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Cruising Flight Performance!Robert Stengel, Aircraft Flight Dynamics,
MAE 331, 2016
Copyright 2016 by Robert Stengel. All rights reserved. For educational use only.http://www.princeton.edu/~stengel/MAE331.html
http://www.princeton.edu/~stengel/FlightDynamics.html
•! Definitions of airspeed•! Performance parameters•! Steady cruising flight conditions•! Breguet range equations•! Optimize cruising flight for
minimum thrust and power•! Flight envelope
Learning Objectives
Reading:!Flight Dynamics !
Aerodynamic Coefficients, 118-130!
1
Review Questions!!! What is “static margin”?!!! Is the airplane’s pitching moment sensitivity to angle of
attack linear?!!! What factors are most important in defining the
airplane’s pitching moment sensitivity to angle of attack ?!
!! Which is more important: stability or control?!!! Is the airplane’s yawing moment sensitivity to sideslip
angle linear?!!! What effect does the wing dihedral angle have on
airplane stability?!!! Why would an airplane have a “twin tail”?!!! What are “ventral fins”, and why do airplanes have/not
have them?! 2
U.S. Standard Atmosphere, 1976
http://en.wikipedia.org/wiki/U.S._Standard_Atmosphere 3
Dynamic Pressure and Mach Number! = airdensity, functionof height= !sealevele
"#h
a = speed of sound= linear functionof height
Dynamic pressure = q ! !V 2 2Mach number =V a
4
Definitions of Airspeed
•! Indicated Airspeed (IAS)
•! Calibrated Airspeed (CAS)*
Airspeed is speed of aircraft measured with respect to air massAirspeed = Inertial speed if wind speed = 0
IAS = 2 pstagnation ! pambient( ) "SL =2 ptotal ! pstatic( )
"SL
!2qc"SL
, with qc ! impact pressure
CAS = IAS corrected for instrument and position errors
=2 qc( )corr #1
!SL
* Kayton & Fried, 1969; NASA TN-D-822, 1961 5
Definitions of Airspeed
True Airspeed (TAS)*
Equivalent Airspeed (EAS)*
Airspeed is speed of aircraft measured with respect to air massAirspeed = Inertial speed if wind speed = 0
EAS = CAS corrected for compressibility effects =2 qc( )corr #2
!SL
V ! TAS = EAS !SL
!(z)= IAScorrected
!SL!(z)
Mach number
M =TASa
* Kayton & Fried, 1969; NASA TN-D-822, 19616
Checklist!"!IAS?!"!CAS?!"!EAS?!"!TAS?!"!M?!
7
Flight in the Vertical Plane!
8
Longitudinal Variables
9
Longitudinal Point-Mass 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%CL12#V 2S "mgcos$
mV!h = " !z = "vz =V sin$!r = !x = vx =V cos$
V = velocity = Earth-relative airspeed = True airspeed with zero wind
! = flight path angleh = height (altitude)r = range
•! Assume thrust is aligned with the velocity vector (small-angle approximation for !!)
•! Mass = constant
10
Conditions for Steady, Level Flight
0 =CT ! CD( ) 1
2"V 2S
m
0 =CL12"V 2S ! mg
mV!h = 0!r = V
•! Flight path angle = 0•! Altitude = constant•! Airspeed = constant•! Dynamic pressure = constant
•! Thrust = Drag
•! Lift = Weight
11
Power and ThrustPropeller
Turbojet
Power = P = T !V = CT12"V 3S ! independent of airspeed
Thrust = T = CT12!V 2S ! independent of airspeed
Throttle Effect
T = Tmax!T = CTmax!TqS, 0 " !T " 1
12
Typical Effects of Altitude and Velocity on Power and Thrust
•! Propeller
•! Turbojet
13
Models for Altitude Effect on Turbofan Thrust
From Flight Dynamics, pp.117-118
Thrust = CT V ,!T( ) 12" h( )V 2S
= ko + k1V#( ) 1
2" h( )V 2S!T , N
ko = Static thrust coefficient at sea levelk1 = Velocity sensitivity of thrust coefficient
! = Exponent of velocity sensitivity= "2 for turbojet
#T = Throttle setting, 0,1( )$ h( ) = $SLe
"%h , $SL = 1.225 kg /m3, % = 1/ 9,042( )m"1 14
Models for Altitude Effect on Turbofan Thrust
From AeroModelMach.m in FLIGHT.m, Flight Dynamics, http://www.princeton.edu/~stengel/AeroModelMach.m
Atmos(-x(6)) : 1976 U.S. Standard Atmosphere function-x(6)= h = Altitude, m
airDens = ! = Air density at altitude h, kg/m3
u(4)= "T = Throttle setting, 0,1( )
[airDens,airPres,temp,soundSpeed] = Atmos(-x(6));Thrust = u(4) * StaticThrust * (airDens / 1.225)^0.7 * (1 - exp((-x(6) – 17000)/2000));
(airDens / 1.225)^0.7 * (1 - exp((-x(6) – 17000)/2000))
Empirical fit to match known characteristics of powerplant for generic business jet
15
Thrust of a Propeller-Driven
Aircraft
T = !P!I
PengineV
= !net
PengineV
Efficiencies decrease with airspeedEngine power decreases with altitude
Proportional to air density, w/o supercharger
With constant rpm, variable-pitch propellerwhere!P = propeller efficiency!I = ideal propulsive efficiency!netmax
" 0.85 # 0.9
16
Reciprocating-Engine Power and Specific Fuel Consumption (SFC)
•! Engine power decreases with altitude–! Proportional to air density, w/o supercharger–! Supercharger increases inlet manifold pressure,
increasing power and extending maximum altitude
17
P h( )PSL
= 1.132 ! h( )!SL
" 0.132
Anderson (Torenbeek)
SFC ! Independent of Altitude
Advance Ratio
J = VnD
from McCormick
Propeller Efficiency, ""P, and Advance Ratio, J
Effect of propeller-blade pitch angle
whereV = airspeed, m / sn = rotation rate, revolutions / sD = propeller diameter, m
18
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 McritDecrease with increasing altitude
!m = !mair + !mfuel
!o = pstag pambient( )(" #1)/" ; " = ratio of specific heats $1.4
!t = turbine inlet temp. freestream ambient temp.( )% c = compressor outlet temp. compressor inlet temp.( )
from Kerrebrock
19
Performance Parameters
Lift-to-Drag Ratio
Load Factor
LD = CL
CD
n = LW = L mg ,"g"s
Thrust-to-Weight Ratio TW = T mg ,"g"s
Wing Loading WS , N m2 or lb ft 2
20
Checklist!"!Flight variables?!"!Propeller vs. jet propulsion?!"!Variation with airspeed and altitude?!"!Advance ratio?!"!Wing loading?!
21
!!iiss""rriiccaall FFaacc""iidd•! Aircraft Flight Distance Records
http://en.wikipedia.org/wiki/Flight_distance_record
http://en.wikipedia.org/wiki/Flight_endurance_record•! Aircraft Flight Endurance Records
Rutan/Scaled Composites Voyager
Rutan/Virgin Atlantic Global Flyer
22
Steady, Level Flight!
23
Trimmed Lift Coefficient, CL
•! Trimmed lift coefficient, CL–! Proportional to weight and wing loading factor–! Decreases with V2
–! At constant true airspeed, increases with altitude
W = CLtrim
12!V 2"
#$%&' S = CLtrim
qS
CLtrim= 1qW S( ) = 2
!V 2 W S( ) = 2 e"h
!0V2
#$%
&'(W S( )
24
! = 1/ 9,042 m, inverse scale height of air density
Trimmed Angle of Attack, !!•! Trimmed angle of attack, !!
–! Constant if dynamic pressure and weight are constant
–! If dynamic pressure decreases, angle of attack must increase
! trim =2W "V 2S #CLo
CL!
=
1qW S( )#CLo
CL!
25
Thrust Required for Steady, Level Flight
26
Thrust Required for Steady, Level FlightTrimmed thrust
Ttrim = Dcruise =CDo
12!V 2S
"
#$
%
&'+(
2W 2
!V 2S
Minimum required thrust conditions
!Ttrim!V
= CDo"VS( )# 4$W
2
"V 3S= 0
Necessary Condition: Slope = 0
Parasitic Drag Induced Drag
27
Necessary and Sufficient Conditions for Minimum
Required Thrust
CDo!VS( ) = 4"W
2
!V 3S
Necessary Condition = Zero Slope
Sufficient Condition for a Minimum = Positive Curvature when slope = 0
! 2Ttrim!V 2 = CDo
"S( ) + 12#W2
"V 4S> 0
(+) (+) 28
Airspeed for Minimum Thrust in Steady, Level Flight
Fourth-order equation for velocityChoose the positive root
VMT =2!WS
"
#$
%
&'
(CDo
Satisfy necessary condition
V 4 = 4!CDo
" 2
#
$%&
'(W S( )2
29
Lift, Drag, and Thrust Coefficients in Minimum-Thrust Cruising Flight
CLMT= 2!VMT
2WS
"#$
%&'
=CDo
(= CL( ) L/D( )max
Lift coefficient
CDMT= CDo
+ !CLMT2 = CDo
+ !CDo
!= 2CDo
" CTMT
Drag and thrust coefficients
30
Achievable Airspeeds in Constant-Altitude 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 values with maximum thrust or power
Back Side of the Thrust Curve
31
Power Required for Steady, Level Flight
32
P = T x V
Power Required for Steady, Level Flight
Trimmed power
Ptrim =TtrimV = DcruiseV = CDo
12!V 2S
"
#$
%
&'+2(W 2
!V 2S)
*+
,
-.V
Minimum required power conditions
!Ptrim!V
= CDo
32"V 2S( ) # 2$W
2
"V 2S= 0
Parasitic Drag Induced Drag
33
Airspeed for Minimum Power in Steady,
Level Flight
•! Fourth-order equation for velocity–! Choose the positive root
VMP =2!WS
"
#$
%
&'
(3CDo
•! Satisfy necessary condition CDo
32
!V 2S( ) = 2"W2
!V 2S
•! Corresponding lift and drag coefficients CLMP
=3CDo
!CDMP
= 4CDo 34
Achievable Airspeeds for Jet in Cruising Flight
Tavail = CDqS = CDo
12!V 2S"
#$%&' +
2(W 2
!V 2S
Thrust = constant
CDo
12!V 4S"
#$%&' (TavailV
2 + 2)W2
!S= 0
V 4 ! 2TavailCDo
"SV 2 + 4#W 2
CDo"S( )2
= 0
4th-order algebraic equation for V35
Achievable Airspeeds for Jet in Cruising Flight
Solutions for V2 can be put in quadratic form and solved easily
V2 ! x; V = ± x
V 4 ! 2TavailCDo
"SV 2 + 4#W 2
CDo"S( )2
= 0
x2 + bx + c = 0
x = ! b2± b
2"#$
%&'2
! c =V 2
36
Available thrust decreases with altitudeStall limitation at low speed
Mach number effect on lift and drag increases thrust required at high speed
Thrust Required and Thrust Available for a Typical Bizjet
Tmax (h) = Tmax (SL)! SL( )e"#h!(SL)
$
%&
'
()
x
= Tmax (SL) e"#h$% '(
x= Tmax (SL)e
" x#h
Empirical correction to force thrust to zero at a given altitude, hmax. c is a convergence factor.
Tmax (h) = Tmax (SL)e! x"h 1! e! h!hmax( ) c#$ %&
37
Typical Simplified Jet Thrust Model
Thrust Required and Thrust Available for a Typical Bizjet
Typical StallLimit
38
Checklist!"!Thrust required vs. airspeed and altitude?!"!Minimum-thrust cruise vs. minimum-power
cruise?!"!Power/thrust available for climb?!"!Mach effect?!
39
Flying Qualities Becomes a Science!
Chapter 3, Airplane Stability and Control, Abzug and Larrabee!
•! What are the principal subject and scope of the chapter?!
•! What technical ideas are needed to understand the chapter?!
•! During what time period did the events covered in the chapter take place?!
•! What are the three main "takeaway" points or conclusions from the reading?!
•! What are the three most surprising or remarkable facts that you found in the reading?!
40
The Flight Envelope!
41
Flight Envelope Determined by Available Thrust
•! Flight ceiling defined by available climb rate
–! Absolute: 0 ft/min–! Service: 100 ft/min–! Performance: 200 ft/min
Excess thrust provides the ability to accelerate or
climb
•! All altitudes and airspeeds at which an aircraft can fly –! in steady, level flight –! at fixed weight
42
Additional Factors Define the Flight Envelope
!! Maximum Mach number!! Maximum allowable
aerodynamic heating!! Maximum thrust!! Maximum dynamic
pressure!! Performance ceiling!! Wing stall!! Flow-separation buffet
!! Angle of attack!! Local shock waves
Piper Dakota Stall Buffethttp://www.youtube.com/watch?v=mCCjGAtbZ4g
43
Boeing 787 Flight Envelope (HW #5, 2008)
Best Cruise Region
44
Lockheed U-2 “Coffin Corner”
45
Stall buffeting and Mach buffeting are limiting factors
Narrow corridor for safe flight
Climb Schedule
Air Commerce Act of 1926•! Airlines formed to carry mail and passengers:
–! Northwest (1926)–! Eastern (1927), bankruptcy–! Pan Am (1927), bankruptcy–! Boeing Air Transport (1927), became United (1931)–! Delta (1928), consolidated with Northwest, 2010–! American (1930)–! TWA (1930), acquired by American–! Continental (1934), consolidated with United, 2010
Boeing 40
Ford Tri-Motor Lockheed Vega
http://www.youtube.com/watch?v=3a8G87qnZz4
!!iiss""rriiccaall FFaacc""iiddss
46
Commercial Aircraft of the 1930sStreamlining, engine cowlings
Lockheed 14 Super Electra, Boeing 247
47
Douglas DC-1, DC-2, DC-3
Comfort and Elegance by the End of the Decade
Boeing 307, 1st pressurized cabin (1936), flight engineer, B-17 pre-cursor, large dorsal fin (exterior and interior)
Sleeping bunks on transcontinental planes (e.g., DC-3)Full-size dining rooms on flying boats
48
Seaplanes Became the First TransOceanic Air Transports
•! PanAm led the way–! 1st scheduled TransPacific flights(1935)–! 1st scheduled TransAtlantic flights(1938)–! 1st scheduled non-stop Trans-Atlantic flights (VS-44, 1939)
•! Boeing B-314, Vought-Sikorsky VS-44, Shorts Solent•! Superseded by more efficient landplanes (lighter, less drag)
http://www.youtube.com/watch?v=x8SkeE1h_-A
49
Checklist!"!Flight envelope?!
50
Optimal Cruising Flight!
51
Maximum Lift-to-Drag Ratio
CL( )L /Dmax =CDo
!= CLMT
LD = CL
CD=
CL
CDo+ !CL
2
! CLCD
( )!CL
=1
CDo+ "CL
2 #2"CL
2
CDo+ "CL
2( )2= 0
Satisfy necessary condition for a maximum
Lift-to-drag ratio
Lift coefficient for maximum L/D and minimum thrust are the same
52
Airspeed, Drag Coefficient, and Lift-to-Drag Ratio for L/Dmax
VL/Dmax =VMT =2!
WS
"#$
%&'
(CDo
CD( )L /Dmax = CDo+ CDo
= 2CDo
L / D( )max =CDo
!
2CDo
=1
2 !CDo
Maximum L/D depends only on induced drag factor and zero-lift drag coefficient
Induced drag factor and zero-lift drag coefficient are functions of Mach number
Airspeed
Drag Coefficient
Maximum L/D
53
Cruising Range and Specific Fuel Consumption
0 = CT !CD( ) 12"V 2S m
0 = CL12"V 2S !mg#
$%&'( mV
•! Thrust = Drag
•! Lift = Weight
•! Thrust specific fuel consumption, TSFC = cT•! Fuel mass burned per sec per unit of thrust
!mf = !cTTcT :kg skN
54
!h = 0!r =V
•! Level flight
•! Power specific fuel consumption, PSFC = cP•! Fuel mass burned per sec per unit of power
!mf = !cPPcP :kg skW
Breguet Range Equation for Jet Aircraft
drdm
= dr dtdm dt
=!r!m= V
!cTT( ) = ! VcTD
= ! LD
"#$
%&'
VcTmg
Rate of change of range with respect to weight of fuel burned
Range traveled
Range = R = dr0
R
! = " LD
#$%
&'(
VcT g
#$%
&'(Wi
Wf
!dmm
Louis Breguet, 1880-1955
dr = ! LD
"#$
%&'
VcTmg
dm
55
Maximum Range of a Jet Aircraft Flying at
Constant Altitude
Range is maximized when
Range = ! CL
CD
"#$
%&'
1cT g
"#$
%&'
2CL(SWi
Wf
)dmm1 2
=CL
CD
"
#$%
&'2cT g
"#$
%&'
2(S
mi1 2 !mf
1 2( )
Vcruise t( ) =2W t( )
CL! hfixed( )S
At constant altitude and SFC
CL
CD
!
"#$
%&= maximum
B-727
56
Breguet Range Equation for Jet Aircraft
For constant true airspeed, V = Vcruise, and SFC
R = ! LD
"#$
%&'VcruisecT g
"#$
%&'ln m( ) mi
m f
= LD
"#$
%&'VcruisecT g
"#$
%&'ln mi
mf
"
#$%
&'
= VcruiseCL
CD
!"#
$%&
1cT g
!"#
$%&ln mi
mf
!
"#$
%&
!! Vcruise(CL/CD) as large as possible!! Respect Mcrit!! ## as small as possible!! h as high as possible
MD-83
57
Maximize Jet Aircraft Range Using Optimal Cruise-Climb
Vcruise = 2W CL!S
!R!CL
"! Vcruise
CLCD
( )!CL
=
! VcruiseCL
CDo+ #CL
2( )$
%&&
'
())
!CL
= 0
Assume 2W t( ) ! h( )S = constanti.e., airplane climbs at constant TAS as fuel is burned
58
Maximize Jet Aircraft Range Using Optimal Cruise-Climb
! Vcruise CL CDo+ "CL
2( )#$ %&!CL
= 2W'S
! CL1/2 CDo
+ "CL2( )#$ %&
!CL
= 0
Optimal values: (see Supplemental Material)
CLMR=
CDo
3!: Lift Coefficient for Maximum Range
CDMR= CDo
+CDo
3= 4
3CDo
Vcruise!climb = 2W t( ) CLMR" h( )S = a h( )Mcruise-climb
a h( ) : Speed of sound; Mcruise-climb : Mach number 59
Step-Climb Approximates Optimal Cruise-Climb
!! Cruise-climb usually violates air traffic control rules!! Constant-altitude cruise does not!! Compromise: Step climb from one allowed altitude
to the next as fuel is burned
60
!!iiss""rriiccaall FFaacc""iidd
Breguet 890 Mercure
Breguet Atlantique
•! Louis Breguet (1880-1955), aviation pioneer•! Gyroplane (1905), flew vertically in 1907•! Breguet Type 1 (1909), fixed-wing aircraft•! Formed Compagnie des messageries
aériennes (1919), predecessor of Air France•! Breguet Aviation manufactured numerous
military and commericial aircraft until after World War II; teamed with BAC in SEPECAT
•! Merged with Dassault in 1971
Breguet 14
SEPECAT Jaguar
61
Checklist!"!Specific fuel consumption?!"!Breguet equation?!
"! Constant altitude?!"! Cruise-climb?!
62
Next Time:!Gliding, Climbing, and
Turning Flight!Reading:!
Flight Dynamics !Aerodynamic Coefficients, 130-141, 147-155!
63
Conditions for gliding flightParameters for maximizing climb angle and rate
Review the V-n diagramEnergy height and specific excess power
Alternative expressions for steady turning flightThe Herbst maneuver
Learning Objectives
##uupppplleemmeennttaall MMaa$$rriiaall
64
Achievable Airspeeds in Propeller-Driven
Cruising Flight
Pavail = TavailV
V 4 ! PavailVCDo
"S+ 4#W 2
CDo"S( )2
= 0
Power = constant
Solutions for V cannot be put in quadratic form; solution is more difficult, e.g., Ferrari s method
aV 4 + 0( )V 3 + 0( )V 2 + dV + e = 0
Best bet: roots in MATLAB
Back Side of the Power
Curve
65
P-51 Mustang Minimum-Thrust
Example
VMT =2!WS
"
#$
%
&'
(CDo
=2!1555.7( ) 0.947
0.0163=76.49!
m / s
Wing Span = 37 ft (9.83m)Wing Area = 235 ft 2 (21.83m2 )
Loaded Weight = 9,200 lb (3,465 kg)CDo
= 0.0163! = 0.0576
W / S = 39.3 lb / ft 2 (1555.7 N /m2 )
Altitude, mAir Density, kg/m^3 VMT, m/s
0 1.23 69.112,500 0.96 78.205,000 0.74 89.1510,000 0.41 118.87
Airspeed for minimum thrust
66
P-51 Mustang Maximum L/D
Example
VL /Dmax = VMT =76.49!
m / s
Wing Span = 37 ft (9.83m)Wing Area = 235 ft (21.83m2 )
Loaded Weight = 9,200 lb (3,465 kg)CDo
= 0.0163! = 0.0576
W / S = 1555.7 N /m2
CL( )L /Dmax =CDo
!= CLMT
= 0.531
CD( )L /Dmax = 2CDo= 0.0326
L / D( )max =1
2 !CDo
= 16.31
Altitude, mAir Density, kg/m^3 VMT, m/s
0 1.23 69.112,500 0.96 78.205,000 0.74 89.1510,000 0.41 118.87 67
Breguet Range Equation for Propeller-Driven Aircraft
drdw
=!r!w=
V!cPP( )
= !V
cPTV= !
VcPDV
= !LD"
#$
%
&'1
cPW
Rate of change of range with respect to weight of fuel burned
Range traveled
Range = R = dr0
R
! = "LD#
$%
&
'(1cP
#
$%
&
'(
Wi
Wf
! dww
Breguet 890 Mercure
68
Breguet Range Equation for Propeller-Driven Aircraft
For constant true airspeed, V = Vcruise
R = ! LD"
#$
%
&'1cP
"
#$
%
&'ln w( )Wi
Wf
=CL
CD
"
#$
%
&'1cP
"
#$
%
&'ln
Wi
Wf
"
#$$
%
&''
Range is maximized when
CL
CD
!
"#
$
%&= maximum= L
D( )max
Breguet Atlantique
69
P-51 Mustang Maximum Range (Internal Tanks only)
W = CLtrimqS
CLtrim= 1qW S( )
= 2!V 2 W S( ) = 2 e"h
!0V2
#$%
&'(W S( )
R = CL
CD
!
"#
$
%&max
1cP
!
"#
$
%&ln
Wi
Wf
!
"##
$
%&&
= 16.31( ) 10.0017!
"#
$
%&ln
3,465+6003,465
!
"#
$
%&
=1,530 km (825 nm( )
70
Maximize Jet Aircraft Range Using Optimal Cruise-Climb
! Vcruise CL CDo+ "CL
2( )#$ %&!CL
= 2w'S
! CL1/2 CDo
+ "CL2( )#$ %&
!CL
= 0
Optimal values:
CLMR=
CDo
3!: CDMR
= CDo+CDo
3= 4
3CDo
2w!S
= Constant; let CL1/2 = x, CL = x
2
""x
xCDo
+ #x4( )$
%&&
'
())=CDo
+ #x4( )* x 4#x3( )CDo
+ #x4( )2 =CDo
* 3#x4( )CDo
+ #x4( )2
71
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