Aircraft Propellers

J. Gordon Leishman

45 Aircraft Propellers

Introduction

The modern propeller reflects more than a century of steady development in aeronautical engineering. Since the earliest days of powered flight, propellers have been continually refined in aerodynamic design, structural configuration, and manufacturing methods, directly improving aircraft performance and efficiency. Today, companies such as Hartzell, Sensenich, and McCauley produce highly efficient propellers using composite materials, carefully optimized airfoil sections, and blade geometries tailored to specific operating conditions. Many modern blades incorporate planform sweep and advanced tip shapes to reduce compressibility effects and noise at higher rotational speeds. These contemporary designs, exemplified by Figure 1, reflect sustained progress in aerodynamics, materials, and manufacturing techniques and remain an important element of overall aircraft performance.

A state-of-the-art 5-bladed propeller manufactured by Hartzell Propeller, Inc. from advanced materials.

The earliest known use of propellers or rotors dates back to around 500 BC in ancient China, where children played with bamboo dragonflies. These were simple wooden toys consisting of a stick and blades that, when spun rapidly between the hands, would lift into the air. In the 1480s, Leonardo da Vinci sketched the “aerial screw,” as shown in Figure 2. This concept was an aerodynamic device derived from an Archimedes water screw, which some consider a precursor to the modern helicopter rotor. Although never built at 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 their axis of rotation, a force called thrust.

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

In the 19th century, George Cayley made significant contributions to the understanding of the principles of flight, experimenting with gliders and recognizing the need for a powerplant and propulsion mechanism if sustained flight were ever to be achieved. At least one of his airplane concepts and an airship, the latter shown in the illustration in Figure 3, indicated that the means of propulsion was “airscrews,” or propellers. The power source was unspecified, although the only option available at the time was a steam engine. A steam engine of the era would have been far too heavy (boiler, engine, condenser, fuel, water, etc.) and too low-powered for any aeronautical use. The use of “screw” propellers for propulsion was nothing new, as they had been used in marine applications since the 1830s. Nevertheless, propeller blade designs suitable for efficient operation in air had not yet been determined. In the 1870s, Thomas Moy built the Aerial Steamer, an early powered aircraft with two six-bladed propellers driven by a steam engine. The blades exhibited a geometric twist, with pitch becoming finer toward the blade tips. Like the Cayley concept, it did not fly, but it was a step forward in critical thinking about how an airplane might be propelled efficiently.

A Cayley airship concept that was intended to use twin “airscrews” for forward 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 found no theory or analysis capable of describing or predicting propeller performance. At the time, the state of the art was trial and error. They then set out to study the aerodynamics of propellers in a wind tunnel of their own design, demonstrating that long, thin (i.e., high-aspect-ratio) blades with nose-down twist (called washout) along their span yielded the best propulsive efficiency. By 1908, they had improved their propeller designs for the Wright Flyer III, enabling much longer flights with greater range. Their “raked” propeller had a swept leading edge and more blade area toward the tip, as shown in Figure 4. This design yielded further efficiency gains, enabling greater conversion of the limited engine power into thrust. The broader propeller tips were also structurally stiffer, thereby preventing buzzing and fluttering, a shortcoming of many early propeller designs.

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

During WWI, propeller technology advanced rapidly to meet the demand for more capable airplanes. Initially, fixed-pitch wooden propellers carved from laminated spruce blocks were standard, as shown in Figure 5. However, as engine power and aircraft speeds increased, the limitations of wooden propellers quickly became evident, including cracking and delamination. They were also highly vulnerable to combat damage, and a single bullet through a wooden blade often caused catastrophic failure.

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

While wood was easy to work with and, with sufficient skill, 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 damage resistance, but it also introduced challenges, including added weight and the need for more complex, expensive manufacturing processes. One of the earliest aircraft to use steel propellers was the Handley Page O/400 bomber, which significantly improved its performance and reduced its vulnerability in the demanding conditions of long-range bombing missions under enemy fire.

With the formation of dedicated research institutions such as the NACA and the Royal Aircraft Establishment (RAE), researchers systematically quantified the aerodynamic principles governing propeller performance. From the 1920s to the 1950s, NACA conducted wind-tunnel tests and collected detailed propeller performance data. Engineers experimented with various blade planforms, including tapered and hyperbolic shapes, as well as specific tip shapes and airfoil sections better suited to propellers. The RAE also researched propellers extensively, focusing on theoretical aerodynamic analyses, mainly by Hermann Glauert ^[2], supported also by wind tunnel measurements. These propeller developments laid the groundwork for more sophisticated aircraft designs, contributing to the rapid evolution of aviation technology.

Materials such as duralumin, an aluminum alloy developed by Alfred Wilm in Germany before World War I and patented in 1911, helped make propellers lighter and stronger. Engineers soon began developing variable-pitch aluminum propellers, allowing pilots to adjust the blade pitch angles for optimal flight performance. This significantly improved propeller efficiency and airplane performance over a broader range of airspeeds. In the 1930s, Hamilton Standard introduced the first practical controllable pitch propeller; see Figure 6. During takeoff and climbout, the pilot could use a lower blade pitch for optimal performance, then shift to a higher blade pitch for efficient cruising. This innovation soon led to the constant-speed propeller, which automatically adjusted blade pitch during flight using a governor ^[3] to maintain a selected rotational speed while allowing the propeller to operate efficiently over a much broader range of flight conditions.

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

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-blade propellers was another crucial development. Initially, two-blade and three-blade propellers were standard. Nevertheless, engineers soon realized they needed four- and five-blade designs to handle the power output of increasingly powerful piston engines. Additional blades also provided smoother operation, reducing engine and propeller vibrations and improving the airplane’s overall performance at higher airspeeds. As engine power levels increased further after the advent of supercharging and turbocharging, counter-rotating tandem propellers were used to prevent a single propeller from becoming prohibitively large and thereby absorbing all available engine power.

Iconic WWII aircraft, such as the British Supermarine Spitfire and the American North American P-51 Mustang, exemplified advancements in propeller and engine technology. The Spitfire, with its powerful Rolls-Royce Merlin engine and Rotol four-bladed aluminum propeller, as shown in Figure 7, achieved exceptional performance in attainable airspeed and operational altitude. Later Spitfire models were equipped with a five-bladed propeller powered by a version of the Rolls-Royce Griffon engine, representing the pinnacle of piston-engine and propeller performance. When fitted with a four-bladed Hamilton Standard propeller, the P-51 Mustang became renowned for its long range, speed, and effectiveness as an escort fighter.

The Rotol four-blade propeller was introduced to improve performance by increasing thrust and efficiency.

These technological improvements in propeller design 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 propeller-driven airplanes enabled more effective long-range bombing campaigns, strategic airlift, 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.

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

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 Figure 8. Airplanes such as the Lockheed C-130 Hercules, the modern Airbus A400M Atlas, and the C-130J Super Hercules have since used advances in turboprop technology to improve efficiency, versatility, and reliability. Today, the turboprop remains a preferred propulsion system for many aircraft and appears on high-performance single-engine aircraft such as the Pilatus PC-12 and the Cessna Caravan 208.

A four-bladed propeller powered by a turboshaft engine blended power and propulsive efficiency.

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

Most modern propellers feature swept leading edges and thin airfoil sections, as illustrated in Figure 9. Scimitar propellers, which have highly swept blades made primarily of composite materials, were developed in the 1980s. Like the swept wings used on high-speed aircraft, these propeller blades delay the onset of drag rise and wave drag, allowing them to remain efficient propulsors at higher flight speeds and Mach numbers. They also reduce noise, a significant concern for all modern aircraft because of increasingly stringent ICAO noise standards. Many modern turboprop aircraft use this swept-blade propeller design, including the ATR 72, Bombardier Dash 8, and Pilatus PC-12.

The swept-blade propellers of a C-130J Super Hercules military transport. The blades are parked in their fully feathered condition.

Ongoing developments in propeller design for electric-powered airplanes focus on improving efficiency and reducing noise, particularly for urban air mobility (UAM) and unoccupied aerial vehicles (UAVs). Engineers are exploring innovative blade geometries and lightweight materials to develop quiet, compact propeller systems that meet the specific requirements of drones, electric aircraft, and eVTOL concepts. Goals include optimizing the flight performance of multi-rotor systems and variable-pitch propellers by leveraging advances in battery technology and manufacturing techniques. While the tipping point toward electric propulsion over traditional approaches is likely decades away, technological developments are progressing rapidly.

Learning Objectives

Appreciate the historical development of propeller technology from the days of the Wright brothers to the present.
Understand the basic operating principles of a propeller, including methods for calculating its thrust, required power, and efficiency.
Be familiar with momentum theory and blade element theory when calculating propeller performance.
Know how to read and interpret a propeller performance chart.
Understand the operational advantages of variable-pitch propellers compared to fixed-pitch designs.
Know how to select a propeller to meet a performance requirement, including for low-Reynolds-number propellers.

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’s shape creates a pressure difference between the upper and lower surfaces, generating lift (see Figure 10). The forward component of the aerodynamic force produces the propeller’s thrust, with the axial component of lift contributing positively and the axial component of blade drag opposing it. The engine supplies torque to the propeller shaft, which overcomes the drag forces on the blades and maintains their rotation. Sedulous propeller design maximizes thrust by optimizing blade geometry, angle-of-attack distribution, and rotational speed, ensuring the engine’s power is converted into propulsive force with maximum efficiency.

Each rotating blade produces lift, resulting in net thrust. Torque (and thus power) is required to accomplish this and is delivered via the propeller shaft.

Velocity Triangle & Blade Twist

A notable design feature of propeller blades is their significant spanwise twist, i.e., geometric washout. Each blade is composed of an airfoil section. Because flow velocity varies along the blade, the twist ensures the local angle of attack at each section stays near its most aerodynamically efficient value across the entire blade span. In practice, most sections should have angles of attack between about ${5^\circ}$ and ${8^\circ}$ , or otherwise near their angle of attack for the peak lift-to-drag ratio, i.e., $\alpha = \alpha_{\rm opt}$ . This yields maximum thrust for minimum torque and shaft power.

To understand why a propeller blade must be significantly twisted, consider the following analysis. Let $R$ be the radius of the propeller and let $y$ be the spanwise distance from the axis of rotation to a blade element, as shown in Figure 11. The corresponding nondimensional radius is $r = y/R$ , and the aerodynamically passive root cutout extends to $r_0$ . The angular velocity is $\Omega$ , and the tangential (in-plane) velocity increases linearly with $y$ , i.e., $U_T = \Omega \, y$ . The local geometric blade pitch angle, $\beta(r)$ , is defined relative to the plane of rotation.

Definition of the blade element for a propeller and the sectional velocity diagram.

Assume that the axial velocity $V_a$ through the propeller disk is uniform, such that $V_a \approx \text{constant}$ , consistent with the actuator disk assumption, although in reality both the axial and tangential velocity components vary with radius. The total relative velocity (also known as the relative wind) at the blade section $r$ is then

(1) $\begin{equation*} U(r) = \sqrt{(\Omega \, y)^2 + V_a^2} = \sqrt{(\Omega \, R \, r)^2 + V_a^2} \end{equation*}$

The inflow angle is

(2) $\begin{equation*} \phi(r) = \tan^{-1}\!\left( \frac{V_a}{\Omega \, y} \right) = \tan^{-1}\!\left( \frac{V_a}{\Omega \, R \, r} \right) \end{equation*}$

and so the sectional angle of attack is

(3) $\begin{equation*} \alpha(r) = \beta(r) - \phi(r) \end{equation*}$

Notice that the axial velocity is the sum of the freestream velocity (airspeed) ${V_{\infty}}$ and the induced velocity $v_i$ , i.e.,

(4) $\begin{equation*} V_a = V_\infty + v_i \end{equation*}$

where $V_a$ is the axial velocity through the propeller disk. In general, the twist derivation should be expressed in terms of $V_a$ , with $V_a \approx V_\infty$ for forward flight and $V_a = v_i$ for static conditions. For a propeller in its normal working state during flight, $v_i \ll V_\infty$ , whereas for static thrust (hover), $V_\infty = 0$ , so that $V_a = v_i$ . In either case, $V_a$ is assumed to be uniform across the propeller disk, consistent with the Rankine-Froude actuator disk assumption, which is also a design goal aimed at minimizing losses.

Under the assumption of a constant aerodynamic angle of attack, i.e., $\alpha = \alpha_{\rm opt}$ , which represents an idealized design condition, it follows that

(5) $\begin{equation*} \beta(r) = \alpha_{\rm opt} + \tan^{-1}\!\left(\frac{V_a}{\Omega R r}\right) \end{equation*}$

This is the more general twist relation. For small inflow angles, the approximation $\tan^{-1} x \approx x$ gives

(6) $\begin{equation*} \beta(r) \approx \alpha_{\rm opt} + \left(\frac{V_a}{\Omega R}\right)\frac{1}{r} \end{equation*}$

which can be recast in terms of the tip pitch angle, $\beta_{\mathrm{tip}}$ , as

(7) $\begin{equation*} \beta(r) \approx \alpha_{\rm opt} + \frac{\beta_{\mathrm{tip}}-\alpha_{\rm opt}}{r} \end{equation*}$

because $\alpha_{\rm opt}$ is a constant. This expression shows why the blade twist is approximately hyperbolic, with the dominant radial dependence proportional to $1/r$ . The exact twist distribution, however, is not purely hyperbolic because the arctangent relation, induced velocity, Mach number, Reynolds number, section thickness, camber, and structural constraints all affect the final blade geometry, as shown in Figure 12. 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, section thickness-to-chord ratio, and camber (i.e., zero-lift angle of attack, $\alpha_0$ ), the best angle of attack is not necessarily a constant. The net twist varies with the intended application, but the net washout angles may be as large as 40 degrees over the blade span; these angles are significantly larger than those used on airplane wings.

A propeller blade typically exhibits hyperbolic twist along its span to maintain efficient operation of the blade-section angles of attack.

Fixed-Pitch Versus Variable Pitch

A fixed-pitch propeller has an inherently narrow operating envelope because its blade angles are optimized for only one flight condition. If the pitch is selected to provide acceptable thrust at takeoff and during climb, then at higher forward speeds the blade section angles of attack become progressively smaller, reducing both thrust and propulsive efficiency. Conversely, if the pitch is selected for efficient cruise, the blade sections operate at excessively high angles of attack at low forward speeds, producing high torque and relatively low efficiency during takeoff and climb.

This behavior is illustrated in Figure 13, which shows typical propeller efficiency curves as a function of advance ratio. This behavior is more conveniently interpreted using the advance ratio, $J$ , which is defined as

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

where ${V_\infty}$ is the flight airspeed, $n$ is the rotational speed (rev/s) of the propeller, and $D$ is its diameter. Each blade-pitch setting produces a distinct efficiency curve with a peak occurring over a relatively narrow range of advance ratio.

Typical variation of propeller efficiency with advance ratio. Each pitch angle yields peak efficiency only over a limited operating range.

A fine-pitch (low-pitch) propeller reaches its peak efficiency at low advance ratios corresponding to low forward speed and high rotational speed, making it suitable for takeoff and climb. However, its efficiency decreases rapidly as airspeed increases. A coarse-pitch (high-pitch) propeller reaches peak efficiency at higher advance ratios corresponding to cruise flight conditions but performs poorly at low speeds. A medium-pitch setting provides a compromise between these two extremes but cannot maintain peak efficiency over the entire flight envelope.

The underlying reason for this behavior can be understood from the velocity triangles discussed previously and as also shown in the accompanying Figure 14. As flight speed increases, the relative inflow angle at each blade section changes, altering its effective angle of attack. Because the blade twist distribution is fixed, only the overall blade pitch can be changed to restore the sections to near-optimal angles of attack. A fixed-pitch propeller cannot make this adjustment, so its efficiency inevitably falls away from the design condition.

Velocity-triangle interpretation of blade-pitch control. By increasing blade pitch with increasing airspeed, the blade sections can be maintained near their optimal angles of attack, preserving thrust and efficiency.

The solution to this limitation is to progressively vary the average blade pitch angle during flight. A fine (low) pitch is used for takeoff and climb to produce high thrust at low airspeed, while a coarser (higher) pitch is used for cruise to maintain efficient operation at higher advance ratios. By regulating blade pitch this way, the propeller can operate near peak efficiency over a much broader flight envelope. Therefore, aircraft equipped with a variable-pitch or constant-speed propeller can maintain high thrust and propulsive efficiency over a wider airspeed range. A constant-speed propeller is a variable-pitch propeller equipped with a governor that automatically adjusts the blade pitch to maintain a selected rotational speed, $n$ , despite changes in airspeed or engine power; if the propeller tends to overspeed, the governor increases blade pitch, and if it slows down, the pitch is reduced, allowing the propeller to return to the commanded value of $n$ . This lets the blades operate near their optimal angles of attack across a broad range of flight conditions while reducing pilot workload.

The propeller’s maximum thrust is limited by the onset of stall, just as a regular wing is limited by its angle-of-attack capability. However, because the local incident Mach number varies from the root to the tip of the propeller, and because an airfoil’s stall angle of attack decreases with increasing Mach number, the allowable angle of attack before stall decreases from the root to the tip. To this end, designers use relatively thin airfoil sections with higher lift-to-drag ratios and higher drag-divergence Mach numbers. Nevertheless, a stall will eventually occur if the blade pitch is increased too much, accompanied by a loss of thrust, an increase in required power, and a marked drop in propulsive efficiency.

If an engine fails in flight, a feathering propeller may be rotated toward a high-pitch position so that the blade chords are nearly aligned with the relative airflow, thereby reducing propeller drag, as shown in Figure 15. Feathering systems are commonly used on multi-engine airplanes and some single-engine turboprops. Depending on the design, springs, counterweights, oil pressure, or combinations of these mechanisms may move the blades toward feather following a loss of engine power or governor oil pressure. Not all constant-speed propellers are capable of feathering. Many constant-speed propellers used on single-engine piston airplanes instead move toward low pitch following a loss of oil pressure. A fixed-pitch propeller cannot be feathered, and its effect on glide performance depends on its pitch, whether it remains windmilling or stops, and the aerodynamic characteristics of the installation.

The propeller blades may sometimes be set to a negative pitch to produce negative thrust, thereby acting as an aerodynamic brake during landing. However, this latter feature is usually found only in high-performance turboprop aircraft.

From the front, a stopped, feathered propeller presents less frontal area to the flow and experiences minimal drag.

Types of Propellers

Many propeller types are in use, ranging from two-blade to four- or more-blade designs, some with fixed pitch and others with variable pitch, as shown in the photographs of Figure 16. In addition, some propellers may have swept blades, a design feature that reduces compressibility drag at the tips when the aircraft operates at higher airspeeds, similar to the design used on the wings of supersonic aircraft. As engine power increases, more blades (or a greater blade area) are required to convert that power into useful thrust. In some cases, to prevent the propeller diameter from becoming too large, counter-rotating propellers may be used to absorb the high power available from the engine shaft and generate thrust.

Propellers come in many types, including those with different numbers of blades. Counter-rotating coaxial propellers suit high-power applications.

Increasing the number of blades can be beneficial when a propeller must absorb more power within a limited diameter, because the blade loading can be distributed over more lifting surfaces. This can reduce loading on each blade and may help limit diameter, tip Mach number, and noise. However, adding blades does not automatically improve propulsive efficiency. More blades also increase blade-profile drag, weight, cost, hub complexity, and the risk of blade-to-blade interference. The propeller’s net efficiency depends on diameter, disk loading, rotational speed, tip Mach number, blade-section Reynolds numbers, activity factor, solidity, and blade pitch. Today, there is an increasing emphasis on reducing propeller noise, which has led engineers designing modern propellers, including for general aviation aircraft, to use more blades and smaller diameters, resulting in lower tip Mach numbers.

What is the difference between propeller power and torque?

Torque, denoted by $Q$ , is the rotational force applied to the propeller shaft, while angular velocity, $\Omega$ , is the rate at which the propeller rotates, measured in radians per second. Power, $P$ , is the rate of energy transfer, given by $P = Q \, \Omega$ . High torque at low rotational speed yields lower power, whereas lower torque at high rotational speed can still generate significant power. Efficient propeller performance depends on balancing $Q$ and $\Omega$ to achieve optimal thrust and efficiency across different flight conditions. For example, consider a propeller with high torque but lower rotational speed: if $Q$ = 50 Nm and $\Omega$ = 100 rad/s, then the power is $P = Q \, \Omega$ = 50 $\times$ 100 = 5000 W = 5 kW. Alternatively, if the propeller operates at lower torque but higher speed, say $Q$ = 25 Nm and $\Omega$ = 200 rad/s, the power remains the same at $P$ = 25 $\times$ 200 = 5000 W = 5 kW.

Momentum Theory of the Single Propeller

The Rankine propeller theory,^[4] also known as the Rankine-Froude theory,^[5] provides a fundamental framework for understanding propeller performance. This theory models the propeller as an idealized actuator disk that imparts momentum to the air, creating a pressure jump across the disk. Physically, 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, changing its momentum and kinetic energy. Consequently, the force on the propeller, produced by a pressure difference between opposite sides of the propeller disk, is opposite to the force on the fluid flow; i.e., the thrust $T$ is directed upstream.

Flow Model

The flow model for the single propeller problem is shown in Figure 17. The upstream and downstream sections labeled 1 and 2 bound the control volume. Remember that, by convention, $d\vec{S}$ points outward from the control volume when using the integral form of the conservation equations. The upstream or freestream (undisturbed) velocity is ${V_{\infty}}$ , and the pressure there is $p_{\infty}$ . It will be assumed to be uniform, which is reasonable 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 change only with downstream distance.

Control volume used for the momentum analysis of a single thrusting propeller. Notice the pressure jump over the propeller disk.

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 $V_{\infty} + v$ , where $v$ is the induced velocity at the propeller disk. The static pressure will also change there. The propeller acts on the air, increasing its momentum and kinetic energy, so the downstream velocity is $V_{\infty} + w$ . In the slipstream, the static pressure also recovers to ambient conditions; that is, the pressure downstream slowly returns to $p_{\infty}$ .

Application of the Conservation Laws

The conservation of mass requires a 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

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

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

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

where the direction of $F_{\rm flow}$ is downstream, and the thrust on the propeller, $T$ , is in the opposite direction (pointing left in Figure 17) so that

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

The conservation of energy states that the shaft power delivered to the propeller must equal the rate of increase of kinetic energy of the slipstream, i.e.,

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

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

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

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 useful propulsive power, and the $T \, v$ term, which is the work done on the air or the induced power loss. The induced power appears as kinetic energy left in the slipstream, reducing the fraction of shaft power converted into useful propulsive power.

Induced Velocity

The induced velocity in the plane of the propeller can be found by substituting Eq. 11 into Eq. 13, which gives

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

Expanding out and simplifying gives

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

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

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

For the next step, take Eq. 11 and substitute in Eq. 9 (mass flow rate) and the connection between ${v}$ and ${w}$ ( $w = 2 v$ ) to get

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

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

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

and expanding out gives

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

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

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

for which there are 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

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

or nondimensionally as

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

Therefore, for any given propeller producing thrust $T$ , its induced velocity $v$ decreases rapidly with increasing airspeed, i.e., $v \rightarrow 0 \mbox{~~for larger values of~} V_{\infty}$ . In the design of a propeller in its normal propulsive state, the induced losses can often be ignored compared to the profile losses.

Power Required

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

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

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

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

This power equation can also be written nondimensionally as

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

Notice that in the limiting case when the airspeed, ${V_{\infty}}$ , becomes high, then $P \rightarrow T \, V_{\infty} \mbox{~~for larger values of~} V_{\infty}$ , 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, ${P_0}$ , will be required to overcome the viscous (shear) drag of the blades so that the total power required, $P_{\rm tot}$ , can be written as

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

Propulsive Efficiency

The propeller’s propulsive efficiency 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

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

which shows 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.,

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

The profile power losses, denoted by ${P_0}$ , depend on the blade sections and the net blade area, which is measured by the propeller’s solidity or activity factor.

Activity Factor & Solidity

A parameter called the activity factor, $AF$ , has long been used in propeller design to measure the amount and radial distribution of blade area. It is not simply a blade-area ratio, because the blade area near the tip is aerodynamically more influential than that near the root. For this reason, the activity factor weights the chord distribution by the cube of the nondimensional radius.

Using $x=y/R$ as the nondimensional radius, one common NACA-style definition of the total activity factor is

(29) $\begin{equation*} AF = \frac{100{,}000}{16} \, N_b \int_{x_0}^{1} \frac{c(x)}{D} \, x^3 \, dx \end{equation*}$

where $N_b$ is the number of blades, $c(x)$ is the local blade chord, $D=2R$ is the propeller diameter, and $x_0$ is the nondimensional root cutout, often taken to be about 0.15. Written in terms of the dimensional radial coordinate $y$ , this same definition becomes

(30) $\begin{equation*} AF = \frac{100{,}000 \, N_b}{32 R^5} \int_{y_0}^{R} c(y) \, y^3 \, dy \end{equation*}$

where $y_0=x_0R$ . This form makes clear that the activity factor is dimensionless. A higher activity factor generally indicates that the propeller has a greater aerodynamically effective blade area, especially toward the outer part of the blade, and can absorb more power at a given diameter and rotational speed.

Definitions and Nomenclature

Note that both $d$ and $D$ have historically been used to denote the propeller diameter and may be used interchangeably. In much of the NACA propeller literature, $D$ is the preferred symbol for diameter. The activity factor must also be interpreted carefully, as different authors may report either a per-blade activity factor or a total activity factor for all blades. Likewise, propeller efficiency may be denoted by $\eta$ or $\eta_p$ , with $\eta_p$ preferred here.

Solidity, denoted by $\sigma$ , is a related but distinct dimensionless parameter. It is the ratio of the total blade planform area to the propeller disk area. If $\bar{c}$ is the arithmetic mean chord over the active blade span, then

(31) $\begin{equation*} \sigma = \frac{N_b \, \bar{c}(R-y_0)}{\pi R^2} \end{equation*}$

where

(32) $\begin{equation*} \bar{c} = \frac{1}{R-y_0}\int_{y_0}^{R} c(y) \, dy \end{equation*}$

if the mean is taken over the active blade span. If the root cutout is neglected, so that $y_0=0$ , then this expression reduces to

(33) $\begin{equation*} \sigma = \frac{N_b \, \bar{c}}{\pi R} = \frac{2 N_b \, \bar{c}}{\pi D} \end{equation*}$

Equivalently, the total solidity may be written directly as

(34) $\begin{equation*} \sigma = \frac{N_b}{\pi R^2} \int_{y_0}^{R} c(y) \, dy \end{equation*}$

A local solidity may also be defined for a blade element at radius $y$ by comparing the blade chord to the circumference of the annulus swept by that blade element, i.e.,

(35) $\begin{equation*} \sigma(y) = \frac{N_b \, c(y)}{2 \pi y} \end{equation*}$

This local definition is often more useful in blade-element theory than a single overall solidity value.

The activity factor and solidity are related measures of blade area, but they are not the same. Solidity measures the total blade area relative to the disk area, whereas the activity factor weights the blade chord distribution toward the tip and is more closely associated with a propeller’s ability to absorb power.

Wake Swirl Effects

Incorporating wake swirl into momentum theory entails adding a tangential velocity component to the otherwise axially aligned slipstream. Rankine’s actuator disk theory considers only the axial-momentum balance and assumes steady, inviscid, incompressible flow, although Glauert also considered this term. When swirl is present, the tangential velocity component and the associated angular momentum of the downstream flow must be accounted for because they affect the power requirement and propulsive efficiency. Although blade-element or blade-element momentum theory best captures these effects, momentum theory can provide a useful approximation.

Propeller swirl refers to the tangential, or rotational, velocity component in the downstream wake behind the propeller, which imparts a spiral or corkscrew-like motion to the slipstream, as shown in Figure 18. This swirl arises because the rotating blades exert aerodynamic forces on the flow with a tangential component, imparting angular momentum to the slipstream. In real flows, viscosity produces profile losses, but wake swirl primarily originates from the torque applied to the flow. The resulting average tangential velocity component in the wake is denoted here by $\bar{w}_t$ . The higher the torque transmitted to the flow, the greater the swirl component. Swirl reduces propulsive efficiency because some of the shaft power is converted into rotational kinetic energy in the wake rather than into useful axial momentum.

Swirl refers to the tangential, or rotational, velocity component in the wake behind the plane of propeller rotation, which slightly decreases propulsive efficiency.

Recall that the thrust $T$ generated by a propeller with a pure axial velocity increment, $w$ , is given by

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

where $\overbigdot{m}$ is the mass flow rate through the propeller. If an average swirl velocity, $\bar{w}_t$ , is also present in the far wake, then the rate of increase of kinetic energy of the slipstream may be approximated as

(37) $\begin{equation*} P = \frac{1}{2} \overbigdot{m} \left( (V_{\infty} + w)^2 - V_{\infty}^2 + \bar{w}_t^{\,2} \right) \end{equation*}$

which exceeds the power required for a purely axial change in momentum through the propeller.

The tangential velocity component, $\bar{w}_t$ , may be related approximately to the angular momentum imparted to the slipstream. If the torque applied to the flow is $Q$ , then

(38) $\begin{equation*} Q = \frac{dH}{dt} \end{equation*}$

where $H$ is the flow’s angular momentum about the propeller axis. Using a representative mean aerodynamic radius, $\bar{y}$ , the torque may be approximated as

(39) $\begin{equation*} Q \approx \overbigdot{m} \, \bar{y} \, \bar{w}_t \end{equation*}$

so that

(40) $\begin{equation*} \bar{w}_t \approx \frac{Q}{\overbigdot{m} \, \bar{y}} \end{equation*}$

For a simple estimate, the mean aerodynamic radius may be taken as

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

where $R$ is the propeller radius, so that

(42) $\begin{equation*} \bar{w}_t \approx \frac{Q}{\overbigdot{m} \left(\dfrac{2}{3} R\right)} \end{equation*}$

The shaft power associated with the propeller torque is $P = Q \, \Omega$ , where $\Omega$ is the angular velocity of the propeller. Therefore, for a given shaft power and rotational speed, the torque applied to the flow is

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

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

Propeller Performance Coefficients

The thrust coefficient for a propeller is a nondimensional thrust and is defined as

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

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 (rps). If the propeller’s rotational angular velocity is $\Omega$ , then $n = \Omega/2\pi$ . The rotational speed is often measured in revolutions per minute (rpm), so rpm = 60 $\times$ rps = 60 ${n}$ .

For a propeller, a nondimensional airspeed called the advance ratio, given the symbol ${J}$ , is defined by

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

Finally, the power coefficient for a propeller is defined as

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

where ${P = \Omega \, Q}$ .

Notice that the definitions of propeller thrust coefficient, advance ratio, and power coefficient differ from those used for helicopter rotors. It is essential not to confuse them. The table below clarifies the differences, where ${d}$ = 2 $R$ .

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

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

and for the induced power, then

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

The corresponding propulsive efficiency (neglecting profile power) is then

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

Momentum Theory of the Tandem Propeller

Tandem coaxial propellers are mounted on the same axis and rotate in opposite directions, enabling a more compact design than separate, non-coaxial propellers. Additionally, the opposing rotation of the propellers cancels out torque reaction effects, which can be significant when large amounts of power and torque are transmitted from the engine. In a tandem propeller system, two propellers are aligned along the same axis but operate at different stations and effectively work on the same slipstream. The second propeller acts on the accelerated flow produced by the first, thereby increasing the flow’s velocity and generating additional thrust.

Therefore, there are two conditions of interest: 1. The rear propeller is sufficiently separated to operate within the front propeller’s fully-developed slipstream velocity. 2. Both propellers are located so close that they share the same induced velocity.

Separated Tandem Propellers

In a separated tandem propeller configuration, the rear propeller operates within the front propeller’s contracted slipstream. The two propellers cannot be analyzed simply as independent propellers because the front propeller increases the axial velocity entering the rear propeller. Thus, the rear propeller operates with a different inflow velocity from the front propeller. Momentum theory can provide a first approximation of this interaction by accounting for the increased axial velocity at the rear propeller. A more complete analysis must also account for effects such as wake swirl, nonuniform inflow, and viscous losses, typically using blade-element theory.

For the first propeller, the single-disk momentum result may be written as

(50) $\begin{equation*} T_1 = 2 \varrho_{\infty} A_1 (V_{\infty}+v_1)v_1 \end{equation*}$

where $v_1$ is the induced velocity at the plane of the first propeller. The corresponding ideal power is

(51) $\begin{equation*} P_1 = T_1 (V_{\infty}+v_1) \end{equation*}$

The rear propeller sees an effective inflow velocity greater than ${V_{\infty}}$ . If this inflow is denoted by $V_2$ , then an idealized momentum-theory estimate for the second propeller is

(52) $\begin{equation*} T_2 = 2 \varrho_{\infty} A_2 (V_2+v_2)v_2 \end{equation*}$

and

(53) $\begin{equation*} P_2 = T_2 (V_2+v_2) \end{equation*}$

where $v_2$ is the additional induced velocity at the second propeller. The effective inflow $V_2$ depends on the slipstream contraction, the axial velocity increment, the swirl, and the spacing between the two propellers. In the simplest idealized estimate, $V_2$ may be approximated by the axial velocity in the wake of the first propeller, but this approximation neglects nonuniformity and swirl-recovery effects.

The total thrust and ideal power are then

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

and

(55) $\begin{equation*} P_{\rm total}=P_1+P_2 \end{equation*}$

with additional profile losses included as

(56) $\begin{equation*} P_{\rm total}=P_1+P_2+P_{0,1}+P_{0,2} \end{equation*}$

The corresponding overall propulsive efficiency is

(57) $\begin{equation*} \eta_{\rm total} = \frac{(T_1+T_2)V_{\infty}} {P_1+P_2+P_{0,1}+P_{0,2}} \end{equation*}$

This expression emphasizes that the useful propulsive power depends on the net thrust acting on the aircraft and the aircraft’s flight speed, ${V_{\infty}}$ , rather than on the local inflow speed at the rear propeller.

Tandem Propellers with Shared Induced Velocity

Consider two tandem propellers positioned closely enough to share the same induced velocity increment, $v$ , as shown in Figure 19. The flow velocity at the propeller planes is $V_{\infty} + v$ . The analysis is simplified by assuming that the induced velocities are equal for both propellers. The mass flow rate through the tandem propeller system is

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

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

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

In this idealized limiting case, the two closely spaced propellers are treated as producing two equal pressure jumps across the same effective actuator disk. The total induced velocity at the shared disk is $v$ , and the corresponding far-wake velocity increment is $w=2v$ . Therefore, the total thrust of the tandem system is

(59) $\begin{equation*} T_{\rm total} = \overbigdot{m} \, w = 2 \overbigdot{m} \, v \end{equation*}$

If the two propellers are assumed to share the loading equally, then

(60) $\begin{equation*} T_1 = T_2 = \frac{T_{\rm total}}{2} = \overbigdot{m} \, v \end{equation*}$

This represents an idealized limit in which the closely spaced tandem propellers act approximately as a single actuator disk, with the total pressure rise divided equally between the two propellers.

The ideal power required by each propeller is then

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

and

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

so that, for equal loading,

(63) $\begin{equation*} P_{\rm total} = P_1+P_2 = T_{\rm total}(V_{\infty}+v) = 2 \overbigdot{m} \, v (V_{\infty}+v) \end{equation*}$

The efficiency of the tandem propeller system should be assessed from the ratio of the useful propulsive power of the complete system to the total shaft power supplied to both propellers. The useful propulsive power is based on the net thrust acting on the aircraft and the aircraft’s flight speed, i.e.,

(64) $\begin{equation*} P_{\rm useful}=T_{\rm total}V_{\infty} \end{equation*}$

where

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

Including profile power losses on both propellers, the overall efficiency is therefore

(66) $\begin{equation*} \eta_{\rm total} = \frac{T_{\rm total}V_{\infty}} {P_1+P_2+P_{0,1}+P_{0,2}} \end{equation*}$

For the ideal equal-loading shared-induced-velocity case,

(67) $\begin{equation*} P_1+P_2 = T_{\rm total}(V_{\infty}+v) \end{equation*}$

so that

(68) $\begin{equation*} \eta_{\rm total} = \frac{T_{\rm total}V_{\infty}} {T_{\rm total}(V_{\infty}+v)+P_{0,1}+P_{0,2}} \end{equation*}$

At higher airspeeds, where $v/V_{\infty}\rightarrow 0$ , this tends toward

(69) $\begin{equation*} \eta_{\rm total} \rightarrow \dfrac{1} {1+\dfrac{P_{0,1}+P_{0,2}}{T_{\rm total}V_{\infty}}} \end{equation*}$

This total-system efficiency is the meaningful comparison. Defining separate propulsive efficiencies for the front and rear propellers is less useful, as the rear propeller operates in the accelerated, possibly swirling wake of the front propeller.

Comparing Efficiencies

A common issue in evaluating tandem versus single-propeller designs is the basis for comparison, such as constant net thrust, equal power input, or equivalent solidity (i.e., the ratio of blade area to disk area). Within the framework of momentum theory and its inherent assumptions, no configuration is inherently superior for a given thrust and disk loading. At higher airspeeds, the profile power required by the blades becomes the dominant factor in determining net propulsive efficiency. Although counter-rotating propellers can recover some energy from wake swirl, this effect is generally secondary, though it can become significant in high-power or high-disk-loading applications.

When comparing propulsive efficiencies, it is essential to account for the added complexity of managing power distribution between propellers, as well as the overall weight efficiency of the propulsion system. Depending on the design choices and operating conditions, a tandem propeller system may outperform or underperform a well-optimized single propeller. The primary advantage of a tandem configuration is its ability to absorb the same power as a single large propeller while using a smaller overall diameter. Ultimately, the effectiveness of any propeller system depends on careful design and optimization to meet the required performance goals, subject to constraints such as size, weight, and cost. Generally, more blades increase weight and cost.

General Propeller Performance

Blade-element theory is a more accurate method for predicting propeller performance and helps design propeller blades for optimal efficiency. Developing a computer program to calculate propeller performance using the blade-element method is relatively straightforward. The principle used here is to calculate the angle of attack and the corresponding lift and drag for each propeller blade section, as shown in Figure 20. All blades are twisted along their span from root to tip to achieve good aerodynamic and propulsive efficiency. Therefore, the local blade pitch varies from section to section 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 differ across 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 axial inflow velocity through the disk. In the simplest analysis, this axial velocity is approximated by the freestream velocity, ${V_{\infty}}$ , but more generally it is ${V_{\infty}}+v$ , where $v$ is the induced velocity. The span position from the rotational axis is denoted by ${y}$ . This combination yields what is typically called the velocity triangle, which helps visualize and calculate the relative flow angles.

In elementary blade-element analyses of propellers operating at moderate or high advance ratios, the induced velocity, $v$ , is often small compared to the freestream velocity, i.e., $v \ll V_{\infty}$ , and may be neglected as a first approximation. However, this approximation becomes increasingly poor at low airspeeds, high disk loadings, during climb, and under static-thrust conditions. In those cases, a better approximation for the average induced velocity over the propeller disk can be obtained using Eq. 21, i.e.,

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

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

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

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

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.,

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

where ${\alpha}$ is measured from the zero-lift angle, 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, the Reynolds number too. The lift on a blade element will be

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

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

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

When the lift and drag on all the sections of the propeller blade are obtained, the net thrust of the propeller is found by resolving the local aerodynamic forces in the axial direction and then integrating along the blade span, i.e.,

(75) $\begin{equation*} T = N_b \int_{y_0}^{R} \left(dL \cos \phi - dD \sin \phi \right) \end{equation*}$

where $N_b$ is the number of propeller blades, and $R$ is the propeller’s radius. The term $dL \cos \phi$ is the axial component of the sectional lift, whereas $dD \sin \phi$ is the axial component of the sectional profile drag, which opposes thrust. In first-order analyses, the drag contribution to thrust is sometimes neglected, but it should be retained when profile drag is significant, especially for low-Reynolds-number propellers. Additionally, it is reasonable to assume the problem is axisymmetric; 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

(76) $\begin{equation*} P = \Omega N_b \int_{y_0}^{R} y \left(dL \sin \phi + dD \cos \phi \right) \end{equation*}$

where $dD$ denotes the local profile drag, which depends on the airfoil shape used on the blade and the incident Mach number. Notice the inclusion of the moment arm ${y}$ to obtain the torque (torque is a moment, the product of a force and a distance, or “arm”), and power is the product of torque and angular velocity. In this case, however, the in-plane component of the lift, $dL \sin \phi$ , contributes to the torque and hence to the power required to rotate the blades. This term reflects the inclination of the local aerodynamic force relative to the propeller plane, while $dD \cos \phi$ is the direct contribution of sectional profile drag to torque.

If the section drag coefficient is known, then

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

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.,

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

where $a_{\infty}$ is the local speed of sound, i.e., $C_d = C_d(M_{\rm hel})$ . Of course, if the helical Mach number exceeds the drag-divergence Mach number of the airfoil sections, the propeller’s efficiency will decrease rapidly.

Numerical Implementation

The propeller blade can be discretized into $N$ segments, as shown in Figure 21. To give any reasonable definition of the spanwise aerodynamic loads, $N$ should be between 30 and 100. Each segment is then small enough to be treated as having constant flow properties. Let ${y_i}$ denote the spanwise position of the $i$ -th segment, where $i = 1, \ldots, N$ , and let $\Delta y$ be the spanwise width of each segment.

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

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

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

and so the inflow angle is

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

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

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

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

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

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

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

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

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

which, for simplicity of illustration here, corresponds to a rectangular-rule approximation.

MATLAB code to implement blade element theory for propeller analysis. The inclusion of the inflow velocity is left to the reader.

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)
D = 2 * R; % Propeller diameter (m)
N = 100; % Number of segments
c = 0.1; % Chord length (m)
C_l_alpha = 5.7; % Lift-curve slope (per radian)
C_d = 0.01; % Profile drag coefficient

% Discretize the blade span
y0 = 0.15 * R; % Root cutout
y = linspace(y0, R, N);
dy = y(2) – y(1);

% Initialize variables
n = Omega / (2 * pi); % Rotational speed (rev/s)
J = V_inf / (n * D); % 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);

% Local velocity triangle
V = sqrt((Omega * y).^2 + V_inf^2);
phi = atan(V_inf ./ (Omega * y));
alpha = beta – phi;
alpha_0 = 0; % Zero-lift angle in radians; replace as appropriate for the airfoil section

% Lift coefficient
C_l = C_l_alpha * (alpha – alpha_0);

% 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) – dD .* sin(phi));
P = Omega * N_b * sum(y .* (dL .* sin(phi) + dD .* cos(phi)));

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

% Plot the results
figure;
plot(rad2deg(beta_range), C_T, ‘-o’);
xlabel(‘Blade pitch angle, \beta (deg)’);
ylabel(‘Thrust Coefficient, C_T’);
title(‘Thrust Coefficient vs. Blade Pitch Angle’);

figure;
plot(rad2deg(beta_range), C_P, ‘-o’);
xlabel(‘Blade pitch angle, \beta (deg)’);
ylabel(‘Power Coefficient, C_P’);
title(‘Power Coefficient vs. Blade Pitch Angle’);

% 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 terms of the thrust coefficient ${C_T}$ , power coefficient ${C_P}$ , and propulsive efficiency $\eta_p$ , analogous to airfoil and wing aerodynamic coefficients.^[6] All propellers have their characteristics quantified in this manner, with 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 propeller blade sections but the reference pitch at 75% of the blade span (i.e., at $y = 0.75 R$ ), which is a geometric quantity that can be measured.

Helical Pitch and Slip

Often, propeller pitch is measured in length units rather than as an angle. This refers to the geometric helical pitch traced by the reference blade section during one revolution, similar to a screw thread. Hence, the old name for a propeller is an airscrew. As illustrated in Figure 22, the geometric helical pitch is the axial distance that a blade reference section would advance in one complete revolution if it followed its geometric helix without slip. 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. However, the actual advance per revolution is usually smaller than the geometric pitch because the propeller operates in a yielding fluid flow and experiences induced velocity, profile, and wake losses; this difference is called slip.

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

(85) $\begin{equation*} \text{\small Circumference} = 2\pi \, y \end{equation*}$

At the blade tip, where $y = R$ , this circumference is

(86) $\begin{equation*} 2\pi R = \pi d \end{equation*}$

where $d=2R$ is the propeller diameter. Simultaneously, the propeller advances axially by the actual advance per revolution, $P_A$ , which is

(87) $\begin{equation*} P_A = \frac{V_{\infty}}{n} \end{equation*}$

where ${V_{\infty}}$ is the airspeed and $n$ is the rotational speed in revolutions per second.

The advance helix angle is defined by

(88) $\begin{equation*} \tan \phi_h = \frac{\text{\small Axial movement per revolution}}{\text{\small Circumferential movement per revolution}} \end{equation*}$

Substituting terms gives

(89) $\begin{equation*} \tan \phi_h = \frac{P_A}{2\pi \, y} = \frac{V_{\infty}}{2\pi \, n \, y} \end{equation*}$

For a reference span station, say $y_{\text{ref}} = 0.75R$ , which is the conventional reference location for catalog data, the advance helix angle becomes

(90) $\begin{equation*} \tan \phi_h = \frac{V_{\infty}}{2\pi \, n \, (0.75R)} = \frac{V_{\infty}}{0.75 \pi \, n \, d} \end{equation*}$

This advance helix angle should not be confused with the geometric blade pitch angle, $\beta$ , or the aerodynamic inflow angle, $\phi$ , used in the blade-element velocity triangle.

A concept sometimes used to describe a propeller’s efficiency is slip, which quantifies the difference between the geometric pitch and the actual advance through the air. The geometric pitch, $P_D$ , is the theoretical distance the propeller would advance in one revolution if the blade reference section followed its geometric helix without slip. The actual advance, $P_A$ , is the distance the propeller actually progresses through the air per revolution. The propeller slip ratio, $S$ , is defined as

(91) $\begin{equation*} S = \frac{P_D - P_A}{P_D} = 1 - \frac{P_A}{P_D} \end{equation*}$

A high slip ratio indicates a larger difference between geometric pitch and actual advance, whereas a low slip ratio indicates that the actual advance is close to the geometric pitch. While the slip ratio is sometimes used as an indicator of propeller performance, it has limited quantitative aerodynamic significance and should not be confused with 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, thereby facilitating communication and comparison among pilots, mechanics, and manufacturers. Diameter and pitch are expressed in inches. For example, the Sensenich 69CK propeller has a 69-inch diameter and is available with helical pitch values ranging from 42 to 58 inches, offering options such as 69 x 42, 69 x 44, and so on. While metric conversions are used internationally, general aviation favors inches for compatibility and ease of use.

Advance Ratio

The propeller advance ratio, ${J}$ , is a fundamental aerodynamic parameter that helps determine a propeller’s efficiency and thrust production under different operating conditions. It connects forward motion, rotational speed, and propeller design, serving as a crucial nondimensional parameter in propeller and rotor analysis. The advance ratio, ${J}$ , is defined as

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

where ${V_{\infty}}$ is the airspeed (in units of m/s or ft/s), $n$ is the rotational speed of the propeller (revolutions per second, rps), and $d$ = diameter of the propeller (in units of m or ft). Physically, the advance ratio is the distance the propeller travels per revolution, expressed in diameters; it is a nondimensional distance, or advance.

Thrust Coefficient

Recall that the thrust coefficient for a propeller is defined as

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

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

Representative ${C_T}$ versus ${J}$ results for a propeller are shown in Figure 23, with a separate curve for each reference blade pitch. Notice that the results from the blade element method discussed previously (i.e., the “theory”) agree well with the measurements.

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

Power Coefficient

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

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

where $P$ is 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 Figure 24. Again, the theory agrees well with the measurements.

Representative propeller performance curves show the power coefficient, $C_P$ , as a function of advance ratio, ${J}$ .

Propulsive Efficiency

Propeller efficiency measures how effectively the propeller converts available power into useful thrust. It is 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

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

which is a nondimensional statement that the propeller’s efficiency is the ratio of useful power to input power. Note that the symbols $\eta$ and $\eta_p$ denote propeller efficiency, with $\eta_p$ preferred in most contexts to avoid confusion with the propulsion system’s overall efficiency.

Using the previous definitions of ${C_T}$ and $C_P$ then

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

The corresponding representative $\eta_p$ versus ${J}$ results for a propeller are shown in Figure 25, again with one curve for each reference pitch angle.

Representative propeller performance curve of propulsive efficiency, ${\eta}$ , versus advance ratio, ${J}$ .

For a given propeller operating at a given blade pitch $\beta_{75}$ and rotational speed $n$ , propulsive efficiency increases with forward airspeed to a maximum, 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 operate aerodynamically only over relatively small ranges of angle of attack, i.e., local blade section angles of attack between 2 and 14 degrees, depending on the local Mach and Reynolds numbers. The local sectional angles of attack on the propeller depend not only on the propeller’s rotational speed and airspeed (or advance ratio) but also on how the propeller is twisted along its span. So, as airspeed changes, $n$ (propeller speed, in revolutions per second) is assumed to be constant. Then, the blade pitch must increase to maintain the most efficient angles of attack across the propeller. By gradually increasing blade pitch, the propeller can achieve peak efficiency over a much wider range of airspeeds, which is the purpose of a continuously variable-pitch, or “constant speed,” propeller.

Reference Blade Pitch

The reference geometric blade pitch angle of a propeller, often denoted $\beta_{75}$ , is measured at the section located at the 3/4 radius from the rotational axis, i.e., at $r = 0.75$ , where $r = y/R$ is the nondimensional radius. The pitch angle is taken between the local chord line of that section and the plane of rotation. It should be noted that it is a purely geometric quantity and does not represent the aerodynamic angle of attack. At that station, a material point on the blade traces a helix whose geometric helical pitch is given by

(97) $\begin{equation*} P_D(0.75)=2\pi\, \left( \dfrac{3}{4} \right)\,R\,\tan\beta_{75}= \dfrac{3\pi}{2}\,R\,\tan\beta_{75} \end{equation*}$

Given a reported helical pitch value at 3/4 radius, the corresponding geometric pitch angle is

(98) $\begin{equation*} \beta_{75}=\tan^{-1}\left(\frac{P_D(0.75)}{\dfrac{3\pi}{2}\,R}\right) \end{equation*}$

The 3/4-radius station matters because it connects directly to blade-element theory and propeller testing. Much of the useful aerodynamic loading occurs in the outer portion of the blade, but not exactly at the tip, where compressibility, tip losses, and structural details become important. A reference station near $r/R = 0.75$ is a useful representative location for specifying blade pitch.

This idea can be illustrated by considering simple radial weightings of the blade loading. The sectional velocity combines the rotational and axial components, i.e.,

(99) $\begin{equation*} U(r)^2=(\Omega \, R)^2\big(r^2+a^2\big) \end{equation*}$

where

(100) $\begin{equation*} a \equiv \frac{V_a}{\Omega \, R}=\frac{J}{\pi}+\lambda \quad \text{and} \quad \lambda \equiv \frac{v_i}{\Omega \, R} \end{equation*}$

Neglecting induced inflow corresponds to $\lambda\to 0$ , so that $a\to J/\pi$ . For a blade with uniform chord and approximately constant sectional aerodynamic coefficients, the sectional aerodynamic force scales approximately with $U^2$ . A thrust-weighted representative radius may be estimated as

(101) $\begin{equation*} r_T^{\ast}=\frac{\displaystyle\int_{0}^{1} U(r)^2\,r\,dr}{\displaystyle\int_{0}^{1} U(r)^2\,dr} \end{equation*}$

Substituting $U^2=(\Omega \, R)^2(r^2+a^2)$ gives

(102) $\begin{equation*} r_T^{\ast}=\frac{\dfrac{1}{4}+\dfrac{a^2}{2}}{\dfrac{1}{3}+a^2} =\frac{3}{4}\,\frac{1+2a^2}{1+3a^2} \end{equation*}$

In the small-inflow limit $a\to 0$ ,

(103) $\begin{equation*} r_T^{\ast}=\frac{3}{4} \end{equation*}$

For torque, and hence power at constant rotational speed, the sectional aerodynamic force is multiplied by the moment arm $r$ . The corresponding power-weighted representative radius is therefore

(104) $\begin{equation*} r_P^{\ast} = \frac{\displaystyle\int_0^1 U(r)^2\,r^2\,dr} {\displaystyle\int_0^1 U(r)^2\,r\,dr} \end{equation*}$

which, in the small-inflow limit, gives

(105) $\begin{equation*} r_P^{\ast} = \frac{\displaystyle\int_0^1 r^4\,dr} {\displaystyle\int_0^1 r^3\,dr} = \frac{4}{5} \end{equation*}$

Thus, under these simple assumptions, the representative radius is $3/4$ radius when weighted by thrust and $4/5$ radius when weighted by power. The conventional $3/4$ -radius station is particularly well-suited as a representative location for blade loading and geometric pitch, while remaining close to the region that dominates power absorption. These results are not universal because real propellers have root cutouts, nonuniform chord and loading distributions, induced inflow, profile drag, compressibility effects, Reynolds-number variation, and finite-blade effects.

Further Discussion of Propeller Performance

With this understanding of the flow at the blade section and how the propeller creates thrust, the curves of ${C_T}$ , $C_P$ , and $\eta_p$ , 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 operate close to the stall point. 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. Consequently, thrust is maintained while blade drag decreases, significantly increasing propulsive efficiency.

At the lowest angles of attack, the blades produce little lift, and the propeller produces little thrust. However, there is a range of airspeeds (assuming blade pitch does not change) for which good propulsive efficiency is obtained. Therefore, the best aerodynamic efficiency will be achieved only when all or most blade sections operate at or near the angles of attack that yield their best lift-to-drag ratio, typically between 2 and 8 degrees, depending on the incident Mach number. At higher airspeeds (or high ${J}$ ), blade sections experience reduced angles of attack and higher helical Mach numbers, which decreases thrust and efficiency unless the blade pitch increases further. Eventually, the blade pitch cannot be mechanically increased to improve efficiency, so efficiency decreases.

Note on Propulsive Efficiency

The results also explain the differences in propulsive efficiency of fixed-pitch versus constant-speed propellers, as shown in Figure 26. Notice that if the propeller pitch is fixed, its propulsive efficiency increases slowly with airspeed, reaches a maximum, and then decreases rapidly. The relatively low efficiency of a fixed-pitch propeller at low airspeeds results in poor takeoff and climb performance for the aircraft. Further increases in airspeed beyond the peak-efficiency speed will cause propeller efficiency to decrease precipitously. This behavior reduces the power delivered to the airstream as useful work, thereby setting an upper limit on the airplane’s achievable airspeed.

Compared to a fixed-pitch propeller, 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 to a fine pitch for takeoff. This provides good propulsive efficiency and low airspeed, resulting in markedly better takeoff and climb performance for the airplane. As airspeed increases, the blade pitch can be adjusted to maintain a constant rpm schedule, thereby allowing the propeller to closely track the peak-efficiency envelope. This is why airplanes with constant-speed propellers exhibit superior overall flight performance and can cruise at much higher airspeeds, as shown in Figure 26. A constant-speed propeller also lets the engine operate at a selected rotational speed across a range of flight conditions, improving operating efficiency and reducing pilot workload.

Check Your Understanding #1 – Calculating propeller performance

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

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 ${J}$ , 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 ( $^{\circ}$ )	$\eta_p$	${J}$	${V_{\infty}}$ (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

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

The propeller operates at an equivalent ISA density altitude of 8,000 ft. 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

$P = \varrho_{\infty} \, n^3 \, d^5 \, C_P = 0.001869 \times \left( \frac{2000}{60} \right)^3 \times 7.0^5 \, C_P = 1.163\times 10^6 \, C_P~\mbox{ft-lb/s}$

Converting to horsepower by dividing by 550 gives

$P = 2115.31 \, C_P~\mbox{hp}$

where the conversion to horsepower (hp) is obtained 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 airspeed and ${J}$ increase, the propeller tip speed may approach supersonic conditions. If this condition occurs, the propeller loses propulsive efficiency and produces more noise. This efficiency loss requires more power to produce the same thrust, increasing fuel consumption and adversely affecting overall flight performance. Supersonic airflow generates shock waves, dramatically increasing propeller noise. Propeller noise is a significant concern in aviation, affecting passenger comfort, community acceptance, and regulatory compliance.

Let $V_{\rm hel}$ denote the helical tip velocity, defined as 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 ${d}$ (= 2 $R$ ) is

(106) $\begin{equation*} V_{\Omega \, R} = \Omega \, R = \Omega \left( \frac{d}{2} \right) \end{equation*}$

and so the helical tip speed of the propeller is

(107) $\begin{equation*} V_{\rm hel} = \sqrt{ V_{\Omega \, R}^2 + V_{\infty}^2} \end{equation*}$

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

(108) $\begin{equation*} M_{\rm hel} = \frac{V_{\rm hel} }{a_{\infty}} \end{equation*}$

where $a_{\infty}$ is the speed of sound at the atmospheric conditions at which the propeller operates.

The measurements shown in Figure 27, for a high-speed NACA propeller with thin blades, eventually reach supersonic helical-tip Mach numbers. In this case, the measurements were taken at a constant blade pitch to examine variations in rotational speed. Notice that propulsive efficiency drops rapidly (from a peak of about 0.95) at higher airspeeds when rotational tip speeds are kept high. Compressibility losses increase as the helical-tip Mach number approaches 0.8. 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.

Measurements of propulsive efficiency on a NACA high-speed propeller as a function of helical tip Mach number.

Propeller Noise

Propeller noise is a significant contributor to the acoustic signature of propeller-driven aircraft, particularly during takeoff, climb, and high-speed cruise flight. It originates from unsteady aerodynamic forces on the blades, the periodic displacement of air by the rotating blade volume, and compressibility effects at high tip speeds. The dominant sources are thickness noise, loading noise, compressibility noise, and, at sufficiently high tip Mach numbers, shock-associated noise originating near the blade tips.

Thickness noise arises from the periodic displacement of air by the rotating blades. It is mainly associated with blade volume and radiates strongly at the blade-passing frequency and its harmonics. Loading noise is generated by the unsteady aerodynamic forces on the blades, primarily lift and drag. It is usually most intense during high-thrust phases such as takeoff and climb. Thickness noise is often modeled as a monopole-type source, whereas loading noise is commonly modeled as a dipole source. Together, these mechanisms produce the prominent tonal peaks in the propeller noise spectrum.

As the tip Mach number approaches or exceeds about 0.85, compressibility effects become increasingly important, and localized shock waves may form near the blade tips. This condition produces a significant increase in noise amplitude, harshness, and high-frequency content. The noise may become impulsive in character and highly directional. The schlieren image in Figure 28 illustrates this phenomenon for $M_{\rm tip} = 0.87$ , showing spiraling acoustic waves forming at the blade tips. These waves form a rotating wave pattern analogous to the Mach wave system produced by a body moving supersonically. The tip vortex and turbulent blade wake are also visible and contribute additional high-frequency acoustic content.

Schlieren flow visualization image showing the formation of a spiraling noise wave at the tip of a propeller.

Tip Mach Number Scaling

The far-field acoustic power radiated by a propeller increases rapidly with blade tip speed. A useful first-order scaling comes from considering the acoustic power radiated by a rotating aerodynamic force. From classical compact-dipole aeroacoustic theory, the radiated acoustic power scales approximately as

(109) $\begin{equation*} W \ \propto \ \frac{F^2 \, \Omega^2}{\varrho_\infty \, a_\infty^3} \end{equation*}$

where $W$ is the radiated acoustic power, $\varrho_\infty$ is the freestream air density, $F$ is a representative aerodynamic force on the blade, $\Omega$ is the angular velocity of the propeller, and $a_\infty$ is the freestream speed of sound. For a geometrically fixed propeller operating at comparable loading, the representative aerodynamic force may be written as

(110) $\begin{equation*} F \sim \varrho_\infty \, \Omega^2 \, R^4 \end{equation*}$

where $R$ is the blade radius and the factor $R^2$ supplies the characteristic blade area. Substitution into the acoustic-power scaling gives

(111) $\begin{equation*} W \ \propto\ \frac{ \left( \varrho_\infty \, \Omega^2 \, R^4 \right)^2 \Omega^2 }{ \varrho_\infty \, a_\infty^3 } = \varrho_\infty \frac{ \Omega^6 \, R^8 }{ a_\infty^3 } \end{equation*}$

Using the rotational tip Mach number

(112) $\begin{equation*} M_{\rm tip} = \frac{\Omega \, R}{a_\infty} \end{equation*}$

then gives

(113) $\begin{equation*} W \ \propto\ \varrho_\infty \, a_\infty^3 \, R^2 \, M_{\rm tip}^{\,6} \end{equation*}$

For a fixed propeller geometry and atmospheric conditions, this reduces to the commonly used trend

(114) $\begin{equation*} W \ \propto\ M_{\rm tip}^{\,6} \end{equation*}$

This result shows why propeller noise is so sensitive to blade tip speed. Because sound pressure level is logarithmic, the corresponding change in SPL is more appropriately written as

(115) $\begin{equation*} \Delta \text{SPL} \approx 10 \, m \log_{10}\left(\frac{M_{{\rm tip},2}}{M_{{\rm tip},1}}\right) \end{equation*}$

where $m$ is an empirical exponent. The value of $m$ is not universal, and it depends on blade count, blade loading, observer location, operating condition, and the noise component being considered. NACA propeller-noise data provide a useful example of this dependence, as shown in Figure 29.

Overall sound pressure level as a function of propeller tip Mach number for propellers with 3, 4, 6, and 8 blades. (Data adapted from NACA propeller-noise charts.)

The data in Figure 29 show that a 3-bladed propeller yields a fitted value of $m \approx 6$ , consistent with the simple acoustic-power scaling developed above. The fitted slopes correspond to an acoustic-power scaling of the form $W \ \propto \ M_{\rm tip}^{m}$ , or equivalently $\Delta \mathrm{SPL} \approx 10m\log_{10}(M_{{\rm tip},2}/M_{{\rm tip},1})$ . However, the fitted exponent increases with blade count, with representative values of about $m = 8.5$ , 11.7, and 15.2 for the 4-, 6-, and 8-bladed cases, respectively. This trend occurs because increasing the number of blades substantially reduces the low-Mach-number noise level, while the curves tend to converge as $M_{\rm tip}$ approaches unity. The convergence at higher tip Mach numbers suggests that compressibility effects, shock formation near the blade tips, and stronger high-harmonic content become increasingly important. Therefore, the $M_{\rm tip}^{6}$ scaling should be interpreted as a useful engineering trend for some lower-speed propeller-noise conditions, not as a universal law for all propellers.

Blade-Passing Frequency & Noise Reduction

The dominant frequency of propeller noise corresponds to the blade-passing frequency, defined as

(116) $\begin{equation*} f_{\rm bpf} = N_b \, f_{\rm rot} \end{equation*}$

where $N_b$ is the number of blades and $f_{\rm rot}$ is the shaft rotational frequency. Tonal components associated with thickness and loading noise are usually strongest at the blade-passing frequency and its harmonics. Increasing the number of blades increases the blade-passing frequency and can reduce the loading on each blade for a given total thrust, which helps explain the lower noise levels observed in the low-Mach-number region of Figure 29. However, adding blades does not eliminate the strong influence of the tip Mach number. As the tips approach transonic conditions, compressibility effects and shock-associated noise can dominate, and the benefit of additional blades may diminish.

Broadband noise arises from turbulence ingestion, unsteady blade loading, tip-vortex formation, and distorted inflow conditions. The radiation pattern is highly directional, with thickness and loading noise exhibiting distinct directivity patterns that depend on blade geometry, operating conditions, and observer location. For this reason, propeller noise reduction cannot be achieved by a single design change. Limiting tip speed is usually the most important way to avoid shock-induced impulsive noise. This can be accomplished by reducing rotational speed, limiting the diameter, or using swept, thinner blade tips to delay compressibility effects. Additional noise reductions may be achieved by increasing the blade count, reducing per-blade loading, optimizing sweep, twist, taper, and airfoil section selection, and minimizing installation-related inflow distortion. Pusher configurations require particular care because the propeller may ingest fuselage or wing wakes, which can increase unsteady loading and noise. Tractor propellers operating in cleaner inflow generally have lower noise for a given thrust.

Check Your Understanding #2 – Calculating a propeller’s helical tip Mach number

Refer back to the example in Check Your Understanding #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 propeller’s rotational speed at the tip and the freestream (airspeed).

Show solution/hide solution.

Let $V_{\rm hel}$ denote the helical tip velocity, defined as 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 under 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 note that at airspeeds above 400 ft/s, the propeller blade tips would likely operate near or just beyond the critical Mach number (i.e., the onset of transonic flow), which, for thin propeller blade tips, is approximately 0.8. Under these conditions, the propeller will likely experience a loss of propulsive efficiency.

Propeller Selection Process

Propeller selection must match the aircraft’s mission, performance goals, and propulsion system. Key factors include thrust requirements across flight phases, powerplant type (piston, turboprop, or electric), cruise efficiency, and installation constraints. The propeller diameter significantly affects thrust, power consumption, and efficiency. Larger diameters enhance low-speed performance but may be limited by ground clearance, airframe geometry, or high tip speeds. At higher rpm, blade tips must remain subsonic to avoid compressibility losses and the rapid noise increase associated with tip Mach number.

The blade count affects the thrust a propeller can produce at a given diameter. More blades let the disk absorb more power, which helps high-power engines or designs constrained by ground clearance or airframe layout. However, additional blades add weight, drag, and mechanical complexity, especially in variable-pitch systems. Blade geometry must also be optimized spanwise to maintain efficiency across varying inflow conditions. Key factors include the airfoil section, chord distribution, and twist, which typically follow a hyperbolic pattern from the root to the tip.

Variable-pitch or constant-speed propellers maintain propulsive efficiency by adjusting blade angles in flight and are standard on high-performance piston and turboprop aircraft. Material selection also matters. Wood is lightweight and inexpensive, but it is limited to low-power use of up to 200 shp (149.4 kW). In contrast, aluminum is at least three times as strong but approximately six times as heavy by volume. Composite blades combine low weight with high strength, allowing for thinner, more efficient shapes in high-speed turboprop applications.

The propeller must also match the engine’s power and torque characteristics. Piston engines operate efficiently at lower rpm (less than 3,000) and can often drive propellers directly. Turboprop engines typically require a reduction gearbox between the turbine and the propeller because the turbine operates efficiently at much higher rotational speeds than the propeller.^[7] Electric motors can also drive propellers directly because they produce useful torque over a broad speed range. Small UAV propellers may operate at several thousand rpm, whereas larger propellers generally require lower rotational speeds to control tip Mach number and noise. Additional concerns for propeller integration include vibration, noise, and inflow distortion, particularly in pusher configurations affected by an upstream wake. A successful design will balance these factors to deliver efficient, reliable, and quiet propulsion that suits the aircraft and its mission.

Steps for Propeller Selection

The following are the specific steps that are involved in the selection of a propeller:

Define the mission profile and flight envelope. These requirements may include the maximum takeoff weight, cruise speed and altitude, climb performance, and powerplant characteristics and limitations such as available power and allowable rotational speed (rpm) range. Airframe installation constraints, such as ground clearance or nacelle position, must also be considered.
Estimate thrust and power requirements. Use aircraft drag estimates and mission analysis to determine the required thrust during takeoff, climb, and cruise. Combine this with available shaft power from the powerplant to calculate the required propeller efficiency.
Select a candidate propeller diameter and rotational speed. Select a diameter that provides sufficient disk area for thrust without exceeding the tip Mach number limit of about 0.85 at maximum propeller rpm, minimizing compressibility losses.
Choose or design a blade geometry. For custom designs, one can use blade-element theory (BET) or similar methods to determine the optimal twist and planform (chord) distributions. For off-the-shelf options, consult the manufacturer’s data sheets and performance characteristics (e.g., $J$ curves) to determine the thrust coefficient, power coefficient, and efficiency, which will help select the propeller.
Calculate the advance ratio and match it to the performance requirements. Use the $J$ parameter to determine the aircraft’s operating point, then compare it with the propeller performance curves. Generally, one aims to operate the propeller near peak efficiency at cruise while still ensuring sufficient thrust and propulsive efficiency at lower airspeeds, such as takeoff.
Determine whether variable pitch is required. If the flight conditions span a wide range of speeds or power settings, a variable-pitch propeller may be required to maintain propulsive efficiency. Fixed-pitch or ground-adjustable designs are simpler but can be optimized for only one flight condition.
Evaluate structural and material choices. Consider blade loads, centrifugal forces, vibration, and the possibilities of material fatigue. Common materials include wood, aluminum, and composites, each with distinct trade-offs in weight, strength, durability, and cost.
Integrate with the engine and airframe. Ensure compatibility with the engine’s torque curve and power output. It may be necessary to account for airframe-induced inflow distortion, especially in pusher configurations. Validate hub mounting, spinner clearance, and “ram” airflow to the cooling system.
Test and refine the selection. After preliminary design or selection, verify the propeller’s performance through ground runs and flight testing. Normally, aircraft manufacturers instrument the powerplant and aircraft to record sufficient parameters to verify or revise the preliminary propeller design, as needed, based on flight-test data.

Speed-Power Coefficient (Weick) Method

The speed-power coefficient, $C_S$ , is a nondimensional parameter introduced by Fred Weick at NACA to characterize propeller performance independently of physical dimensions, such as diameter or blade pitch. Weick originally used the $C_S$ coefficient to develop empirical performance maps of propeller efficiency, thereby reducing or eliminating the need for wind-tunnel or shaft-torque (i.e., dynamometer) measurements. This approach remains useful today, particularly when evaluating geometrically similar propellers or analyzing flight data from smaller propeller-driven aircraft. While the advance ratio $J$ is central to propeller selection and aerodynamic design, the $C_S$ coefficient is often more practical for interpreting in-flight performance, particularly under less-than-ideal operating conditions. For fixed-pitch propellers, which are efficient only over a narrow range of conditions, the $C_S$ value at the intended operating conditions helps identify the most appropriate propeller. This is because it links flight speed, power, and rpm to the known or predicted performance characteristics of both the powerplant and the aircraft.

Definition and Derivation

Weick^[8] noted that while the ${C_T}$ , $C_P$ , and $J$ 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 $V_{\infty} /n \, d$ 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 ${V_{\infty}}$ and $n$ but have different diameters, ${d}$ , and, therefore, other values of $V_{\infty} / n \, d$ .”

To this end, Weick showed that a new dimensionless propeller coefficient involving $P$ , $n$ , and ${V_{\infty}}$ , but not the propeller diameter, may be obtained from $C_P$ and $J = V_{\infty}/(n d)$ . Starting from

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

and

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

then

(119) $\begin{equation*} \frac{J^5}{C_P} = \frac{(V_{\infty}/n d)^5}{P/(\varrho_{\infty} n^3 d^5)} = \frac{\varrho_{\infty} V_{\infty}^5}{P n^2} \end{equation*}$

The speed-power coefficient is usually taken as the fifth root of this quantity, i.e.,

(120) $\begin{equation*} C_S = \frac{J}{C_P^{1/5}} = \left(\frac{\varrho_{\infty} \, V_{\infty}^5}{P \, n^2}\right)^{1/5} = V_{\infty}\left(\frac{\varrho_{\infty}}{P \, n^2}\right)^{1/5} \end{equation*}$

where $\varrho_{\infty}$ is the air density, ${V_{\infty}}$ is the true airspeed of the propeller through the air, $P$ is the shaft power available to the propeller, and $n$ is the propeller’s rotational speed in revolutions per second. Notice that the numerical value of $C_S$ must be calculated using a consistent system of units.

It should be apparent that $C_S$ is called a “speed-power” coefficient because it includes the rotational and forward speeds, as well as the power the propeller needs to absorb.^[9] The diameter and pitch selection of a propeller is based on finding the optimal balance between efficiency and operational constraints under specific airplane design conditions. The power required for flight depends on the aircraft and its operating conditions. Because a fixed-pitch propeller operates most efficiently at a single point in its performance envelope, the $C_S$ coefficient can help determine the best-matched propeller for the desired airspeed and power available at the propeller’s shaft.

Practical Usage and Application

The speed-power coefficient $C_S$ has several practical applications, including the selection of fixed-pitch propellers, in-flight performance testing, and aircraft performance analysis, particularly when complete aerodynamic data are unavailable. In many aircraft test programs, thrust is not directly measurable, but airspeed, rpm, shaft power, and air density are usually obtainable. Because the $C_S$ coefficient incorporates both power and air density, it naturally accounts for variations in throttle setting and density altitude, thereby reducing test data across different flight conditions to a more consistent, useful form.

Wind Tunnel Testing

The value of the $C_S$ coefficient can first be explained with reference to propeller performance measurements made in the wind tunnel. Propeller testing in the tunnel often involves a series of geometrically similar designs, with pitch and airspeed varied in uniform increments. For example, the NACA tested propellers in its 20-foot wind tunnel using a system that enabled continuous pitch adjustments across a wide range of angles and airspeeds until the blade tips exceeded their critical Mach numbers. High helical-tip Mach numbers typically limit propeller performance in terms of efficiency, noise, or both. For a fixed-pitch propeller with a given blade design, the available range of pitch angles is typically limited to two or three.

If performance data are available for candidate propellers, the corresponding $C_S$ value can also be calculated and compared with the efficiency curves. Start with the definition

(121) $\begin{equation*} C_S = \left( \frac{\varrho_\infty \, V_\infty^5}{P \, n^2} \right)^{1/5} \end{equation*}$

Using

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

and

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

then

(124) $\begin{equation*} C_S = \left( \frac{\varrho_\infty (J n d)^5} {C_P \varrho_\infty n^3 d^5 \, n^2} \right)^{1/5} = \left(\frac{J^5}{C_P}\right)^{1/5} = \frac{J}{C_P^{1/5}} \end{equation*}$

The propulsive efficiency is

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

and, using the standard definitions of ${C_T}$ , $C_P$ , and $J$ , this becomes

(126) $\begin{equation*} \eta_p = \frac{C_T J}{C_P} \end{equation*}$

Therefore,

(127) $\begin{equation*} J = \frac{\eta_p C_P}{C_T} \end{equation*}$

and substituting into the expression for $C_S$ gives

(128) $\begin{equation*} C_S = \frac{J}{C_P^{1/5}} = \frac{\eta_p C_P}{C_T C_P^{1/5}} = \left(\frac{\eta_p}{C_T}\right) C_P^{4/5} \end{equation*}$

This form allows $C_S$ to be evaluated directly from the standard propeller “ $J$ curves,” provided that ${C_T}$ , $C_P$ , and $\eta_p$ are taken at the same value of $J$ .

An example of this type of analysis is shown in Figure 30. The curves in the plot show $C_S$ as a function of advance ratio $J$ for each blade pitch, along with the corresponding curves of propeller efficiency $\eta_p$ . The “curve of best efficiency” traces the locus of operating points at which each propeller configuration is most efficient. By computing $C_S$ for a desired flight speed, shaft power, and rpm, one can determine the corresponding $J$ and select a propeller diameter that places the powerplant-aircraft system near the optimal propeller curve. Alternatively, for a given propeller and $J$ value, using $C_S$ allows the power $P$ and revolutions per second $n$ available from a given powerplant to match this propeller.

Plot showing $C_S$ as a function of advance ratio $J$ , along with curves of propeller efficiency at various blade pitch angles.

If the desired advance ratio is known or can be determined, then the corresponding propeller diameter is computed using

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

The blade pitch is then chosen so that the propeller exhibits peak efficiency near the target $C_S$ , with a deliberate bias toward lower values to avoid performance degradation at takeoff and climb. This trade-off is necessary because fixed-pitch propellers cannot be adjusted in flight. Alternatively, the propeller speed, $n$ , can be chosen for a given propeller diameter, i.e.,

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

Regarding wind tunnel testing, the $C_S$ coefficient provides no additional information for propeller selection, since the diameter and pitch required to meet a given level of performance can be determined entirely from the “ $J$ -curves.” What it does do, however, is verify that the selected propeller matches the power available from the powerplant at a given airspeed and density altitude, and that the corresponding values of the $J$ and $C_S$ coefficients are within the normal operating range of that propeller and are close to the line of best propulsive efficiency. This approach ensures that the selected propeller uses the available power efficiently, without overloading or underloading.

Flight Testing

In flight testing, the speed-power coefficient $C_S$ provides a valuable way to assess propeller performance when the propeller “ $J$ curves” are unavailable. Its utility lies in the fact that all of the constituent terms of $C_S$ , i.e.,

(131) $\begin{equation*} C_S = V_{\infty} \sqrt[5]{\frac{\varrho_{\infty}}{P \, n^2}} \end{equation*}$

can be obtained from basic flight instrumentation. The airspeed is measured using a Pitot-static system and corrected as needed for instrument, position, and density effects to obtain the true airspeed, ${V_\infty}$ . The air density $\varrho_\infty$ is computed from ambient pressure and temperature measurements using the equation of state, or from pressure-altitude and temperature measurements using the ISA model.

The shaft power $P$ can be estimated from powerplant data using performance charts or “decks” in the form of numerical data. For example, in piston engines, the shaft power required to compute $C_S$ is estimated from engine performance data that correlate manifold pressure, engine rpm, pressure altitude, and outside air temperature with brake power (e.g., bhp) output. These values are measured from the aircraft’s engine instruments and, in post-flight analysis, are cross-referenced against charts or tables provided by the engine manufacturer. This indirect method, while approximate, is sufficiently accurate for trend analysis and performance comparison. For electric propulsion, the electrical input power, measured from voltage and current, must be converted to the mechanical shaft power delivered to the propeller before computing $C_S$ . This conversion requires an estimate or calibration of the motor, ESC, and wiring efficiencies, so that

(132) $\begin{equation*} P_{\rm shaft} \approx \eta_{\rm elec} \, V_{\rm batt} \, I \end{equation*}$

where $\eta_{\rm elec}$ is the net electrical-to-shaft efficiency. Using uncorrected electrical input power will understate the value of $C_S$ and may make the propeller appear more heavily loaded than it actually is.

By evaluating $C_S$ at a sequence of flight-test points for different values of $J$ , engineers can assess a propeller’s suitability, evaluate off-design behavior, and ensure proper matching between the propeller and powerplant, all without needing detailed propeller “ $J$ -curves.” As shown in Figure 31, which presents notional flight-test data, the $C_S$ and $J$ values obtained for each propeller should form a consistent trend if the measurements are repeatable. However, a smooth or tightly clustered trend does not by itself establish that the propeller is well matched to the powerplant and aircraft. Suitability must be assessed by comparing the measured operating points with known or estimated efficiency characteristics, particularly the region of best propulsive efficiency. Significant scatter may indicate measurement uncertainty or inconsistent operating conditions. Multiple flights with each propeller should be conducted to establish repeatable trends.

Calculations of $J$ and $C_S$ from flight-test data can assess how well a propeller matches the powerplant and the aircraft.

Pusher Propellers

Tractor propellers, mounted at the front of the aircraft and operating in undisturbed freestream flow, are the most common configuration in aircraft design. However, numerous aircraft have been developed with pusher propellers, in which the propeller is located aft of the fuselage or wing. In this arrangement, the propeller operates in the airframe’s wake, which can affect its aerodynamic performance. Today, pusher configurations are frequently employed in uncrewed aerial vehicles (UAVs) and drones, as illustrated in Figure 32.

The wake generated by the fuselage affects a pusher propeller’s aerodynamic performance because it ingests air altered by upstream boundary-layer growth and, in some cases, by separated flow regions. The main aerodynamic consequence is a distorted inflow profile across the propeller disk, with the inner region typically experiencing a significant velocity deficit. This reduces the effective mass flow rate through the propeller and changes the local angles of attack on the blades. In severe cases, ingestion of a fuselage wake can introduce unsteady, turbulent structures into the flow, potentially degrading propeller performance and increasing noise.

There are very few direct measurements comparing the performance of pusher propellers with that of tractor propellers under the same or similar operating conditions. Fred Weick alludes to wind tunnel tests conducted by the Royal Aircraft Establishment (RAE) in 1922. The results, summarized in Figure 33, suggest that even with relatively large fuselage sizes relative to the propeller diameter, the effects on maximum propulsive efficiency of pusher versus tractor propellers are minimal, perhaps on the order of 2%. Nevertheless, the wide range of fuselage shapes and downstream wake profiles encountered in practical design indicates that additional modeling effort is warranted to more precisely quantify potential interference effects.

Measurements of the maximum efficiency of tractor and pusher propellers. From: Weick, F., “Aircraft Propeller Design,” 1930.

Flow Model

Consider a flow model of a pusher propeller, as shown in Figure 34, where the inflow into the propeller disk is composed of two concentric zones: 1. An inner disk of radius $b$ that is subject to a velocity deficit $\delta$ , i.e., $V_\infty' = V_\infty \, (1 - \delta)$ . The parameter $2b$ denotes the characteristic wake width or spread, typically comparable to the fuselage diameter. 2. An outer annular region from $y = b$ to $y = R$ that is exposed to undisturbed freestream flow, i.e., $V_\infty' = V_\infty$ . In this model, each region is treated as an independent actuator disk segment, where $A_1 = \pi \, b^2$ and $A_2 = \pi \, (R^2 - b^2)$ denote the inner and outer areas, respectively. Assume both regions have the same average induced velocity $v$ at the propeller plane.

Thrust

The thrust on each region is

(133) $\begin{equation*} T'_1 = 2 \varrho \, A_1 \, v \left( (1 - \delta) V_\infty + v \right) \quad \text{and} \quad T'_2 = 2 \varrho \, A_2 \, v \left( V_\infty + v \right) \end{equation*}$

Therefore, the total thrust is

(134) $\begin{equation*} T' = T'_1 + T'_2 = 2 \varrho \, v \bigg( A_1 \left(1 - \delta \right) V_\infty + A_2 V_\infty + v \left(A_1 + A_2 \right) \bigg) \end{equation*}$

Notice that $A$ = $A_1 + A_2 = \pi R^2$ , so this previous result becomes

(135) $\begin{equation*} {T'} = 2 \varrho \, v \left( V_\infty A \left( 1 - \delta \, \frac{A_1}{A} \right) + v A \right) \end{equation*}$

The corresponding thrust in the clean inflow case (no velocity deficit) is

(136) $\begin{equation*} T = 2 \varrho \, v \left( V_\infty A + v A \right) \end{equation*}$

and so the thrust ratio (distorted inflow versus clean) is

(137) $\begin{equation*} \frac{T'}{T} = \frac{V_\infty \left( 1 - \delta \, \dfrac{A_1}{A} \right) + v}{V_\infty + v} \end{equation*}$

which for ${v \ll V_\infty}$ , i.e., a propeller at typical thrust levels in the normal working state, becomes

(138) $\begin{equation*} \frac{T'}{T} \approx 1 - \delta \, \left( \frac{b}{R} \right)^2 \end{equation*}$

Power and Efficiency

Inflow distortion also affects the power required by a pusher propeller. If part of the propeller disk operates in the velocity deficit behind a fuselage or other upstream body, then the propeller ingests air with less axial kinetic energy than the undisturbed freestream. In an idealized actuator-disk sense, this wake ingestion can reduce the kinetic energy loss associated with propulsion and may yield a small apparent improvement in propulsive efficiency. This result is consistent with the measurements summarized by Weick, which show that pusher propellers can achieve maximum efficiencies slightly higher than or comparable to those of tractor propellers for certain fuselage-propeller arrangements.

However, this conclusion requires careful interpretation. The benefit does not come from the pusher propeller being intrinsically more efficient than a tractor propeller. Rather, it arises because the propeller may recover part of the kinetic-energy deficit already present in the wake of the upstream body. In practice, the same wake also introduces nonuniform inflow, unsteady blade loading, turbulence ingestion, possible separated flow, increased vibration, and increased noise. These effects can offset or reverse any ideal wake-ingestion benefit.

Therefore, a pusher propeller may show a small installed-efficiency advantage in some configurations, especially when it is well matched to the upstream wake. In other installations, the distorted inflow may reduce efficiency and increase noise. The simplified thrust estimate above is useful for showing how a velocity deficit modifies the effective disk loading, but the installed efficiency of a pusher propeller is ultimately an airframe-propeller integration problem that must be assessed using wind-tunnel measurements, flight testing, or coupled aerodynamic analysis.

Low Reynolds Number Propellers

Low-Reynolds-number propellers are designed to operate efficiently in flow regimes where the Reynolds number based on the blade sectional chord, often evaluated near the 75% radius station, is typically from about $10^4$ to several $10^5$ . These operating conditions are typical of small aircraft, UAVs, and drones. In this regime, blade sectional drag is higher than at chord Reynolds numbers above 10⁶, which significantly reduces thrust and lowers the propeller’s overall propulsive efficiency.

As the Reynolds number drops below about one million ( $Re = 10^6$ ), viscous boundary-layer effects, transition, laminar separation bubbles, and separation begin to strongly influence the lift and drag characteristics of airfoil sections. Figure 35 shows the profound effects of operating airfoils at lower Reynolds numbers, ranging from 20,000 to 3,000,000. At the highest Reynolds number, $3 \times 10^6$ , the lift coefficient varies almost linearly with angle of attack, which is typical of airfoil behavior at higher Reynolds numbers. However, as the Reynolds number drops below a million, the lift and drag curves become much rounder and eventually nonlinear with angle of attack. Indeed, these effects start to become pronounced even at Reynolds numbers of 500,000.

Aerodynamic coefficients of a NACA 4412 airfoil when operating at different Reynolds numbers.

The corresponding drag polar for these conditions offers another useful way to summarize the effects of the Reynolds number on aerodynamic characteristics. Notice again the profound effects of reducing the Reynolds number below 500,000, which deleteriously affect the lift-to-drag ratio, especially below 50,000. For extremely low Reynolds numbers below $10^4$ , lift-to-drag ratios below 5 are expected, at least for “conventional” airfoil sections.

Therefore, the impact of these airfoil section effects on propeller performance at low chord Reynolds numbers is profound. For example, Figure 36 illustrates the efficiencies of a two-bladed propeller at “full scale” and of two smaller ones operating at lower tip Reynolds numbers, based on the average chord. Notice the marked drop in maximum efficiency at lower Reynolds numbers, as well as the range of Reynolds numbers over which the system can operate efficiently. Most of these effects on propulsive efficiency result from the higher sectional drag coefficients at these low Reynolds numbers.^[10] However, there is also a reduction in maximum sectional lift values, which limits thrust production. These considerations are essential for propeller applications in most UAVs and drones, which typically achieve propulsion efficiencies of 50%- 60%, compared with the 80% or higher achieved by “full-scale” propellers.

Efficiency curves of a two-bladed propeller at “full-scale” and small (low blade tip Reynolds number) scales.

Unique low-Reynolds-number airfoil designs are typically employed to minimize sectional drag on smaller propellers, much as those used on fixed wings, but these designs must be carefully tailored. A larger solidity (blade area) also helps generate the necessary thrust under these conditions, perhaps at the expense of some loss in propulsive efficiency. Designing low-Reynolds-number airfoils for propellers typically involves using advanced computational fluid dynamics (CFD) simulations to analyze and optimize performance. These simulations help engineers understand the flow behavior around propeller blades and refine design parameters. Wind tunnel testing is 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, including the drone’s size and weight, aerodynamic and motor characteristics, and desired flight performance, such as airspeed, range, and endurance. A VTOL-capable drone may use separate propeller sets for lift and propulsion (i.e., a hybrid concept). In this setup, the lifting propellers are switched off during forward flight, and a conventional fixed wing generates lift to support the drone’s weight. In some cases, propellers may provide both lift and propulsion, as in a tiltrotor or tilting-wing concept, which entails additional compromises when selecting the optimal propeller. Assuming such propellers are fixed-pitch only, the selected propeller(s) will inevitably be a compromise.

Steps for Propeller Selection

The following are the basic steps to select the appropriate propeller(s) for low-Reynolds-number operation. Without access to a test rig and a wind tunnel, the process will necessarily involve trial and error, sound judgment, and a degree of common sense. Fortunately, at this smaller scale, propellers are inexpensive, allowing the final decision to be made through flight testing with a selection of different propellers until the best performance is achieved.

Propeller diameter. To determine propeller requirements, consider the drone’s weight, aerodynamic efficiency (e.g., lift-to-drag ratio), estimated motor specifications, battery type and voltage, and desired flight characteristics. Heavier drones generally require greater disk area, more blade area, higher rotational speed, or some combination of these to generate the required thrust and power. For lifting propellers, larger diameter and lower disk loading are usually beneficial, whereas the appropriate pitch depends on the intended operating condition, motor characteristics, and allowable rpm. A VTOL drone requires sufficient thrust from its lifting propellers to overcome its own weight and maintain lift.
Calculate the thrust and power requirements. A multirotor must produce a total static thrust at least equal to its weight to hover, but practical designs require additional thrust margin for climb, control authority, gust response, maneuvering, and degraded battery performance. A total thrust-to-weight ratio of about 2:1 is a common design target for agile quadcopters, although endurance-oriented vehicles may use a smaller margin. For example, if a quadcopter has a mass of 1 kg, its weight is about 9.81 N, so a 2:1 design target corresponds to a total static thrust of about 19.6 N, or about 4.9 N per motor-propeller pair. The power requirements can be estimated using momentum theory, starting with an initial estimate of the required propeller diameter.
Match to motor. Match the propeller’s initial size and pitch to the motor’s recommended operating range, considering the motor’s power rating, voltage, current limit, thermal limit, and speed constant, usually denoted by $K_V$ in rpm/V. The $K_V$ value is not a power rating; it indicates the approximate no-load rotational speed per volt applied to the motor. Different propellers can dramatically affect the motor rpm, current draw, thrust, efficiency, and thermal margin, as well as the drone’s speed, maneuverability, and endurance.
Establish the type of propeller. 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 achieve the highest propulsive efficiency at the propeller’s operating ${J}$ for a given combination of required thrust, propeller rpm or rps, and airspeed. To this end, propeller charts, i.e., ${C_T}$ versus ${J}$ , $C_P$ versus ${J}$ , and $\eta_p$ versus ${J}$ , should be consulted. The speed-power coefficient, $C_S$ , should be evaluated.
Review small-scale propeller measurements. They are available in chart form and as data files for a comprehensive range of drone propellers. Measurements are available in the static-thrust (hover) condition and in forward flight over a range of ${J}$ values. Remember that small propellers often have a relatively low hover figure of merit and lower propulsive efficiencies than larger ones, typically 50% to 60%, depending on Reynolds number, blade geometry, rpm, loading, and operating conditions. This lower efficiency increases power demand, reducing flight duration for a given battery charge.
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 generally best to stay within the recommended ranges, although other propellers may also work.
Decide on the propeller geometry. Propellers are specified by diameter and pitch. For example, a propeller marked 20 x 14 denotes a diameter of 20 inches and a helical pitch of 14 inches; the propeller can also be specified in cm. Propellers are available in standard sizes, such as 12 inches and 14 inches. Lifting (hovering) propellers will have relatively low pitch settings, whereas propulsive propellers will have higher pitch settings. For a fixed diameter and rotational speed, higher flight speeds generally require a higher pitch to keep the blade sections near their efficient angles of attack. However, the appropriate pitch must still be matched to the motor power, rpm, required thrust, and desired operating range.
Decide on the propeller material. Plastic propellers are lightweight and inexpensive but less durable. Carbon-fiber propellers offer better aerodynamic performance but cost significantly more. It is often advisable to purchase and test three inexpensive propellers initially, perhaps with higher and lower pitches than the first choice, before settling on a carbon-fiber propeller.

Hovering (Static Thrust) Propeller

Propellers generally have lower hovering figures of merit than helicopter rotors, partly because of their typically higher disk loadings and suboptimal blade designs. The power requirements can be decomposed into an induced component that depends strongly on thrust and a profile component associated with blade drag. For a fixed geometry and rotational speed, the profile component is often treated as approximately independent of thrust, although it can vary with Reynolds number, blade pitch, and operating condition.

Induced Power

From momentum theory, the induced velocity at the propeller’s disk is

(139) $\begin{equation*} v_i = \sqrt{\frac{T}{2 \varrho_{\infty} \, A}} \end{equation*}$

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

(140) $\begin{equation*} {P_i} = T \, v_i = T \sqrt{\frac{T}{2 \varrho_{\infty} \, A}} = T \sqrt{\frac{2 T}{\varrho_{\infty} \, \pi \, d^2}} \end{equation*}$

Expressing this result in terms of the propeller thrust coefficient, i.e., $T = C_T \, \varrho_{\infty} \, n^2 \, d^4$ , gives

(141) $\begin{equation*} v_i = \sqrt{\frac{2 C_T \, \varrho_{\infty} \, n^2 \, d^4}{\varrho_{\infty} \, \pi \, d^2}} = n \, d \sqrt{\frac{2 C_T}{\pi}} \end{equation*}$

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

(142) $\begin{equation*} C_{P, \text{ideal}} = \frac{P_i}{\varrho_{\infty} \, n^3 \, d^5} = \frac{\sqrt{2} \, C_T^{3/2}}{\sqrt{\pi}} \end{equation*}$

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

(143) $\begin{equation*} C_{P_i} = \kappa \frac{\sqrt{2} \, C_T^{3/2}}{\sqrt{\pi}} \end{equation*}$

Profile Power

The profile power $P_0$ is the power required to overcome the aerodynamic drag of the propeller blades as they rotate through the air. In the simplified analysis used here, the section drag coefficient is assumed constant, so the profile power is treated as independent of thrust. For a propeller in hover, assuming constant blade chord $c$ , constant section drag coefficient $C_{d_{0}}$ , the differential profile power on a blade segment at radius $y$ is given by

(144) $\begin{equation*} dP_0 = \dfrac{1}{2} \, \varrho \, \Omega^3 \, y^3 \, c \, C_{d_{0}} \, dy \end{equation*}$

Multiplying by the number of blades $N_b$ and integrating from the root ( $y = 0$ ) to the tip ( $y = R$ ) yields the total profile power, i.e.,

(145) $\begin{equation*} P_0 = \int_0^R N_b \, dP_0 = \dfrac{1}{8} \, \varrho \, N_b \, c \, C_{d0} \, \Omega^3 \, R^4 \end{equation*}$

The nondimensional profile power coefficient, using the propeller convention adopted in this chapter, is defined as

(146) $\begin{equation*} C_{P_0} = \frac{P_0}{\varrho \, n^3 \, d^5} \end{equation*}$

and substituting $\Omega = 2\pi n$ and $d = 2R$ gives

(147) $\begin{equation*} C_{P_0} = \frac{N_b \, c \, C_{d_0} \, \pi^3}{16 \, d} \end{equation*}$

The solidity of the propeller is

(148) $\begin{equation*} \sigma = \frac{N_b \, c}{\pi R} = \frac{2 \, N_b \, c}{\pi d} \end{equation*}$

and substituting into the expression for $C_{P_0}$ gives

(149) $\begin{equation*} C_{P_0} = \frac{\sigma \, C_{d_0} \, \pi^4}{32} \end{equation*}$

Total Power

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

(150) $\begin{equation*} C_P = C_{P_i} + C_{P_0} = \kappa \frac{\sqrt{2} \, C_T^{3/2}}{\sqrt{\pi}} + \frac{\sigma C_{d_0} \, \pi^4}{32} \end{equation*}$

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

(151) $\begin{equation*} C_P = \left( \frac{1}{FM} \right) \frac{\sqrt{2} \, C_T^{3/2}}{\sqrt{\pi}} \end{equation*}$

As shown in the power-polar diagram in Figure 37, with suitable values of $C_{d_0}$ and $\kappa$ (or $FM$ ), the modified momentum theory can predict hovering performance well. The low Reynolds number, combined with the typical (and excessive) spanwise blade twist and non-optimal chord distribution used on these propellers, generally results in relatively poor hovering performance. In this case, representative values such as $\kappa = 1.35$ and $C_{d_0} = 0.2$ agree reasonably well with these measurements over the plotted range. Equivalently, an effective figure of merit of about $FM \approx 0.5$ may be used as a compact empirical representation near the operating condition of interest. Notice that increasing propeller pitch produces more thrust for the same rpm; the resulting measurements then move up the curve to higher values of ${C_T}$ and $C_P$ . At higher rpm, the blade Reynolds numbers increase, which can improve the section’s aerodynamic characteristics. In nondimensional form, this effect may appear as a modest increase in the thrust coefficient or a reduction in the power coefficient at a given operating condition, although the dimensional thrust and power generally increase with rpm.

Representative power polar for a hovering propeller, i.e., the static thrust condition. The measurements were made by sweeping the rotor rpm.

Worked Example #3 –Drone in hovering flight

A VTOL quad-rotor drone has a mass of 7.5 kg at takeoff, including the battery, camera, and other components. Estimate the required propeller and motor size. Assume MSL ISA conditions.

Show solution/hide solution.

As a general rule of thumb, the propeller(s) must provide enough thrust to lift a mass of 7.5 $\times$ 2 = 15 kg, so 147.15 N of thrust is required to ensure sufficient takeoff and maneuver performance. For example, for a quadcopter, each motor must be attached to a propeller to provide sufficient thrust to overcome a 3.75 kg load, i.e., 36.8 N prt rotor.
Estimate the propeller’s diameter and pitch to produce the required thrust. The thrust equation is $T = \varrho_{\infty} \, n^2 \, d^4 \, C_T$ , where in this case $T$ = 36.8 N (thrust per propeller), ${C_T}$ = 0.12 (nominal thrust coefficient for a low pitch propeller as shown in the figure above), $\varrho_{\infty}$ = 1.225 kg/m ${^{3}}$ (standard sea-level air density), $n$ = rpm/60, where the assumed loaded propeller speed is 8,880 rpm for this preliminary estimate, and ${d}$ = propeller diameter (m). Rearranging to solve for ${d}$ gives
$d = \left( \frac{T}{C_T \, \varrho_{\infty} \, n^2} \right)^{\dfrac{1}{4}}$

and substituting the values gives

$d = \left( \frac{36.8}{0.12 \times 1.225 \times \left(\dfrac{8,880}{60}\right)^2} \right)^{\dfrac{1}{4}} = 0.33 \text{ m} =12.9 \text{ inches}$

Propellers are typically available only in standard sizes; the next size up is 13 or 14 inches.
The momentum theory can be used to estimate the net power, which will be
${ P_{\rm req} = \frac{N_r}{FM} \frac{T^{3/2}}{\sqrt{2 \varrho_{\infty} A}} }$

where $N_r$ is the number of rotors (= 4 in this case), $A = \pi d^2/4$ , and the figure of merit, $FM$ , for a small rotor can be assumed to be 0.6. Inserting the numbers gives

$P_{\rm req} = \frac{4}{0.6} \frac{36.8^{3/2}}{\sqrt{2 \times 1.225 \times \pi \times 0.33^2/4}} = 3.25 \text{ kW}$

so about 800 W per motor.
Select the motor in the 800 W power class. Motors are marketed with a $K_V$ rating, and most also specify a required battery voltage and current range. For example, a motor may have $K_V = 800~\text{rpm/V}$ , meaning that its no-load speed is approximately 800 rpm per volt applied. If powered by an 11.1-Volt LiPo battery, it will spin at up to 8,880 rpm. At an assumed speed of about 8,880 rpm, an 800- $K_V$ motor on an 11.1-V battery yields the correct no-load speed, although the loaded speed will be lower. Alternatively, a lower- $K_V$ motor may be appropriate if a higher-voltage battery is used. The final choice must be checked against the motor’s current and thermal limits, as well as the measured thrust with the selected propeller.
The motor’s datasheet may also suggest a range of propeller sizes and helical pitches that fully absorb the motor’s power and give good propulsive efficiency. A propeller for good hovering performance typically has a helical pitch of approximately 4 to 8 inches, depending on the required thrust.
Test the drone with the selected propeller(s) to ensure proper operation. If necessary, propellers with slightly different pitches may be required. Suppose a test stand with a load cell or even a digital spring balance is available. In that case, thrust and battery energy consumption can be measured with different propellers to determine the most suitable one before the first flight tests. Note that the best lifting propellers have a low helical pitch.
While more detailed performance calculations may ultimately be required, the steps here outline a rapid method for sizing the propellers.

Propulsion Propeller

Selecting a propeller for a drone to meet its propulsion requirements must be performed more rigorously and requires access to propeller charts, specifically the thrust coefficient ${C_T}$ versus advance ratio $J$ , the power coefficient $C_P$ versus $J$ , and the efficiency $\eta_p$ versus $J$ . Such charts may be derived from wind-tunnel tests or from calculations using a propeller model (code). Drone propellers, with their high rpm and lower airspeeds (compared to a conventional airplane), operate at relatively low $J$ values, typically less than unity.

Figure 38 shows wind-tunnel measurements for a small-scale propeller at three pitch values that meet the propulsive requirements. At low values of $J$ , especially for higher-pitch propellers, some blade sections may operate at excessive angles of attack and may stall. The propeller may still produce substantial static or low-speed thrust, but its propulsive efficiency is low because the useful propulsive power, $T V_\infty$ , is small. As airspeed builds and $J$ increases, the blade section angles of attack decrease, the stalled regions diminish, and the propulsive efficiency increases. 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 both the lift produced by the blades and the net thrust of the propeller. Eventually, thrust drops to zero, and at sufficiently high ${J}$ for a given blade pitch, the propeller generates negative thrust, creating drag.

Small-scale 11-inch-diameter, low-Reynolds-number propeller performance charts for thrusting propellers at different pitch values, measured in a wind tunnel. $Re_{\rm tip} \approx 10^5$ .

Small-scale 11-inch-diameter, low-Reynolds-number propeller performance charts for thrusting propellers at different pitch values, measured in a wind tunnel. $Re_{\rm tip} \approx 10^5$ .

When selecting a specific fixed-pitch propeller, the goal is to operate it at or near its optimal propulsive efficiency. The advantage of nondimensional coefficients, including the speed-power coefficient, is that propeller data can often be compared across geometrically similar propellers, provided the advance ratio, blade Reynolds numbers, tip Mach number, and blade geometry remain sufficiently similar. At small scales, Reynolds-number effects can be significant, so direct scaling with diameter must be used with caution. Oversized propellers may require values of $n$ that are too low to keep the blade sections operating efficiently, whereas smaller propellers must operate at much higher values of $n$ to provide the required thrust. Higher blade pitch values generally increase the value of ${J}$ at which the maximum propulsive efficiency is obtained. The process is inevitably iterative, based on the power required for flight and, hence, the drone’s drag, with the final propeller pitch representing a compromise. Although the design may be based on a single principal operating condition, it should still retain an adequate efficiency margin over the expected airspeed range. In addition to $J$ , the $C_S$ coefficient helps ensure the selected propeller matches the powerplant and aircraft it powers.

Worked Example #4 – Propeller selection for a drone in forward flight

Consider selecting a propeller for a drone with a motor that turns at 7,000 rpm and delivers 0.8 kW of shaft power. The required cruise airspeed is 45 mph. The initial choice is an 11-inch propeller with a 3.8-inch pitch, with the 11-inch diameter imposed by a geometric constraint. Assume MSL ISA conditions and the availability of propeller performance data.

Show solution/hide solution.

The first step is to compute the speed-power coefficient, which is defined as

(152) $\begin{equation*} C_S = \left( \frac{\varrho_{\infty} \, V_{\infty}^5}{P \, n^2} \right)^{1/5} \end{equation*}$

Substituting the known values, $\varrho_{\infty}$ = 1.225 kg/m $^{3}$ , ${V_{\infty}}$ = 20.12 m/s, $P$ = 800 W, and $n$ = 7,000/60 = 116.67 rps, gives

(153) $\begin{equation*} C_S = \left( \frac{1.225 \times 20.12^5}{800 \times 116.67^2} \right)^{1/5} = 0.821 \end{equation*}$

Using the $C_S$ curves and the propeller efficiency curves, the calculated speed-power coefficient can be compared with the available propeller data. The shaded zones represent where the values of $C_S$ and $J$ are compatible.

The associated advance ratio for a candidate propeller operating near its best efficiency can then be read from the corresponding performance curves. If, for example, the data indicate that the best operating point occurs near $J$ = 0.60, then the required diameter would be

(154) $\begin{equation*} d = \frac{V_{\infty}}{n \, J} = \frac{20.12}{116.67 \times 0.6} = 0.287 \, \text{m} = 28.7 \, \text{cm} = 11.3 \, \text{inches} \end{equation*}$

This result suggests that a propeller approximately 11 inches in diameter is consistent with the imposed geometric constraint. However, the final pitch selection must be based on the actual propeller performance curves at the corrected $C_S = 0.821$ , together with the required thrust, motor current limit, and allowable rpm. The initially chosen 11 x 3.8-inch propeller may be compared with other 11-inch propellers of higher or lower pitch to determine which yields the best efficiency under the desired cruise conditions.

Summary & Closure

Propellers are an effective and robust means of generating thrust, making them widely used in aviation across various propulsion systems. Fixed-pitch propellers are straightforward and lightweight but have inherent efficiency limitations across different flight conditions. Variable-pitch and constant-speed propellers extend the operational range by improving efficiency across a broader range of speeds and altitudes, but they entail increased mechanical complexity, weight, and cost. Whether paired with piston, turboshaft, or electric motors, the propeller must match its specific powerplant to achieve optimal performance.

A key trend in modern airplane design is the shift toward propellers with smaller diameters and more blades. This configuration reduces blade tip speeds and minimizes noise while allowing more power to be absorbed within a constrained diameter. 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 manufacture.

For UAVs and drones, propeller selection plays a crucial role in overall efficiency, endurance, and mission effectiveness. Fixed-pitch propellers are typically preferred for their simplicity and reliability; however, they require careful optimization to achieve peak performance. 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 speed-power coefficient, help scale 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 a range of 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.

5-Question Self-Assessment Quickquiz

For Further Thought or Discussion

Is using a fixed-pitch or constant-speed propeller desirable for a UAV? Discuss.
Why does a swept propeller blade generally improve 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 an aircraft’s overall aerodynamic performance. How does the slipstream interact with other aircraft components?
Explain how tip speed affects propeller noise and efficiency. What design considerations are used to optimize tip speed while staying 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” is 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 how airplane propellers work. It’s good, but there is more to learn.
“Airplane Propellers: Principles and Types” is a World War II-era U.S. War Department training film.

In 2022, students from the University of Maryland built a flying quadrotor drone based on the aerial screw design. Still, it had only novelty value compared with modern quadrotor designs. ↵
Glauert, H. (1935) "Airplane Propellers." In: Durand, W.F., Ed., Aerodynamic Theory, Vol. IV, Division L, Springer, New York, 169–360. ↵
. A good analogy of this concept is a continuously variable transmission in an automobile. ↵
Aspects of Rankine's propeller theory were published in various sources, including Rankine, W. J. M., "On the Mechanical Principles of the Action of Propellers," Transactions of the Institute of Naval Architects, Vol. 6, pp. 13-39, 1865. ↵
Froude, R. "On the Part Played in Propulsion by Differences in Fluid Pressure," Transactions of the Royal Institution of Naval Architects, 30, 390, 1889. ↵
However, it should be noted that the definitions of ${C_T}$ and $C_P$ are different from those used for helicopter rotors. ↵
In a free-turbine turboprop, the propeller is driven by a power turbine that is not mechanically connected to the gas generator. ↵
Weick, F. E., Aircraft Propeller Design, McGraw-Hill Book Co., Inc., New York and London, 1930. ↵
The reciprocal of the coefficient was used by Drzewiecki. Voir: Drzewiecki, S., Théorie générale de l'Hélice propulsive, Gauthier-Villars, Paris, 1920. However, this form spans a wide range of values, making its use impractical. ↵
Some additional losses come from the wake, which experiences a higher fraction of losses from thicker boundary layers trailing into the wake and increased turbulence from shear. ↵

License

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Introduction to Aerospace Flight Vehicles Copyright © 2022–2026 by J. Gordon Leishman is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, except where otherwise noted.

Introduction

Propeller Fundamentals

Velocity Triangle & Blade Twist

Fixed-Pitch Versus Variable Pitch

Types of Propellers

Momentum Theory of the Single Propeller

Flow Model

Application of the Conservation Laws

Induced Velocity

Power Required

Propulsive Efficiency

Activity Factor & Solidity

Wake Swirl Effects

Propeller Performance Coefficients

Momentum Theory of the Tandem Propeller

Separated Tandem Propellers

Tandem Propellers with Shared Induced Velocity

Comparing Efficiencies

General Propeller Performance

Theoretical Development

Numerical Implementation

Propeller Performance Charts

Helical Pitch and Slip

Advance Ratio

Thrust Coefficient

Power Coefficient

Propulsive Efficiency

Reference Blade Pitch

Further Discussion of Propeller Performance

Note on Propulsive Efficiency

Helical Tip Mach Number

Propeller Noise

Tip Mach Number Scaling

Blade-Passing Frequency & Noise Reduction

Propeller Selection Process

Steps for Propeller Selection

Speed-Power Coefficient (Weick) Method

Definition and Derivation

Practical Usage and Application

Wind Tunnel Testing

Pusher Propellers

Flow Model

Thrust

Power and Efficiency

Low Reynolds Number Propellers

Selecting Propellers for a Drone

Steps for Propeller Selection

Hovering (Static Thrust) Propeller

Induced Power

Profile Power

Total Power

Propulsion Propeller

Summary & Closure

License

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