128 Chapter 2 Overview of Propulsion
`
`Photograph)
`
`-. . ,.,..".",.. .
`
`engines that provided a total thrust of 186,000 pounds. The later 400-series, which went
`into service in 1988, develops more than 253,000 pounds of thrust when powered by four
`Pratt and Whitney PW4062 turbofan engines. The PW4062 has a maximum thrust-to-
`weight ratio of more than 6.0 and a minimum thrust-specific fuel consumption of less
`than 0.45 lb-fuelMlb-thrust Even at 40,000 feet with a Mach number of 0.8, the
`-. . . . - ? - .
`.- " .
`..,..<
`.
`YW4UbL has a thrust-speclf~c fuel consumption or less tnan u.o lo-ruevnri~~-tnrusr, wlrn
`four of these engines and a 57,285-gallon fuel capacity, the 747-400 has a maximum
`takeoff weight of 875,000 pounds and a range of more than 8,000 miles.
`During the last three decades of the twentieth century, the continuous improvement
`of turbomachinery components led to the evolution of turbojet and turbofan engines with
`greater thrust-to-weight ratio and higher overall efficiency. This evolution produced
`higher compressor pressure ratios, higher turbine inlet temperatures, and higher bypass
`ratlos. Dunng this period, the small to medium-sized aircraft market also benefited
`tremendously from improvements in turbomachinery technology. Today, the turboshaft
`or turboprop engine is commonly used to power short-range commuter aircrafi like the
`de Havilland Dash 8 shown in Fig. 2.1.16.
`A turboprop engine can be designed to turn at constant rotational speed during most
`phases of aircraft operation. The design is typified by the Allison T56-A-15 engines that
`power the C-130 Hercules shown in Fig 2.1.17. The T56-A-15 turns continuously at
`13,820 rpm, which allows the turbine to maintain its most efficient operating speed.
`Since this speed is too fast for the efficient operation of any propeller, in the C-130
`installation, each engine turns a 162-inch propeller through a gear reduction assembly
`having a total reduction ratio of 13.54 to 1. To maintain the constant rotational speed,
`changes in power requirements are met by changing the fuel flow and propeller pitch.
`
`

`

`2.1. Introduction
`
`Figure 2.1.16. Two Pratt and Whitney PW121, 2000-horsepower turboprop engines power the
`de Havilland Dash 8-100 regionalairliner. (Photograph by John Martin)
`
`129
`
`
`Figure 2.1.17. Four Allison T56-A-15 constant-speed turboprop engines turning at 13,820 rpm
`powerthe C-130H Hercules. With a power-to-weight ratio of 2.65 hp/Ib, the T56-A-15 is capable
`of developing 4,910 horsepower and weighs only 1,850 pounds. (U.S. Air Force)
`
`From the pilot’s point of view, controlling a constant-speed turboprop engine is
`remarkably simple. A single throttle lever controls the engine’s propeller, electrical, and
`fuel systems. This is accomplished through an electromechanical control system. The
`throttle is linked to the fuel control system, which meters the fuel flow to maintain
`constant engine power. A powersetting is established by the position of the throttle, and
`a fuel-trimming system maintains the powerat a constant level by monitoring the turbine
`inlet temperature and changing the fuel flow accordingly. As the fuel flow is changed,
`the automatic control system very precisely adjusts the propeller pitch to maintain
`constant engine rpm.
`
`

`

`130 Chapter 2 Overview of Propulsion
`
`The constant-speed turboprop engine offers excellent fuel efficiency at a relatively
`high power-to-weight ratio. At low subsonic Mach numbers, a thrust-specific fuel
`consumption on the order of 0.2 Ib-fuellhrllb-thrust is attainable and the power-to-weight
`ratio for these engines can be in excess of 2.5 hpllb.
`In addition, the constant-speed
`turboprop engine provides another important operational feature. Since the engine is
`always turning at full speed, response to a change in power demand is almost
`instantaneous. This provides excellent performance characteristics during takeoff,
`landing, and emergency situations.
`In effect, a turboprop engine is an extension of the turbofan engine to higher bypass
`ratios. At low subsonic airspeeds, the modern turboprop provides excellent overall
`efficiency. However, conventional propellers operating at high subsonic Mach numbers
`suffer a substantial loss in efficiency, due to compressibility effects. Turboprops have
`been designed to provide high propulsive efficiency at Mach numbers as high as 0.8, but
`no such design has yet been placed in commercial service. Improvements in the power-
`to-weight ratio and thermodynamic efficiency of turboshaft engines also played a major
`role in the development of modem helicopters.
`In the first 100 years since the maiden flight of the 1903 Wright Flyer, great strides
`have been made in the development of aircraft propulsion systems. A 12-horsepower
`piston engine with a power-to-weight ratio of approximately 0.06 hpllb powered the 1903
`Wright Flyer. Today, modern gas turbine engines have a power-to-weight ratio in excess
`of 2.5 hpllb, and the largest gas turbines develop close to 100,000 horsepower. The 1903
`Wright Flyer first lifted from the Earth with a gross weight of 750 pounds and covered a
`total distance of 120 feet. Today, transport aircraft powered by modern turbofan engines
`have a maximum takeoff weight approaching 1,000,000 pounds and a maximum range of
`more than 8,000 miles. Based on these astounding advancements, it is indeed difficult to
`imagine what the next century holds in store.
`
`2.2. The Propeller
`An airplane propeller has much in common with a finite wing. In fact, each blade of the
`propeller can be viewed as a rotating wing. Like a wing, the cross-section of a propeller
`blade is an airfoil section, as shown in Fig. 2.2.1. Both the wing and the propeller blade
`are designed to produce lift. The lift developed by a wing is usually directed to support
`the weight of the airplane and keep it aloft. The lift developed by a propeller is typically
`aligned more or less with the direction of flight and is intended to support the airplane
`drag and accelerate the airplane in the direction of motion. To a person not familiar with
`the terminology of aeronautics, it might seem strange to say that we can direct lift to
`produce a horizontal force in the direction of motion. However, recall that the
`aerodynamic definition of lift is not a force opposing gravity but a force normal to the
`relative airflow. It is primarily the motion of the airplane through the air that provides
`the relative airflow over an airplane wing. Thus, the lift on a wing is normal to the
`direction of flight. The relative airflow over a propeller blade is provided primarily by
`the rotation of the propeller. Thus, the lift on a propeller blade is directed approximately
`along the axis of rotation. The component of the aerodynamic force that is parallel to the
`axis of rotation is the thrust.
`
`

`

`2.2. The Propeller 131
`
`Figure 2.2.1. Airfoil cross-section of an airplane propeller blade.
`
`Because the velocity of each section of a rotating propeller blade depends on the
`distance of the section from the axis of rotation, a propeller blade usually has much more
`twist or geometric washout than a typical wing. As shown in Fig. 2.2.2, the sections
`close to the axis are moving more slowly and form a larger angle with the plane of
`rotation, while the sections close to the tip are moving faster and form a smaller angle
`with the plane of rotation. The angle that the section zero-lift line makes with the plane
`of rotation will be called the aerodynamic pitch angle. This pitch angle varies with the
`radial distance, r, and will be denoted as P(r).
`The reader should be cautioned that pitch angles are often tabulated relative to the
`section chord line or to a flat lower surface of the airfoil section. However, the
`aerodynamic pitch angle, which is used in this text, is more convenient for aerodynamic
`analysis. If the airfoil section geometry is known and a pitch angle is given in terms of
`some other definition, the aerodynamic pitch angle can be determined. For example,
`
`Section A-A
`
`Y
`
`Section B-B
`Figure 2.2.2. Radial variation in aerodynamic pitch angle along a propeller blade
`
`

`

`132 Chapter 2 Overview of Propulsion
`
`if ,8, is the pitch angle relative to the chord line, the aerodynamic pitch angle, ,8, can be
`written simply as
`
`where a ~ o
`is the zero-lift angle of attack for the airfoil section of the blade that is located
`at radius, r.
`The pitch of a propeller is not commonly specified in terms of a pitch angle, but
`rather in terms of a pitch length, which is usually referred to simply as the pitch. The
`pitch length for a propeller is similar to that for a common screw. In fact, propellers were
`originally called airscren~s. If the propeller were turned through the air without slipping,
`the distance that the propeller would move forward in each revolution is the pitch length.
`In this text, the aerodynamic pitch length is given the symbol A and is defined relative to
`the section zero-lift line as
`
`This geometric relation is shown schematically in Fig. 2.2.3. Here again the reader is
`cautioned that propeller pitch is normally tabulated relative to the section chord line or to
`a flat lower surface of the airfoil section. The aerodynamic pitch, A, is determined from
`the chord-line pitch, A,, according to the relation
`
`A(r) = 2 x r
`
`4. - 2 x r tan a1.0
`2 x r + A,. tan aLO
`
`Figure 2.2.3. The helix that defines aerodynamic pitch length.
`
`

`

`2.2. The Propeller 133
`
`For the special case of a constant-pitch propeller, the pitch length does not vary with
`the radial coordinate, v. Thus, in a constant-pitch propeller, the tangent of the pitch angle
`decreases with increasing radius in proportion to 117. The term constant-pitch should not
`be confused with the term fixed-pitch, which is related to the term variable-pitch. In a
`variable-pitch propeller the propeller pitch can be changed by rotating each blade about
`an axis that runs along the length of the blade. A,fixed-pitch propeller is one in which
`the pitch cannot be varied. As previously stated, a constant-pitch propeller is one in
`which the pitch length does not vary with v.
`Like a wing, each blade of the propeller will be subject to a drag force as well as a
`lift force. Since the propeller blade length is finite, there will be both parasitic drag and
`induced drag. Because drag is defined as the component of the aerodynamic force that is
`parallel to the relative airflow, the drag on a rotating propeller blade produces a moment
`about the propeller axis that opposes the propeller rotation. The torque necessary to
`counter this moment and the power required to sustain the rotation must be provided by
`the airplane's engine. Since the axis of propeller rotation is typically aligned closely with
`the direction of motion, this propeller-induced moment is felt by the airplane as a rolling
`moment, which must be countered with an aerodynamic moment produced by the
`airframe or by another propeller rotating in the opposite direction.
`Maximizing thrust while minimizing the torque necessary to turn a propeller is
`obviously one important aspect of good propeller design. The ratio of the thrust
`developed to the torque required for a propeller is analogous with the lift-to-drag ratio for
`a wing. The torque required to turn the propeller multiplied by the angular velocity is
`called the propeller brake power. It is this power that must be supplied by the engine.
`The thrust developed by the propeller multiplied by the airspeed of the airplane is called
`the propulsive power. This is the useful power that is provided to propel the airplane
`forward against the airframe drag. The ratio of the propulsive power to the brake power
`for a propeller is called the propulsive eficiency. This is one important measure of
`propeller performance. However, it is not the only important measure. The thrust that is
`developed by a propeller when the airplane is not moving is called the static thrust. It is
`important for a propeller to produce high static thrust when accelerating an airplane from
`brake release on takeoff. Since the airspeed is zero for the case of static thrust, the
`propulsive power and the propulsive efficiency are both zero. Thus, propulsive
`efficiency is not a particularly good measure of a propeller's ability to accelerate an
`airplane from a standing start.
`The thrust force and the rolling moment are not the only aerodynamic force and
`moment produced by a rotating propeller. Additional aerodynamic reactions are
`produced if the propeller rotation axis is not aligned perfectly with the direction of flight.
`The angle that the propeller axis makes with the freestream airflow is called the thrust
`angle or the propeller angle of attack. Since the engine and propeller are typically
`mounted directly to the airframe, the propeller angle of attack will change with the
`airplane angle of attack. Because the airplane angle of attack can change during flight,
`the propeller angle of attack can also change.
`When a rotating propeller is at some positive angle of attack relative to the
`freestream flow, there is a component of the freestream in the plane of propeller rotation,
`as seen in Fig. 2.2.4. This component of the freestream changes the relative airflow over
`
`

`

`134 Chapter 2 Overview of Propulsion
`
`each blade of the propeller. It increases the relative airspeed for the downward-moving
`blades and decreases the relative airspeed for the upward-moving blades. Thus, both the
`lift and drag are increased on the downward-moving side of the propeller and decreased
`on the upward-moving side. The difference in thrust generated by the two sides of the
`propeller produces a yawing moment and the difference in circumferential force produces
`a net normal force in the plane of rotation, as seen in Fig. 2.2.4.
`The axial component of the airplane's airspeed will also affect the aerodynamic
`forces and moments acting on a rotating propeller. Since this component of the airspeed
`is normal to the plane of propeller rotation, it changes the angle of attack for the
`individual blades. This acts very much like the downwash on a finite wing, and like
`downwash on a wing, this normal component of airflow changes the aerodynamic forces
`generated on each blade of the propeller.
`Even in the static case, when the airplane is not moving, there is airflow normal to
`the plane of propeller rotation. As is the case with any finite wing, the vorticity shed
`from each blade of the propeller produces downwash along its own length. However, in
`addition to being subjected to its own downwash, each blade of a propeller is "flying
`directly behind" the blade that precedes it in the rotation sequence. The blades of a
`rotating propeller act very much like an infinite series of finite wings, flying in a row one
`behind the other. The downwash on any one such wing would be increased by the
`vorticity shed from every other wing that proceeds it in the flight line. Similarly, the
`downwash on each blade of a rotating propeller is amplified by its proximity to the other
`blades. In the case of a rotating propeller, the total downwash generated by all of the
`propeller blades combined is usually called the propeller's induced velocity.
`
`Figure 2.2.4. Effect of propeller angle of attack on the aerodynamic forces acting on a rotating
`propeller.
`
`

`

`2.2. The Propeller 135
`
`A propeller, or even a common house fan, provides a good means for the student to
`get a feel for the downwash produced by a lifting surface. The downwash produced by a
`rotating propeller or a common house fan is concentrated in a much smaller area than is
`that produced by a moving wing. Because the blades are passing repeatedly through the
`same small section of the fluid, the downwash is amplified with each successive pass.
`This concentrated downwash is readily detected by the human senses. Although the
`downwash induced by a rotating propeller is much stronger than the downwash induced
`by the lifting wing of an airplane, it is conceptually no different in physical origin.
`Similar to the lift on a wing, the thrust developed by a propeller is imparted to the
`individual blades directly through a pressure difference between the upstream and
`downstream sides of the blade. However, also like a wing, the same pressure difference
`is imposed on the fluid as well. The pressure difference acting on the fluid generates
`fluid momentum in a direction opposite to the aerodynamic force exerted on the
`propeller. The velocity increase induced at the plane of the propeller disk is the
`downwash that is usually referred to as the propeller's induced velocity.
`Optimizing the performance of an airplane powered by an engine-propeller
`combination depends on properly matching the propeller with the engine, as well as
`matching the engine-propeller combination with the airframe. Understanding propeller
`performance is critical to this optimization process. For this purpose, it is useful to be
`able to predict the aerodynamic forces and moments acting on a rotating propeller as a
`function of the operating conditions.
`
`2.3. Propeller Blade Theory
`In order to quantitatively understand and predict the performance of a rotating propeller,
`it is necessary to analyze the aerodynamics of the blade in detail. To this end, consider
`the cross-section of the propeller blade that is shown in Fig. 2.3.1. The blade is rotating
`with an angular velocity of w and is advancing through the air with a relative airspeed of
`V,.
`Initially we shall assume that the forward velocity vector is aligned with the axis of
`propeller rotation so that there is no component of the forward airspeed in the plane of
`rotation. The angle that the zero-lift line for the blade section makes with the plane of
`rotation is what we have called the aerodynamic pitch angle, fl. In general, this pitch
`angle varies with the radial distance, r.
`The total downwash angle, zh, for the blade cross-section located at radius r is the
`sum of two parts, the downwash angle that results from the propeller's forward motion,
`E,, and the induced downwash angle, E,,
`
`is named the advance angle, E, is called the induced angle, and zh is
`In this text, E,
`simply referred to as the downwash angle. This downwash angle reduces the angle of
`attack for the airfoil section and tilts the lift vector back, as is shown in Fig. 2.3.1. This
`tilting of the lift vector through the angle zh increases the torque required to turn the
`propeller in much the same way that downwash adds induced drag to a lifting wing.
`From the geometry shown in Fig. 2.3.1, the section thrust and circumferential force are
`
`

`

`136 Chapter 2 Overview of Propulsion
`
`related to the section lift and drag forces through this local downwash angle according to
`the relations
`
`From the axial and circumferential components of induced velocity, VYi and VH,, the
`downwash angle, E,, is determined from the geometry shown in Fig. 2.3.1,
`
`From the known rotational speed and forward airspeed, the advance angle, E,,
`mined directly from this same geometry as
`
`is deter-
`
`Figure 2.3.1. Section forces and velocities acting on a rotating propeller blade.
`
`

`

`Thus, using Eq. (2.3. l), the induced angle, E,, is
`
`E/ ( Y ) = t a n (-1
`
`I Vx + V,,
`W Y - V&
`
`2.3. Propeller Blade Theory 137
`
`-
`
`The aerodynamic angle of attack for the blade section, a,,, measured relative to the
`section zero-lift line is simply the aerodynamic pitch angle, P, less the downwash angle,
`el/,. From Eq. (2.3. l ) , this can be written as
`
`The fluid velocity relative to the blade cross-section located at radius r is designated
`as Vh. This is the vector sum of the velocity created by the rotation of the blade, the
`forward velocity of the propeller, and the propeller induced velocity. The largest
`component of Vh is usually the circumferential component, w r , which results from the
`blade rotation. In addition, there is an axial component of the airspeed, relative to the
`blade section, which results from the propeller's forward airspeed, V,. There is also a
`component of velocity relative to the blade section that results from the propeller's
`induced velocity. The induced velocity results from the same aerodynamic pressure
`difference that produces the lift on the propeller blades.
`In general, the propeller's
`induced velocity will have both an axial component, V,,, and a circumferential
`component, Vo,. The induced velocity will vary with the radial coordinate, r. From the
`geometry shown in Fig. 2.3.1, the total relative airspeed, Vh, at the plane of the blade
`section is given by
`
`Using Eqs. (2.3.8), the section lift for the blade section located at radius Y is
`expressed as
`
`-
`2 -
`2 2 -
`L = i p v b chCL = + P ~
`r chCI.
`
`where c,, is the local section chord length and CL is the local section lift coefficient,
`which is a function of the local section aerodynamic angle of attack given by Eq. (2.3.7).
`Also using Eqs. (2.3.8), the local section drag can be written
`
`

`

`138 Chapter 2 Overview of Propulsion
`
`where ED is the local 2-D section drag coefficient, which includes only parasitic drag.
`The induced drag is included as part of the lift vector.
`In general, the section drag
`coefficient is also a function of the local section angle of attack.
`Applying Eqs. (2.3.9) and (2.3.10) to Eq. (2.3.2), the axial component of section
`force is written
`
`Similarly, from Eq. (2.3.3), the circumferential component of section force is
`
`The thrust per unit radial distance for the full prop-circle is the axial section force per
`blade multiplied by the number of blades, k. Thus, from Eq. (2.3.1 I),
`
`The torque required to turn the propeller is simply the aerodynamic rolling moment, Q,
`that is produced about the axis of propeller rotation. Thus, the torque per unit radial
`distance, for the full prop-circle, is the circumferential section force per blade multiplied
`by the radius and the number of blades. From Eq. (2.3.12), this results in
`
`If the propeller geometry were completely defined, the section chord length and
`section pitch angle would be known functions of the radial coordinate, r. The section lift
`and drag coefficients would be known functions of the radial coordinate, r, and the local
`section angle of attack, a h . If the rotational speed and the forward airspeed are known,
`is known from Eq. (2.3.5). Even so, the two differential equations
`the advance angle, E,,
`expressed in Eqs. (2.3.13) and (2.3.14) contain four unknowns: the thrust, T, the torque,
`!, and the two components of the propeller induced velocity, V,,, and VBI. If these two
`components were known, the downwash angle, E ~ , could be determined from Eq. (2.3.4).
`With only two equations and four unknowns, obviously, additional information is needed
`
`

`

`2.3. Propeller Blade Theory 139
`
`before Eqs. (2.3.13) and (2.3.14) can be solved. For this additional information we will
`now consider the vortex lifting law and the vorticity that is shed from the rotating
`propeller blades.
`Just as is the case for a lifting wing, lift cannot be generated on a rotating propeller
`blade without the simultaneous generation of vorticity. Furthermore, as was previously
`done for a finite wing, the lift on a finite propeller blade can be related to the bound
`vorticity through the vortex lifting law. This requires that for any cross-section of a
`, is related to the fluid density, p, the relative airspeed,
`propeller blade, the section lift,
`Vh, and the total section circulation,
`according to
`
`Thus, using Eqs. (2.3.8) and (2.3.9) with Eq. (2.3.15), the local section circulation for any
`cross-section of a propeller blade can be related to previously defined variables,
`
`Since the lift on a propeller blade is generated directly from a pressure difference
`between the two sides of the blade, the lift must go to zero at the blade tip, where such a
`pressure difference cannot be supported. This fact, combined with Eq. (2.3.16), requires
`that vorticity must be shed from the blade tips of a rotating propeller, just as it is from a
`lifting wing.
`It is this shed vorticity that produces the induced downwash on the
`propeller blades. The bound vorticity does not influence the induced downwash.
`Because of the symmetry of the blades in the prop-circle, each blade receives as much
`upwash from the bound vorticity on other blades as it does downwash.
`The primary difference between the vorticity shed from a propeller blade and that
`shed from a lifting wing is the path that the vorticity takes as it moves downstream.
`Whereas the wingtip vortices shed from a lifting wing move downstream in a fairly linear
`fashion, the blade-tip vortices shed from a rotating propeller follow a helical path, as is
`seen in Fig. 2.3.2. In this photograph, the path of each blade-tip vortex can be visualized
`as a result of contrails, which are caused by moisture condensing in the low-pressure
`region near the core of each vortex. Since the blade-tip vortices all rotate in the same
`direction, the region inside the propeller's helical trailing vortex system is a region of
`very strong downwash. This downwash is the air movement that can be felt directly
`behind a rotating propeller or in front of a common house fan. This cylindrical region of
`strong downwash is called the slipstreanz. In the region just outside the slipstream, there
`is upwash.
`It should be remembered that in aerodynamics terms, "downwash" and
`"upwash" have nothing to do with "up" and "down"
`in the conventional sense.
`Downwash is airflow in a direction opposite to the lift vector, and upwash is airflow in
`the same direction as the lift vector. Thus, the induced upwash in the region outside the
`slipstream is an upstream flow that replaces the air, which has been removed from the
`region forward of the propelled disk by the induced downwash. This flow pattern is
`shown schematically in Fig. 2.3.3.
`
`

`

`140 Chapter 2 Overview of Propulsion
`
`I
`
`Figure 2.3.2. Contl-ails showing the helical path of thc blade-tip vortices shed from a rotating
`propeller. (Photograph by Migucl Snoep)
`
`tic cross-section of the vortex-induced flow in the vicinity of a rotating
`
`

`

`2.3. Propeller Blade Theory 141
`
`Because the vortex lines follow a helical path rather than a circular path, the velocity
`induced by each vortex, which is normal to the vorticity vector, has a component in the
`circumferential direction. Thus, the air in the slipstream rotates as it moves downstream.
`This rotation is not obvious from the feel of the air in the slipstream of a propeller or a
`common house fan. However, by holding a piece of light yarn in the slipstream of a fan,
`this rotation is easily seen.
`In Fig. 2.3.3, an attempt has been made to show separately the velocity induced by
`each section of each blade-tip vortex. At any point in space, the resultant induced
`velocity is the vector sum of the velocity induced by the entire length of all vortex
`filaments in the slipstream. Since the velocity induced by a vortex filament at any point
`in space decreases with the distance from the filament, the closest filaments have the
`greatest influence on the velocity induced at a particular point. However, the induced
`velocity at any point is influenced by the entire helical vortex system. In the slipstream
`downstream from the propeller, the upstream portion of the vortex system as well as the
`downstream portion produces downwash. As a result, downstream from the propeller,
`the induced velocity is greater than it is in the plane of the propeller, where the only
`vorticity is downstream. Far downstream from the propeller, the helical vortex system
`extends essentially from negative infinity to positive infinity, while at the plane of the
`propeller itself, the vortex system extends only from zero to infinity. Thus, far
`downstream from the propeller, the induced velocity is about twice what it is in the plane
`of the propeller disk.
`As one might imagine, computing the velocity induced by the helical vortex system
`trailing downstream from a rotating propeller is considerably more complex than
`computing the velocity induced by the vortex sheet trailing from a finite lifting wing.
`One method for predicting this induced velocity is known as Goldstein's vortex theory.
`To predict the velocity induced in the plane of the propeller disk, Goldstein (1929) made
`two simplifying hypotheses. First, the vortex sheet trailing from a rotating propeller
`blade was assumed to lie along a helical surface of constant pitch. Second, the
`induced velocity was assumed to be normal to the resultant velocity, which is the
`vector sum of the rotation velocity, the forward velocity, and the induced velocity
`itself. These assumptions were not made arbitrarily.
`It can be shown that these
`conditions are both satisfied in the ultimate slipstream of an optimum propeller. This is
`the so-called Betz condition (see Betz 1919). However, Goldstein's assumptions are
`difficult to justify in the plane of a propeller having blades of arbitrary pitch and
`planform shape. Nevertheless, McCormick (1995) states that "studies have been
`performed that support normality at the plane of the propeller" and he has shown that this
`theory gives results that are in reasonable agreement with experimental data.
`From the normality hypothesis, shown in Fig. 2.3.4, the total relative airspeed, V,?, is
`given by
`
`Vh = \
`
`u r
` = - cos 6,
`/ ~ c o s 6 ,
`COS Em
`
`(2.3.17)
`
`Similarly, from Fig. 2.3.4, the induced velocity and its axial and circumferential
`components can be written as
`
`

`

`142 Chapter 2 Overview of Propulsion
`
`Figure 2.3.4. Section fol-ces and velocities acting on a rotating propeller blade under Goldstein's
`hypothesis.
`
`V, = \lw'r'+Y,'sin&; = - sin E;
`
`W Y
`
`COS Em
`
`Vl; =
`
`u r
`C O S E ~ = - sin E; COS(E; + E,)
`COS E,
`
`o r
`Voi = 6 sin&,, = - sin E; sin(&; + E, )
`COS Em
`
`(2.3.18)
`
`(2.3.19)
`
`(2.3.20)
`
`From the hypothesis of constant pitch in the trailing helical vortex sheet, Goldstein's
`vortex theory predicts that the local circumferential component of induced velocity, Vo,,
`in the plane of the propeller disk is related to the local section circulation, I-, by
`
`The proportionality constant, K, known as Goldstein's kappa factor, is available in
`graphical form but has never been presented in closed form.
`
`

`

`A close approximation to Goldstein's result can be obtained from the relation
`
`2.3. Propeller Blade Theory 143
`
`The parameter j'is known as Prandtl's tip loss factor (see Prandtl and Betz 1927), which
`can be expressed as
`
`where /3, is the aerodynamic pitch angle at the propeller blade tip. Using Eq. (2.3.17) in
`Eq. (2.3.16), the section circulation can also be written as
`-
`- COSE;
`T = ;vhchcL = ;orchcL-
`COS Em
`
`After applying Eqs. (2.3.20) and (2.3.24) to Eq. (2.3.22), we obtain the relation
`
`With knowledge of the propeller geometry, the rotational speed, and the forward speed,
`Eq. (2.3.25) contains only a single unknown, the induced angle, E~. This equation is
`easily solved numerically to determine the induced angle as a function of the radial
`coordinate, r.
`Once the induced angle is known as a function of radius, the total thrust developed
`by the propeller is determined by using Eq. (2.3.13) with Eq. (2.3.17) and integrating
`from the hub radius, rh, to the tip radius, r,,
`
`In a similar manner, the total torque required to turn the propeller is found by integrating
`Eq. (2.3.14) with Eq. (2.3.l7),
`
`

`

`144 Chapter 2 Overview of Propulsion
`
`Equations (2.3.26) and (2.3.27) can be integrated numerically to determine the propeller
`thrust and torque. The brake power required to turn the propeller is just the torque
`multiplied by the angular velocity,
`
`As is the case with lift and drag, the aerodynamic forces and moments for a propeller
`are normally expressed in terms of dimensionless coefficients. Whcreas the freestream
`velocity was used as the characteristic velocity in defining the lift and drag coefficients
`for a wing, the freestream velocity is not the most important velocity associated with the
`production of thrust by a rotating propeller. The velocity that has the greatest influence
`on propeller thrust is the rotational velocity of the blades. While there are several
`characteristic dimensions associated with a propeller, the one most commonly used in the
`definition of dimensionless propeller coefficients is the propeller diameter, d,,. Although
`other definitions have occasionally been used, by far the most common definitions for the
`propeller coefficients are:
`
`Thr