Introduction

Propellers

Internal Combustion Engines

Gas Turbine Engines

Chemical Rockets

Non-Chemical Space Propulsion Systems

AER 710 Aerospace Propulsion

C-130

Nieuport N.28C-1

Introduction to the Propeller

• The rotating blade of a propeller shares similar characteristics to a wing passing through the air

• A propeller blade generates thrust F through an aerodynamic lift force component, demands an engine torque Q to overcome aerodynamic drag, and will stall if the local resultant angle of attack of the blade exceeds max

• Additional factors: trailing vortex generation, tip losses, compressibility

Martin MB-2

DH-98 Mosquito

Forces acting on wing airfoil section (above) and propeller blade section (below)

• For evaluation of propeller performance, one can apply a simple analytical approach using the principle of linear momentum conservation, and treating the propeller as an actuator disk where there is a step increase in pressure

Actuator Disk Theory

)VV(VA)VV(mF 033303 Thrust generated by disk:

)pp(AF 121 Alternatively:

211

200 2

1

2

1VpVp

Bernoulli’s eq. applied from upstream to front of disk:

233

222 2

1

2

1VpVp

Similarly, downstream of disk:

)VV)(VV()VV(pp 03032

02

312 2

1

2

1

Noting po = p3 , and V2 = V1, via subtraction one gets:

A3V3 = A1V1

Conservation of mass, incompressible flow:

)VV(VA)pp(AF 0333121

Substituting from earlier:

)VV)(VV()VV(VA

App 0303033

1

312 2

1

which gives the simple result:

and

203

1

VVV

wVV 01

Define propeller-induced velocity w such that:

wVV 203

w)wV(A)VwV)(wV(A)VV(VAF 0100010311 22

and so for thrust,

201

20

2001

20

23 22

2

1

2

1

2

1)wV(wA]V)wV)[(wV(AVmVmP

Ideal power required:

)wV(FP 0

or

Since power from a piston or turboprop engine is relativelyconstant at a given altitude, one can expect the thrust todrop as the airplane picks up airspeed, according to thiscorrelation.

022 012

1 Fw)VA(w)A(

If one wishes to find w as a function of F, from earlier:

1

20

0 2

2

1

2 A

FV

Vw

giving

1

23

2 A

FwFPP

/o

ooo,indo

Ideal static power (Vo = 0):

0

0

0

1

1

V

w)wV(F

FVi,pr

Ideal propeller propulsive efficiency:

1

11

2

qA

Fi,pr

or via substitution (q is dynamic pressure):

i,prS

pr P

FV

Actual propeller propulsive efficiency, in terms of useful(thrust) power and engine shaft power PS :

SP)()wV(FP factor correction0

Correction factor, less than 1, for ideal power estimate:

Variable-pitch propeller better able to approach theideal power requirement, as compared to a fixed-pitchpropeller, in accommodating different flight speedsand altitudes.

Momentum-Blade Element Theory

• Logically, the next level of analysis would look at a given propeller blade’s aerodynamic performance from hub to blade tip

• one can discretize the blade into a finite number of elements, while applying momentum conservation principles

Schematic diagram of a three-bladed propeller, and framework for discretizing an individual blade for analysis

)sin(D)cos(LF ii ddd

Increment of thrust:

22 V)r(VR

Resultant velocity:

)]cos(D)sin(L[rFrQ iiQ dddd

Increment of torque:

rcCVL E d2

1d 2

Increment of lift:

rcCVD dE d2

1d 2

Increment of drag:

22 )V)cos(w())sin(wr(V iiE

Overall resultant velocity:

)V

w(sin

Ri

1

Induced angle of attack:

)(a)(CC ioi

Airfoil lift coefficient:

min,dd CC

Airfoil drag coefficient:

C < C,min

2)CC(kCC min,min,dd C,min < C < C,max

)(kCC max,dd max 1 > max

cosr)(caVB

cosLF ioR d2

dd 2

Via substitutions, increment of thrust:

where B is number of blades.

cosw)coswV(Aw)wV(AF d2d2d 0

Borrowing from actuator disk theory:

cosV)cosVV()rr( RiRi d22

088

2 )(cosr

Bca)

cosr

Bca

cosV

V( o

io

Ri

Equating the above relations, one arrives at:

R

Bc

R

RcB refrefref

2areadisk

area blade

Overall propeller solidity:

r

Bcx

R

Bc

Local solidity:

x = r/R

R

V

)R))(/((

V

nd

VJ

p

22

Advance ratio:

where n is the prop shaft rotation speed (rps).

J

R

V

Nondimensional velocity ratio:

)x

(tan)r

V(tan

11 Also:

TR xVrcosV VT = R

088 22

2 )(Vx

Va)

Vx

Va

x(

T

Roi

T

Roi

Substituting from earlier:

})](Vx

Va)

Vx

Va

x[()

Vx

Va

x({ /

T

Ro

T

Ro

T

Roi

212

222 2882

1

Applicable solution for induced angle of attack via theabove quadratic eq. gives:

42dn

FCT

Propeller thrust coefficient:

53dn

PC S

P

Propeller power coefficient:

QPS

r)]sin(C)cos(C[BcVF idiE d2

1d 2

Incremental thrust no. of blades:

r)]cos(C)sin(C[BcVrP idiES d2

1d 2

Incremental power no. of blades:

)xJ(r

rVVV RE222

2

2222222

FR

CT 42

2

4

Note:

Thrust coefficient:

x)]sin(C)cos(C)[xJ(FR

C idi

x

T

h

d8

d4

221

242

2

SP PR

C53

3

4

Power coefficient:

x)]cos(C)sin(C)[xJ(xPR

C id

x

iSP

h

d8

d4

1222

2

53

3

Momentum-Blade Element Theory (Summary)

• The above equations for CT and CP can be integrated from the hub station (x = xh) to the blade tip (x = 1) using a numerical approach as one moves along the blade of varying and c, calculating the various pertinent parameters (C , Cd, i , etc.) in conjunction

Thrust

Power

Propeller Propulsive Efficiency

• Define as useful thrust power over overall shaft power:

Spr P

FV

JC

C

dnC

VdnC

P

T

P

Tpr

53

42

Also, via substitution:

A variable pitch propeller will have better efficiency over thecourse of the flight mission, relative to a fixed pitch prop.

Chart illustrating propeller propulsive efficiency for an example propeller

Compressibility Tip Loss

• Depending on the blade airfoil section design, drag divergence (compressibility) effects will become evident when the propeller blade’s resultant tip speed VR,tip exceeds a local flow Mach number Matip of around 0.85 (critical value, Macr)

• As a result, one would not typically be cruising at much greater than a flight Mach number Ma of around 0.6

22)(

Maa

ndMatip

)1.0

(100

15 crtipalminpr,nopr

MaMa

Dommasch correlation:

Blade tip Mach number:

Modern high-speed blades may be thinner, and sweptor curved along the blade length, to mitigate the issues with compressibility and compression wavedevelopment at higher local flow Mach numbers

Activity Factor

• Activity factor (AF) is a design parameter associated with the propeller blade’s geometry. The more slender the blade (larger radius, smaller chord), the lower the AF value:

xxd

cAF

hx p

d16

100000 31

pd

cAF 1563

Typically see higher AF props on turboprop engines.

Blade Number• One has the option of setting the number

of blades, B, for a given application. While one has a minimum of 2 blades to choose from, one can presently go as high as around 8 blades on the high-performance end for an unducted propeller

• On occasion, one also sees the use of two contra-rotating rows of blades, to get more thrust delivery from one engine

Photo of Fairey Gannett carrier-borne anti-submarine/AEW aircraft, employing two contra-rotating rows of 4 propeller blades each on a co-axial shaft setup, powered by a 3000-hp Armstrong Siddeley Twin Mamba turboprop engine

Airbus A400M “Atlas”

Helicopter Rotors• helicopter rotors (main and tail) share a

number of similarities with airplane propellers• analysis done above for propellers can be

applied to rotors• orientation of the rotor disk will be somewhat

different from that of the propeller, with respect to the resultant incoming air flow

• Main helo rotor produces lift + thrust

- rotor blade will advance into the air flow when in forward flight, and then retreat during the other half of the rotational cycle

CH-47

- tail rotor primarily controls yaw forces and moments [primarily main-rotor-induced torque] on the helicopter, if only having one main rotor- a tandem-rotor helicopter, with two contra-rotating main rotors, would not need a tail rotor

HH-65 Dolphin

- ducted tail fan is an alternative to the conventional tail rotor

NOTAR

No Tail Rotor (Using Coanda Effect)

• The amount of lift generated by a main rotor is controlled by two means: a) the engine throttle setting for desired level of main rotor rotational speed, and b) collective pitch setting, which sets the angle of incidence of the main rotor blades collectively to produce the desired uniform lifting force on the vehicle (e.g., higher lift required, a higher blade incidence angle setting is needed, for the same rotor rotational speed)

• Rotation of the vehicle’s body in pitch or roll or some combination thereof is largely via the cyclic pitch setting of the main rotor, whereby the individual main rotor blades will have their incidence vary as they complete a given revolution about the vehicle, depending on the desired direction of the rotational moment

Operations of swashplate (item #2, 4 above) for cyclic control

The schematic diagram illustrates a conventional main rotor mast (rotorhead), with the hub above the mast connecting the rotor blades to the drive shaft in a fully articulated design (hinged); a swashplate approach is being used to control the effective main rotor disk deflection and tilt direction thereof

Fully articulated, a.k.a., hinged (horiz. + vert.) rotor head above(vs. rigid, a.k.a., hingeless)

From: Flight International 1986

Bell UH-1C Iroquois (“Huey”)

Rotor mast, Bell UH-1 Iroquois

Hybrid Aircraft Designs• In order to improve range performance

over a conventional helicopter, one will see tilt-wing and tilt-rotor designs for V/STOL (vertical/short takeoff & landing) applications

Tilt-rotor V-22 Osprey

Tilt-wing Canadair CL-84