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Cruising Flight PerformanceRobert Stengel, Aircraft Flight
Dynamics MAE 331, 2010
Copyright 2010 by Robert Stengel. All rights reserved. For educational use only.http://www.princeton.edu/~stengel/MAE331.html
http://www.princeton.edu/~stengel/FlightDynamics.html
U.S. Standard Atmosphere
Air data measurement and computation
Airspeed definitions
Steady, level flight
Simplified power and thrust models
Back side of the power/thrust curve
Performance parameters
Breguet range equation
Jet engine
Propellerdriven
U.S. Standard Atmosphere, 1976
http://en.wikipedia.org/wiki/U.S._Standard_Atmosphere
Dynamic Pressure and Mach Number
! = air density, functionof height
= !sea levele"#h
a = speed of sound, linear functionof height
Dynamic pressure = q =1
2!V 2
Mach number =V
a
Definitions of Airspeed
Definitions of Airspeed
True Airspeed (TAS)*
Indicated Airspeed (IAS)
Calibrated Airspeed (CAS)*
Equivalent Airspeed (EAS)*
Airspeed is speed of aircraft measured withrespect to the air mass Airspeed = Inertial speed if wind speed = 0
IAS = 2 pstag ! pstatic( ) "SL = 2 pt ! ps( ) "SL
CAS = IAS corrected for instrument and position errors
EAS = CAS corrected for compressibility effects
TAS = EAS !SL
!(z)
Mach number
M =TAS
a
* Kayton & Fried, 1969; NASA TND822, 1961
Air Data Measurement andComputation
Air Data System
Air Speed IndicatorAltimeterVertical Speed Indicator
Kayton & Fried, 1969
Air Data ProbesRedundant pitot tubes on F117
Total and static temperature probe
Total and static pressure portson Concorde
Stagnation/static pressure probe
Redundant pitottubes on FougaMagister
Cessna 172 pitot tube
X15 Q Ball
Air Data Instruments(Steam Gauges)
Calibrated Airspeed Indicator Altimeter Vertical Speed Indicator
Variometer/Altimeter
True Airspeed Indicator
Machmeter
1 knot = 1 nm / hr
= 1.151 st.mi. / hr = 1.852 km / hr
Air Data Computation for
Subsonic Aircraft
Kayton & Fried, 1969
Air Data Computation for
Supersonic Aircraft
Kayton & Fried, 1969
The Mysterious Disappearance ofAir France Flight 447 (Airbus A330200)
http://en.wikipedia.org/wiki/AF_447
BEA Interim Reports, 7/2/2009 & 11/30/2009
http://www.bea.aero/en/enquetes/flight.af.447/flight.af.447.php
Suspected Failure of Thales
Heated Pitot Probe
Visual examination showed that the airplane
was not destroyed in flight; it appears to have
struck the surface of the sea in level flight with
high vertical acceleration.
Flight in theVertical Plane
Longitudinal Variables
Longitudinal PointMass
Equations of Motion
!V =
CT cos! " CD( )1
2#V 2S " mg sin$
m%
CT " CD( )1
2#V 2S " mg sin$
m
!$ =CT sin! + CL( )
1
2#V 2S " mgcos$
mV%
CL1
2#V 2S " mgcos$
mV
!h = " !z = "vz = V sin$
!r = !x = vx = V cos$ V = velocity
! = flight path angle
h = height (altitude)
r = range
Assume thrust is aligned with the velocityvector (smallangle approximation for !)
Mass = constant
Steady, Level Flight
0 =
CT ! CD( )1
2"V 2S
m
0 =
CL1
2"V 2S ! mg
mV
!h = 0
!r = V
Flight path angle = 0
Altitude = constant
Airspeed = constant
Dynamic pressure = constant
Thrust = Drag
Lift = Weight
Subsonic Lift and DragCoefficients
CL= C
Lo+ C
L!!
CD= C
Do+ !C
L
2
Lift coefficient
Drag coefficient
Subsonic flight, belowcritical Mach number
CLo, C
L!, C
Do, " # constant
Subsonic
Incompressible
Power and Thrust
Propeller
Turbojet
Power = P = T !V = CT1
2"V 3S # independent of airspeed
Thrust = T = CT1
2!V 2S " independent of airspeed
Throttle Effect
T = Tmax!T = CTmax!TqS, 0 " !T " 1
Typical Effects of Altitude and
Velocity on Power and Thrust
Propeller
Turbojet
Thrust of a PropellerDriven Aircraft
T = !P!I
Pengine
V= !
net
Pengine
V
Efficiencies decrease with airspeed
Engine power decreases with altitude
Proportional to air density, w/o supercharger
With constant rpm, variablepitch propeller
where
!P = propeller efficiency
!I = ideal propulsive efficiency
!netmax " 0.85 # 0.9
Advance Ratio
J =V
nD
from McCormick
Propeller Efficiency, "P,and Advance Ratio, J
Effect of propellerblade pitch angle
where
V = airspeed, m / s
n = rotation rate, revolutions / s
D = propeller diameter, m
Thrust of a
Turbojet
Engine
T = !mV!o
!o"1
#$%
&'(
!t
!t"1
#$%
&'(
)c"1( ) +
!t
!o)c
*
+,

./
1/2
"1012
32
452
62
Little change in thrust with airspeed below Mcrit Decrease with increasing altitude
where
!m = !mair + !mfuel
!o =pstag
pambient
"
#$%
&'
(( )1)/(
; ( = ratio of specific heats * 1.4
!t =turbine inlet temperature
freestream ambient temperature
"
#$%
&'
+ c =compressor outlet temperature
compressor inlet temperature
"
#$%
&' from Kerrebrock
Performance Parameters
LifttoDrag Ratio
Load Factor
LD=CL
CD
n = LW
= Lmg,"g"s
ThrusttoWeight Ratio T W= T
mg,"g"s
Wing Loading WS, N m
2or lb ft
2
Steady, Level Flight
Trimmed CL and !
Trimmed liftcoefficient, CL Proportional to
weight
Decrease with V2
At constantairspeed, increaseswith altitude
Trimmed angle of attack, ! Constant if dynamic pressure
and weight are constant
If dynamic pressure decreases,angle of attack must increase
W = CLtrim qS
CLtrim =1
qW S( ) =
2
!V 2W S( ) =
2 e"h
!0V2
#
$%&
'(W S( )
! trim =2W "V 2S # CLo
CL!
=
1
qW S( ) # CLo
CL!
Thrust Required for
Steady, Level Flight
Trimmed thrust
Ttrim
= Dcruise
= CDo
1
2!V 2S"
#$%&'+ (
2W2
!V 2S
Minimum required thrust conditions
!Ttrim
!V= C
Do"VS( ) #
4$W 2
"V 3S= 0Necessary Condition
= Zero Slope
Parasitic Drag Induced Drag
Necessary and Sufficient
Conditions for Minimum
Required Thrust
!Ttrim
!V= C
Do"VS( ) #
4$W 2
"V 3S= 0
Necessary Condition = Zero Slope
Sufficient Condition for a Minimum = Positive Curvature when slope = 0
! 2Ttrim
!V 2= C
Do"S( ) +
12#W 2
"V 4S> 0
(+) (+)
Airspeed for
Minimum Thrust in
Steady, Level Flight
Fourthorder equation for velocity
Choose the positive root
!Ttrim
!V= C
Do"VS( ) #
4$W 2
"V 3S= 0
VMT
=2
!W
S
"#$
%&'
(CDo
Satisfy necessarycondition V
4=
4!CDo"2
#
$%
&
'( W S( )
2
P51 MustangMinimumThrustExample
VMT
=2
!W
S
"#$
%&'
(CDo
=2
!1555.7( )
0.947
0.0163=76.49
!m / s
Wing Span = 37 ft (9.83m)
Wing Area = 235 ft2(21.83m
2)
Loaded Weight = 9,200 lb (3, 465 kg)
CDo = 0.0163
! = 0.0576
W / S = 39.3 lb / ft2(1555.7 N / m
2)
Altitude, m
Air Density,
kg/m^3 VMT, m/s0 1.23 69.112,500 0.96 78.205,000 0.74 89.1510,000 0.41 118.87
Lift Coefficient in
MinimumThrust
Cruising Flight
VMT
=2
!W
S
"#$
%&'
(CDo
CLMT
=2
!VMT
2
W
S
"#$
%&'=
CDo
(
Airspeed for minimum thrust
Corresponding lift coefficient
Power Required for
Steady, Level Flight
Trimmed power
Ptrim
= TtrimV = D
cruiseV = C
Do
1
2!V 2S"
#$%&'+2(W 2
!V 2S)
*+
,
.V
Minimum required power conditions
!Ptrim
!V= C
Do
3
2"V 2S( ) #
2$W 2
"V 2S= 0
Parasitic Drag Induced Drag
Airspeed for Minimum
Power in Steady,
Level Flight
Fourthorder equation for velocity Choose the positive root
VMP
=2
!W
S
"#$
%&'
(3C
Do
Satisfy necessary condition
!Ptrim
!V= C
Do
3
2"V 2S( ) #
2$W 2
"V 2S= 0
Corresponding lift anddrag coefficients
CLMP
=3C
Do
!
CDMP
= 4CDo
Achievable Airspeeds in Cruising Flight
Two equilibrium airspeeds for a given thrust or power setting
Low speed, high CL, high !
High speed, low CL, low !
Achievable airspeeds between minimum and maximum valueswith maximum thrust or power
Back Side of the
Thrust Curve
Achievable Airspeeds for Jetin Cruising Flight
Tavail
= CDo
1
2!V 2S"
#$%&'+2(W 2
!V 2S
CDo
1
2!V 4S"
#$%&') T