \mbox{~~for larger values of~} V_{\infty} \end{equation*}$

Power Required

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

(16) $\begin{equation*} P = T (V_{\infty} + v ) = \underbrace{T \, V_{\infty}}_{\rm Useful~power} + \underbrace{T \, v}_{\rm Induced~power} = T \, V_{\infty} + \frac{1}{2} \bigg( - T \, V_{\infty} + T \, V_{\infty} \sqrt{1 + \frac{2T}{\varrho_{\infty} A V_{\infty}^2}} \bigg) \end{equation*}$

Notice that the useful power for propulsion is $T \, V_{\infty}$ , and the second term is an irrecoverable induced loss. Rearranging the preceding equation gives

(17) $\begin{equation*} P = T (V_{\infty} + v ) = \frac{T \, V_{\infty}}{2} + \frac{T \, V_{\infty}}{2} \sqrt{1 + \frac{2T}{\varrho_{\infty} A V_{\infty}^2}} = \frac{T \, V_{\infty}}{2} \left( 1 + \sqrt{1 + \frac{2T}{\varrho_{\infty} A V_{\infty}^2}} \right) \end{equation*}$

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

(18) $\begin{equation*} \dfrac{P}{T \, V_{\infty}} = \frac{1}{2} \left( 1 + \sqrt{1 + \frac{2 T}{\varrho_{\infty} A V_{\infty}^2} } \, \right) \end{equation*}$

Notice that in the limiting case when the airspeed, $V_{\infty}$ , becomes high, then

(19) $\begin{equation*} P \rightarrow T \, V_{\infty} \mbox{~~for larger values of~} V_{\infty} \end{equation*}$

meaning that the induced losses tend to zero, and 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, $P_0$ , will be required to overcome the drag of the blades so that the total power required, $P_{\rm tot}$ , can be written as

(20) $\begin{equation*} P_{\rm tot} = P + P_0 \end{equation*}$

Propulsive Efficiency

The propulsive efficiency of the propeller can also be derived from the preceding analysis. The useful power for propulsion is $T \, V_{\infty}$ so the efficiency of the propeller, $\eta_p$ , can be written as

(21) $\begin{equation*} \eta_p = \frac{T \, V_{\infty}}{T (V_{\infty} + v ) + P_0} \end{equation*}$

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 $v \rightarrow 0$ will be dictated by the profile losses on the blades, i.e.,

(22) $\begin{equation*} \eta_p \rightarrow \dfrac{1}{1 + \dfrac{P_0}{T \, V_{\infty}} } \quad \mbox{\small for higher airspeeds} \end{equation*}$

The profile power losses, denoted by $P_0$ , depend on the blade sections and on the blade area, i.e., the solidity or activity factor.

Activity Factor & Solidity

A parameter called the activity factor, $AF$ , was used in the earliest days of airplane propellers. Its definition has originated within the industry rather than in any scientific context. The $AF$ is considered a measure of the propeller’s “aerodynamic activity” or effectiveness in producing thrust and is defined as

$AF =\frac{N_b \, \bar{c}^{~2}}{D^{~2}}$

where $N_b$ is the number of blades, $\bar{c}$ is the reference chord length of the blade, and $D$ is the diameter of the propeller. A higher value of the activity factor indicates that the blades have more lifting area for generating thrust.

Solidity, given the symbol $\sigma$ , is a related dimensionless parameter, which is defined as the ratio of the total blade area to the disk area swept by the propeller, i.e.,

$\sigma = \frac{4 \, N_b \, \bar{c} \, D}{\pi D^{~2}} = \frac{4 N_b \, \bar{c}}{\pi \, D}$

The scientific literature commonly uses total solidity for the entire propeller and “local” solidity for a specific blade section.

Notice that a reference or “mean” chord is used in both parameters. Historically, the definition of the reference chord has been inconsistent, ranging from the local chord value at the 70%, 75%, or 80% radius location to a weighted chord based on the platform distribution. The most appropriate mean chord is a “torque” weighted chord based on

$\bar{c} = \dfrac{4}{R^4} \int_{0.15 R}^{R} c(y) y^3 \, dy$

where $y$ = 0.15 $R$ is a starting point that accounts for the presence of the propeller hub or boss. The reasoning for this $y^3$ weighted chord equation is that the blade sections near the tips are much more aerodynamically effective than those near the root end. In this regard, other definitions of the activity factor have been used, including

$AF = \dfrac{100,000}{R^4} \int_{0.15 R}^{R} c(y) y^3 \, dy$

Therefore, it is important to check the definitions used when dealing with data for propellers in any specific context.

Wake Swirl Effects

Incorporating the swirl in the momentum theory involves adding the tangential component of velocity to the otherwise purely axial flow otherwise considered. Rankine’s actuator disk theory considers only the axial momentum balance, assuming the flow is steady, inviscid, and incompressible. When swirl is included, the average angular velocity component of the downstream flow must now be considered as 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 a propeller, which, in effect, creates a spiral or corkscrew flow. 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, $\bar{w_t}$ , 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 $T$ generated by a propeller with a pure axial velocity component, $w$ , is given by

(23) $\begin{equation*} T = \overbigdot{m} \, w \end{equation*}$

where $\overbigdot{m}$ is the mass flow rate through the propeller. The power $P$ required to generate this thrust, considering the presence of an average swirl velocity, $\bar{w_t}$ , is

(24) $\begin{equation*} P = \frac{1}{2} \overbigdot{m} V_e \end{equation*}$

where the effective velocity, $V_e$ , is given by

(25) $\begin{equation*} V_e = \sqrt{w^2 + \bar{w_t}^2} \end{equation*}$

Therefore, the power required is

(26) $\begin{equation*} P = \frac{1}{2} \overbigdot{m} \sqrt{w^2 + \bar{w_t}^2} \end{equation*}$

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, $\bar{w_t}$ can be derived from the angular momentum imparted by the propeller blades. The torque, $Q$ , generated by the propeller can be calculated using the relationship between torque and power, i.e.,

(27) $\begin{equation*} Q = \frac{P}{\Omega} \end{equation*}$

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

(28) $\begin{equation*} Q = \frac{dL}{dt} \end{equation*}$

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

(29) $\begin{equation*} \bar{y} \approx \frac{2}{3} R \end{equation*}$

where $R$ is the propeller’s radius. Finally, the average tangential velocity, $w_t$ , is

(30) $\begin{equation*} \bar{w_t} = \frac{Q}{\overbigdot{m} \, \bar{y}} = \frac{Q}{\overbigdot{m} \left(\dfrac{2}{3} R\right)} \end{equation*}$

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 are methods that have been used to reduce wake swirl and improve propulsive efficiency.

Propeller Performance Coefficients

The thrust coefficient for a propeller is a non-dimensional thrust^[7]and is defined as

(31) $\begin{equation*} C_T = \frac{T}{\varrho_{\infty} \, n^2 \, d^4} \end{equation*}$

where $T$ is the thrust generated by the propeller, $d$ is the propeller’s diameter, and $n$ is the number of revolutions of the propeller per second. If the rotational angular velocity of the propeller is $\Omega$ , then $n = \Omega/2\pi$ . The rotational speed is often measured in terms of revolutions per minute or rpm, so rpm equals $60 \, n$ .

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

(32) $\begin{equation*} J = \frac{V_{\infty}}{n \, d} \end{equation*}$

Finally, the power coefficient for a propeller is defined as

(33) $\begin{equation*} C_P = \frac{P}{\varrho_{\infty} \, n^3 \, d^5} \end{equation*}$

where $P$ would be the brake power, i.e., the power delivered to the propeller through the driving shaft. In terms of the torque, $Q$ , then $P = \Omega \, Q$ .

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

Rotating-wing system

Thrust coefficient, $C_T$

Power coefficient, $C_P$

Advance ratio

Helicopter rotor

$C_T = \dfrac{T}{\varrho_{\infty} \, A \, \Omega^2 \, R^2}$

$C_P = \dfrac{P}{\varrho_{\infty} \, A \Omega^3 \, R^3}$

$\mu = \dfrac{V_{\infty}}{\Omega \, R}$

Propeller

$C_T = \dfrac{T}{\varrho_{\infty} \, n^2 \, d^4}$

$C_P = \dfrac{P}{\varrho_{\infty} \, n^3 \, d^5}$

$J = \dfrac{V_{\infty}}{n \, d}$

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

(34) $\begin{equation*} \dfrac{v}{V_{\infty}} = \frac{1}{2} \bigg( \sqrt{1 + \frac{8 \, C_T}{\pi \, J^2} - 1 }\bigg) \end{equation*}$

and for the induced power, then

(35) $\begin{equation*} \dfrac{P}{T \, V_{\infty}}= \frac{1}{2} \left( 1 + \sqrt{1 + \frac{8 \, C_T}{\pi \, J^2} } \, \right) \end{equation*}$

The corresponding propulsive efficiency is then

(36) $\begin{equation*} \eta_p = \frac{C_T \, J}{C_p} \end{equation*}$

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 of the flow and additional thrust generation. There are two conditions of interest:

The rear propeller is sufficiently separated that it works in the front propeller’s slipstream and induced velocity field.
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 $V_{\infty}$ to $V_{\infty} + v_1$ , and the second propeller further increases it to $V_{\infty} + v_2$ , the mass flow rates through each propeller are

(37) $\begin{equation*} \overbigdot{m}_1 = \varrho_{\infty} A_1 (V_{\infty} + v_1) \end{equation*}$

and

(38) $\begin{equation*} \overbigdot{m}_2 = \varrho_{\infty} A_2 (V_{\infty} + v_2) \end{equation*}$

Control volume used for the momentum theory analysis of the first type of tandem propeller system.

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

(39) $\begin{equation*} T_1 = \overbigdot{m}_1 v_1 \end{equation*}$

and

(40) $\begin{equation*} T_2 = \overbigdot{m}_2 v_2 \end{equation*}$

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

(41) $\begin{equation*} T_{\text{total}} = T_1 + T_2 \end{equation*}$

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

(42) $\begin{equation*} P_{\text{total}} = T_1 (V_{\infty} + v_1) + T_2 (V_{\infty} + v_2) \end{equation*}$

The addition of the profile power $P_0$ 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

(43) $\begin{equation*} \eta_p = \frac{T \, V_{\infty}}{T (V_{\infty} + v) + P_0} \end{equation*}$

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

(44) $\begin{equation*} \eta_1 = \frac{T_1 V_{\infty}}{T_1 (V_{\infty} + v_1) + P_{0,1}} \end{equation*}$

and

(45) $\begin{equation*} \eta_2 = \frac{T_2 (V_{\infty} + v_1)}{T_2 (V_{\infty} + v_2) + P_{0,2}} \end{equation*}$

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

(46) $\begin{equation*} \eta_{\text{total}} = \frac{(T_1 + T_2) V_{\infty}}{T_1 (V_{\infty} + v_1) + T_2 (V_{\infty} + v_2) + P_{0,1} + P_{0,2}} \end{equation*}$

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 $P_0$ 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

(47) $\begin{equation*} \eta_{\text{total}} \rightarrow \dfrac{1}{1 + \dfrac{P_{0,1} + P_{0,2}}{T \, V_{\infty}} } \quad \mbox{\small for higher airspeeds} \end{equation*}$

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

Tandem Propellers with Shared Induced Velocity

Now consider two tandem propellers positioned closely enough to share the same induced velocity increment $v$ , as shown in the figure below. The flow velocity at the propeller planes is $V_{\infty} + v$ . 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

(48) $\begin{equation*} \overbigdot{m} = \varrho_{\infty} A (V_{\infty} + v) \end{equation*}$

where $A$ is the area of each propeller disk, and it is assumed to be the same for both propellers.

Control volume used for the momentum theory analysis of the second type of tandem propeller.

The thrust generated by each propeller can be expressed as

(49) $\begin{equation*} T_1 = \overbigdot{m} v \end{equation*}$

and

(50) $\begin{equation*} T_2 = \overbigdot{m} v \end{equation*}$

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

(51) $\begin{equation*} T_{\text{total}} = T_1 + T_2 = \overbigdot{m} v + \overbigdot{m} v = 2 \overbigdot{m} v \end{equation*}$

The power required for each propeller is given by

(52) $\begin{equation*} P_1 = T_1 (V_{\infty} + v) = \overbigdot{m} v (V_{\infty} + v) \end{equation*}$

and

(53) $\begin{equation*} P_2 = T_2 (V_{\infty} + v) = \overbigdot{m} v (V_{\infty} + v) \end{equation*}$

Therefore, the total power required is

(54) $\begin{equation*} P_{\text{total}} = P_1 + P_2 = \overbigdot{m} v (V_{\infty} + v) + \overbigdot{m} v (V_{\infty} + v) = 2 \overbigdot{m} v (V_{\infty} + v) \end{equation*}$

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 $V_{\infty}$ . The efficiency of each propeller can be expressed as

(55) $\begin{equation*} \eta_1 = \frac{T_1 V_{\infty}}{T_1 (V_{\infty} + v) + P_{0,1}} \end{equation*}$

and

(56) $\begin{equation*} \eta_2 = \frac{T_2 (V_{\infty} + v)}{T_2 (V_{\infty} + v) + P_{0,2}} \end{equation*}$

Incorporating the profile power $P_0$ associated with the drag of the propeller blades, the overall efficiency of the tandem system can be expressed as

(57) $\begin{equation*} \eta_{\text{total}} = \dfrac{T_{\text{total}} V_{\infty}}{P_{\text{total}} + P_{0,1} + P_{0,2}} \end{equation*}$

which again comes to

(58) $\begin{equation*} \eta_{\text{total}} \rightarrow \dfrac{1}{1 + \dfrac{P_{0,1} + P_{0,2}}{T \, V_{\infty}} } \quad \mbox{\small for higher airspeeds} \end{equation*}$

Comparing Efficiencies

A common question is how a tandem propeller design compares to a single propeller in terms of “efficiency.” The answer lies in the basis used for comparison, e.g., constant net thrust, constant power, or constant solidity (blade area). There is nothing to suggest that one system is better than another, at least within the assumptions and limitations of the momentum theory, the net propulsive efficiency at high airspeeds being dictated by the profile power requirements of the blades.

When comparing efficiencies, consideration must be given to the added complexity of managing the power from both propellers and the overall weight efficiency of the propulsive system. 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 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 here is to calculate 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.

The velocity vectors at any blade element are combined vectorially to give the resulting angle of attack and flow velocity. The resulting angles are generally different at all blade elements 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, $\Omega y$ , of the blade element as it rotates about the shaft and the free-stream velocity, $V_{\infty}$ , where the span position from the rotational axis is denoted by $y$ . This combination yields what is usually called the velocity triangle, which helps in visualizing and calculating the relative flow angles.

In the propeller theory, it is usually assumed that the induced velocity, $v$ , is small compared to the free-stream (flight) velocity, i.e., $v \ll V_{\infty}$ , 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. 13, i.e.,

(59) $\begin{equation*} v = \frac{1}{2} \bigg( -V_{\infty} + \sqrt{V_{\infty}^2 + \frac{2T}{\varrho_{\infty} A}} \bigg) \end{equation*}$

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

Theoretical Development

The local pitch of the blade is $\beta = \beta (y)$ as it varies along the span, so the angle of attack $\alpha$ of any blade section is

(60) $\begin{equation*} \alpha(y) = \beta(y) - \tan^{-1} \left( \frac{\Omega y}{V_{\infty}} \right) = \beta(y) - \phi(y) \end{equation*}$

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.,

(61) $\begin{equation*} C_l(y) = C_{l_{\alpha}} \alpha(y) \end{equation*}$

where $\alpha$ is measured from the zero-lift angle and recognizing that the lift-curve slope $C_{l_{\alpha}}$ 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

(62) $\begin{equation*} dL (y) = \frac{1}{2} \, \varrho_{\infty} \, V^2 \, c \, C_l \, dy \end{equation*}$

where $c$ is the local chord of the blade ( $c \, dy$ is the area of the blade element) and the resultant local velocity at the blade element, $V$ , is

(63) $\begin{equation*} V = \sqrt{ (\Omega y)^2 + V_{\infty}^2 } \end{equation*}$

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.,

(64) $\begin{equation*} T = N_b \int_0^{R} dL \cos \phi \end{equation*}$

where $N_b$ is the number of propeller blades, and $R$ 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

(65) $\begin{equation*} P = \Omega N_b \int_0^{R} (dL \sin \phi + dD \, y) \end{equation*}$

where $dD$ 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 $y$ 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 $dL \sin \phi$ (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

(66) $\begin{equation*} dD = \frac{1}{2} \, \varrho_{\infty} \, V^2 \, c \, C_d \, dr \end{equation*}$

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

(67) $\begin{equation*} M_{\rm hel}(y) = \frac{V}{a} = \frac{ \sqrt{ (\Omega y)^2 + V_{\infty}^2 }}{a_{\infty}} \end{equation*}$

where $a_{\infty}$ is the local speed of sound, i.e., $C_d = C_d(M_{\rm hel}$ ). 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 $N$ segments, the idea being shown in the figure below. To give any reasonable definition of the spanwise aerodynamic loads, $N$ should be between 30 and 100. Each segment is then small enough to assume constant flow properties. Let $y_i$ denote the spanwise position of the $i$ -th segment, where $i = 1, ...., N$ , and let $\Delta y$ be the spanwise width of each segment.

In the numerical solution, the blade is discretized into discrete elements on which the lift and drag are calculated.

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

(68) $\begin{equation*} V_i = \sqrt{ (\Omega y_i)^2 + V_{\infty}^2 } \end{equation*}$

and so the inflow angle is

(69) $\begin{equation*} \phi_i = \tan^{-1} \left( \frac{\Omega \, y_i}{V_{\infty}} \right) \end{equation*}$

The local angle of attack $\alpha_i$ is then obtained using

(70) $\begin{equation*} \alpha_i = \beta(y_i) - \phi_i \end{equation*}$

The local lift coefficient, $C_{l_i}$ , is

(71) $\begin{equation*} C_{l_i} = C_{l_{\alpha}} \alpha_i \end{equation*}$

If the drag coefficient $C_{d_i}$ is known or assumed, e.g., $C_{d_i} = C_{d_{0}}$ , 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 $M_{\rm hel}(y_i)$ . Reynolds number effects can also be included if such data are available.

Computing the lift force, $dL_i$ , and the drag force, $dD_i$ , on each segment gives

(72) $\begin{equation*} dL_i = \frac{1}{2} \, \varrho_{\infty} \, V_i^2 \, c_i \, C_{l_i} \, \Delta y \quad \text{and} \quad dD_i = \frac{1}{2} \, \varrho_{\infty} \, V_i^2 \, c_i \, C_{d_i} \, \Delta y \end{equation*}$

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

(73) $\begin{equation*} T = N_b \, \sum_{i=1}^{N} dL_i \cos \phi_i \quad \text{and} \quad P = \Omega \, N_b \, \sum_{i=1}^{N} (dL_i \sin \phi_i + dD_i y_i) \end{equation*}$

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

MATLAB code to implement the blade element theory analysis of a propeller. The inclusion of the inflow velocity is left as an exercise for the student.

Show code/hide code.

% Constants and input parameters
rho = 1.225; % Air density (kg/m^3)
Omega = 200; % Rotational speed (rad/s)
V_inf = 50; % Free-stream velocity (m/s)
N_b = 3; % Number of blades
R = 1.5; % Propeller radius (m)
N = 100; % Number of segments
c = 0.1; % Chord length (m)
C_l_alpha = 5.7; % Lift-curve slope
C_d = 0.01; % Profile drag coefficient

% Discretize the blade span
y = linspace(0, R, N);
dy = y(2) – y(1);

% Initialize variables
J = V_inf / (Omega * R); % Advance ratio
beta_range = deg2rad(10:1:30); % Range of blade pitch angles in radians
C_T = zeros(size(beta_range));
C_P = zeros(size(beta_range));

for k = 1:length(beta_range)
beta = beta_range(k);

% Initialize variables for each pitch angle
V = sqrt((Omega * y).^2 + V_inf^2);
phi = atan(Omega * y / V_inf);
alpha = beta – phi;

% Lift coefficient
C_l = C_l_alpha * alpha;

% Calculate lift and drag forces on each segment
dL = 0.5 * rho * V.^2 * c .* C_l * dy;
dD = 0.5 * rho * V.^2 * c * C_d * dy;

% Thrust and Power calculation
T = N_b * sum(dL .* cos(phi));
P = Omega * N_b * sum((dL .* sin(phi) + dD .* y) * dy);

% Non-dimensional coefficients
C_T(k) = T / (rho * (Omega^2) * (R^4));
C_P(k) = P / (rho * (Omega^3) * (R^5));
end

% Plot the results
figure;
plot(J, C_T, ‘-o’);
xlabel(‘Advance Ratio, J’);
ylabel(‘Thrust Coefficient, C_T’);
title(‘Thrust Coefficient vs. Advance Ratio’);

figure;
plot(J, C_P, ‘-o’);
xlabel(‘Advance Ratio, J’);
ylabel(‘Power Coefficient, C_P’);
title(‘Power Coefficient vs. Advance Ratio’);

% Output results
fprintf(‘Advance Ratio (J): %.2f\n’, J);
fprintf(‘Thrust Coefficients (C_T):\n’);
disp(C_T);
fprintf(‘Power Coefficients (C_P):\n’);
disp(C_P);

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. 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 $\beta_{75}$ 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 $y = 0.75 R$ ), which is a geometric quantity and can be measured.

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 is known as an airscrew.

Thrust Coefficient

Recall that the thrust coefficient for a propeller is defined as

(74) $\begin{equation*} C_T = \frac{T}{\varrho_{\infty} \, n^2 \, d^4} \end{equation*}$

where $T$ is the thrust generated by the propeller, $d$ is the propeller’s diameter, and $n$ is the number of revolutions of the propeller per second. If the rotational angular velocity of the propeller is $\Omega$ , then $n = \Omega/2\pi$ . Normally, the results for the propeller characteristics are plotted as a function of the advance ratio, $J$ , defined by

(75) $\begin{equation*} J = \frac{V_{\infty}}{n \, d} \end{equation*}$

Representative $C_T$ versus $J$ 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.

Representative propeller performance curves in the form of thrust coefficient, $C_T$ , as a function of advance ratio, $J$ .

Power Coefficient

The corresponding power coefficient for the propeller is defined as

(76) $\begin{equation*} C_P = \frac{P}{\varrho_{\infty} \, n^3 \, d^5} \end{equation*}$

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

Representative propeller performance curves in the form of power coefficient, $C_P$ , as a function of advance ratio, $J$ .

Propulsive Efficiency

Propeller efficiency is a measure of 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

(77) $\begin{equation*} \eta \equiv \eta_p = \frac{T \, V_{\infty}}{P} \end{equation*}$

which is just a non-dimensional statement that the propeller’s efficiency is the ratio of the useful power to the input power. Using the previous definitions of $C_T$ and $C_P$ then

(78) $\begin{equation*} \eta = \frac{T \, V_{\infty}}{P} = \frac{(\varrho_{\infty} \, n^2 \, d^4 \, C_T) \,V_{\infty}} {\varrho_{\infty} \, n^3 \, d^5 \, C_P} = \frac{C_T \, V_{\infty} }{C_P \, n \, d} = \frac{C_T \, J}{C_P} \end{equation*}$

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

Representative propeller performance curve of propulsive efficiency, $\eta$ , as a function of advance ratio, $J$ .

Notice that for a given propeller operated with any given blade pitch $\beta_{75}$ and rotational speed $n$ , 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 $J$ (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, $n$ (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 $C_T$ , $C_P$ , and $\eta$ , as shown previously, can now be explained in greater detail. At low values of $J$ , 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 $J$ 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, and 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 $J$ ) 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 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 means that the usable power delivered to the airstream decreases, effectively setting an upper barrier to the airspeed achievable by the airplane.

Compared to a fixed-pitch propeller, which reaches peak efficiency at a single airspeed, a variable-pitch propeller can maximize efficiency over a wide range of airspeeds.

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 now 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 attached propeller charts, 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, then 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 $J$ , i.e.,

$J = \frac{V_{\infty} }{n \, d} =\frac{V_{\infty} }{(\text{\small rpm}/60) \, d}$

so

$V_{\infty} = J \, (\text{\small rpm}/60) \, d = J (2,000/60) \times 7.0 = 233.34 \, J~\mbox{ft/s}$

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

Blade pitch ( $^{\circ}$ )	$\eta$	$J$	$V_{\infty}$ (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

$C_T = \frac{T}{\varrho_{\infty} n^2 d^4}$

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

$T = \varrho_{\infty} n^2 d^4 C_T$

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 ${^{-3}}$ . Inserting the information gives

$T = \varrho_{\infty} n^2 d^4 C_T = 0.001869 \times \left( \frac{2000}{60} \right)^2 \times 7.0^4 \, C_T = 4986.08 \, C_T$

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

Blade pitch ( $^{\circ}$ )	$J$	$C_T$	$T$
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

$C_P = \frac{P}{\varrho_{\infty} n^3 d^5}$

so the corresponding power needed to drive the propeller is

$P = \varrho_{\infty} n^3 d^5 C_P$

Inserting the known information gives

$P = \varrho_{\infty} n^3 d^5 C_P = 0.001869 \times \left( \frac{2000}{60} \right)^3 \times 7.0^5 \, C_P = 2115.31 \, C_P$

where we have converted to horsepower (hp) by dividing the result in ft-lb s $^{-1}$ by 550. Again, it is best to use a table to show the results, i.e.,

Blade pitch ( $^{\circ}$ )	$J$	$C_P$	$P$ (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 the values of $J$ and airspeed increase, a concern is that the tip speed of the propeller can approach supersonic conditions. If this occurs, the propeller will lose efficiency and produce much 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 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 $V_{\rm hel}$ 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 $R$ or diameter $d$ is

$V_{\Omega R} = \Omega R = \Omega \left( \frac{d}{2} \right)$

and so the helical tip speed of the propeller is

$V_{\rm hel} = \sqrt{ V_{\Omega R}^2 + V_{\infty}^2}$

where $V_{\infty}$ is the forward airspeed. Therefore, the helical Mach number $M_{\rm hel}$ will be

$M_{\rm hel} = \frac{V_{\rm hel} }{a}$

where $a$ is the local speed of sound at the conditions at which the propeller operates. While every propeller will be different, exceeding some limiting critical tip Mach number at the blade tip, which may be around 0.9 for a thin airfoil section, can be expected to decrease propulsive efficiency, increase noise, or both.

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 $V_{\rm hel}$ 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 $d$ is

$V_{\rm tip} = \Omega \left( \frac{d}{2} \right)$

where $\Omega = 2 \pi ( \text{\small rpm}/60 )$ and so the helical tip speed of the propeller is

$V_{\rm hel} = \sqrt{ V_{\rm tip}^2 + V_{\infty}^2}$

where $V_{\infty}$ is the forward airspeed. The helical Mach number $M_{\rm hel}$ will be

$M_{\rm hel} = \frac{V_{\rm hel} }{a}$

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

Blade pitch ( $^{\circ}$ )	$J$	$V_{\infty}$ (ft/s)	$V_{\rm hel}$ (ft/s)	$M_{\rm hel}$
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.

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 drones. In this regime, the blade sectional drag values tend to be higher than at Reynolds numbers above 10⁶, i.e., $C_d = C_d(R\!e_{\rm hel})$ , which significantly impacts the thrust and propulsive efficiency of the propeller.

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 and the range of $J$ values over which it can operate. These considerations are essential for applications to some types of UAVs.

Efficiency curves of a two-bladed proper at large and small scales.

These effects on propulsive efficiency are primarily related to the higher sectional drag values at these low Reynolds numbers and the reduction in maximum sectional lift values. Unique low Reynolds number airfoil designs are typically employed to minimize sectional drag, but these must be designed carefully. A larger solidity (blade area) also assists in generating the necessary thrust under these conditions.

Designing low Reynolds number 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 design assumptions and performance predictions.

Help! How do I select a propeller for a small drone?

Selecting the correct propeller for a small drone that can take off and land vertically involves several interrelated factors, such as the drone’s size, weight, motor performance, desired flight performance characteristics, etc. The following are the basic steps needed to select an appropriate propeller:

To determine the drone’s requirements, consider its weight, motor specifications, battery type and voltage, and desired flight characteristics. Heavier drones will need larger diameter and higher-pitch propellers to generate sufficient lift to overcome their weight.
Match the propeller size and pitch 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.
Calculate the thrust requirements. A drone will need enough thrust to overcome its weight. A thrust-to-weight ratio of at least 2:1 is necessary for good performance. For example, if a drone has a mass of 1 kg, the design thrust will need to be capable of carrying 2 kg, e.g., 500 g per motor for a quadcopter.
Consult the motor specifications. Consider reviewing the motor’s datasheet, which may have a recommended propeller size and pitch. Manufacturers often provide a range of suitable propellers for their motors.
Consider the propeller and propeller material: Propellers are specified in terms of diameter $\times$ pitch. For example, a propeller marked as 10 $\times$ 5 means it has a diameter of 10 inches and a helical pitch distance of 5 inches; the propeller could also be specified as cm units. Plastic propellers are lightweight and inexpensive but less durable. Carbon fiber propellers give better performance but are significantly more expensive. It is often a good choice to buy and test three inexpensive propellers at first, perhaps with a higher and lower pitch compared to your first choice.
Consider using online codes such as eCalc, which can simulate different propeller, motor, and battery combinations to predict performance metrics. This code is by no means perfect, but it will give direction in the choice of a propeller. Review also the propeller measurements made by Professor Selig and his students at the University of Illinois, which are extremely comprehensive and may help in the selection of a propeller.

Example calculation:

The drone weighs 1.5 kg, including the battery, camera, etc.
The thrust required from the propeller(s) must be enough to carry 1.5 kg $\times$ 2 = 3 kg to ensure sufficient takeoff and maneuver performance. For example, for a quadcopter, each motor needs to be attached to a propeller to provide enough thrust to overcome 750 g.
Motors are marketed using a kV (or KV) rating and most also recommend a specific battery requirement. For example, a motor may have an 800 KV rating, which means it will spin at 800 rpm per volt. If the motor supports the use of an 11.1 Volt LiPo battery, then it will spin up to 8,800 rpm.
Estimate the diameter and pitch of the propeller to produce the needed thrust. The motor’s datasheet may suggest a range of propeller sizes and pitches that will fully absorb the motor’s power.
Test the drone with the selected propeller. It may be another propeller of a slightly different pitch is needed. If you are lucky enough to have a test stand, you can measure thurst and battery energy consumption using different propellers.

Summary & Closure

Propellers are relatively simple and robust devices for producing thrust and are widely used in aviation. A fixed-pitch propeller has limited operational characteristics and efficiency, but using a variable-pitch (or constant speed) propeller increases its operating envelope, albeit with some added weight and cost. Propellers are not just used with piston engines; they are also used with turboshaft engines, such as in the case of turboprops. Proper matching of the propeller to the specific engine is essential for optimal performance. The current trend is toward using smaller diameter propellers with more blades, which helps reduce blade tip speeds and noise, a necessary consideration in modern aircraft design. The engineering design challenge is to match the engine and propeller to achieve good overall propulsive efficiency and low specific fuel consumption over a wide range of flight conditions. This goal requires careful consideration of the propeller characteristics to ensure that the optimal combination of efficiency, performance, and low noise is achieved. The most recent advances in propeller technology have used composite propeller blades made from Kevlar or carbon fiber, allowing thinner, swept blades to be made. In addition to being lighter, modern propeller designs have shown substantial gains in propulsive efficiency and noise reduction. Wooden propellers are still made today and often used on lower-performance aircraft because they are lightweight and relatively inexpensive.

5-Question Self-Assessment Quickquiz

For Further Thought or Discussion

Is using a fixed-pitch or constant-speed propeller for a UAV desirable? Discuss.
Why does using a swept blade on a propeller generally help give the propeller better efficiency?
Discuss the factors that may influence the design of a propeller blade, including material selection, shape, and angle of attack.
Explain the concept of propeller slipstream and its influence on the overall aerodynamic performance of an aircraft. How does slipstream interact with other aircraft components?
Explain how tip speed affects propeller noise and efficiency. What design considerations are made to optimize tip speed within acceptable limits?
Explain the phenomenon of propeller torque and its effect on aircraft handling. How do pilots and designers compensate for torque effects during flight?

Other Useful Online Resources

To learn more about propellers, check out these helpful online resources:

Ever wonder how an airplane propeller is made? Take a tour of Hartzell Propeller to learn how propellers work and how they are made.
“The Propeller Explained,” a documentary film.
World’s Largest Wooden Propeller Factory: How It’s Made.
Props to Alaina: The Art of Airplane Propellers.
How Sensenich Wooden Props are made.
Apparently, this is the only video you need to understand airplane propellers! It’s good, but there is more to learn.

In 2022, students from the University of Maryland made a flying drone based on the aerial screw design. ↵
A steam engine of the era would have been far too heavy (boiler, engine, condenser, fuel, water, etc.) and with too little power to be of any aeronautical use. ↵
Glauert, H. (1935) "Airplane Propellers." In: Durand, W.F., Ed., Aerodynamic Theory, Vol. IV, Division L, Springer, New York, 169–360. ↵
An excellent analogy to this would be a continuously variable transmission on an automobile. ↵
. The blade twist itself is fixed and cannot be varied. ↵
Aspects of Rankine's propeller theory were published in various sources, including Rankine, W.J.M. (1865) On the Mechanical Principles of the Action of Propellers. Transaction of the Institute of Naval Architects, 6, 13-39. ↵
Which can be determined from dimensional analysis. ↵

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Introduction to Aerospace Flight Vehicles Copyright © 2022–2025 by J. Gordon Leishman is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, except where otherwise noted.