Fixed Wing Fighter AircraftFlight Performance
Part II
SOLO HERMELIN
Updated: 04.12.12 28.02.15
1
http://www.solohermelin.com
Table of Content
SOLO
2
Aerodynamics
Introduction to Fixed Wing Aircraft Performance
Earth Atmosphere
Mach Number
Shock & Expansion Waves
Reynolds Number and Boundary Layer
Knudsen Number
Flight Instruments
Aerodynamic Forces
Aerodynamic Drag
Lift and Drag Forces
Wing Parameters
Specific Stabilizer/Tail Configurations
Fixed
Wi
ng
Part
I
Fixed Wing Fighter Aircraft Flight Performance
Table of Content (continue – 1)
SOLO
3
Specific Energy
Aircraft Propulsion Systems
Aircraft Propellers
Aircraft Turbo Engines
Afterburner
Thrust Reversal Operation
Aircraft Propulsion Summary
Vertical Take off and Landing - VTOL
Engine Control System
Aircraft Flight Control
Aircraft Equations of Motion
Aerodynamic Forces (Vectorial)
Three Degrees of Freedom Model in Earth Atmosphere
Fixed
Wi
ng
Part
I
Fixed Wing Fighter Aircraft Flight Performance
Table of Content (continue – 2)
SOLO Fixed Wing Fighter Aircraft Flight Performance
4
Parameters defining Aircraft Performance
Takeoff (no VSTOL capabilities)
Landing (no VSTOL capabilities)
Climbing Aircraft Performance
Gliding Flight
Level Flight
Steady Climb (V, γ = constant)
Optimum Climbing Trajectories using Energy State Approximation (ESA)Minimum Fuel-to- Climb Trajectories using Energy State Approximation (ESA)Maximum Range during Glide using Energy State Approximation (ESA)
Aircraft Turn Performance
Maneuvering Envelope, V – n Diagram
Table of Content (continue – 3)
SOLO Fixed Wing Fighter Aircraft Flight Performance
5
Air-to-Air Combat
Energy–Maneuverability Theory
Supermaneuverability
Constraint Analysis
References
Aircraft Combat Performance Comparison
SOLO
This Presentation is about Fixed Wing Aircraft Flight Performance.
The Fixed Wing Aircraft are•Commercial/Transport Aircraft (Passenger and/or Cargo)•Fighter Aircraft
Fixed Wing Fighter Aircraft Flight Performance
Continue from Part I
7
Fixed Wing Fighter Aircraft Flight PerformanceSOLO
The Aircraft Flight Performance is defined by the following parameters
• Take-off distance• Landing distance• Maximum Endurance and Speed for Maximum Endurance• Maximum Range and Speed for Maximum Range• Ceiling(s)• Climb Performance• Turn Performance• Combat Radius• Maximum Payload
Parameters defining Aircraft Performance
8
Performance of an Aircraft with Parabolic PolarSOLO
Assumptions:
•Point mass model.•Flat earth with g = constant.•Three-dimensional aircraft trajectory.•Air density that varies with altitude ρ=ρ(h)•Drag that varies with altitude, Mach number and control effort D = D(h,M,n) and is given by a Parabolic Polar.•Thrust magnitude is controllable by the throttle. •No sideslip angle.•No wind.
α
T
V
L
D
Bx
Wx
Bz
Wz
Wy
By
Aircraft Coordinate System
To understand how different parameters affect Aircraft Performance we start with a Simplified Model, where Analytical Solutions can be obtained. Results for real aircraft will then be presented.
Return to Table of Content
9
SOLO Aircraft Flight Performance
Takeoff
The Takeoff distance sTO is divided as the sum of the following distances:sg – Ground Runsr – Rotation Distancest – Transition Distancesc – Climb Distance to reach Screen Height
ctrgTO sssss +++=
Ground RunV = 0
sg
sTO
sr str
V TORotation
Transition
sCL
θ CL
htr
hobs
R
Takeoff htransition < hobstacle
θ CL
Ground RunV = 0
sg
sTO
sr sobs
V TORotation
Transition
hobs
R
Takeoff htransition > hobstacle
We distinguish between two cases of Takeoff •The Aircraft must passes over an obstacle at altitude hobs..•The obstacle is cleared during the transition phase.
Assume no Vertical Takeoff Capability.
10
Takeoff (continue – 1)
During the Ground Run there are additionaleffects than in free flight, that must be considered:-Friction between the tires and the ground during rolling.-Additional drag due to the landing gear fully extended.-Additional Lift Coefficient due to extended flaps.-Ground Effect due to proximity of the wings to the ground, that reduces the Induced Drag and the Lift.
Ground Run
SOLO Aircraft Flight Performance
Ground run sg
Transition distance st
Climb distance sc
Stall safety
Take-off possible with one engine
Continue take-off if engine fails
after this point
Stop take-off if engine fails before
this point
Acceleration at full power
γ c
Total take-off if distance
VCRVMCGVTVS
hc
L
W
TD
R
μR
The Aircraft can leave the ground when the velocity reaches the Stall Velocity where Lift equals Weight
max,2
02
1Lstallstall CSVLW ρ==
max,0
12
Lstall CS
WV
ρ=
The Liftoff Velocity is 1.1 to 1.2 Vstall.
11
ReactionGroundLWR
gW
RDT
td
VdV
Vtd
xd
−=
−−==
=
/
µ
( )
( )LWDT
gW
Vd
td
LWDT
gWV
Vd
sd xs
−−−=
−−−==
µ
µ/
/
Takeoff (continue – 2)
Average Coefficient of Friction Values μ
Ground Run
SOLO Aircraft Flight Performance
Ground run sg
Transition distance st
Climb distance sc
Stall safety
Take-off possible with one engine
Continue take-off if engine fails
after this point
Stop take-off if engine fails before
this point
Acceleration at full power
γ c
Total take-off if distance
VCRVMCGVTVS
hc
L
W
R
μRD
T
V
12
Takeoff (continue – 3)
SOLO Aircraft Flight Performance
Ground run sg
Transition distance st
Climb distance sc
Stall safety
Take-off possible with one engine
Continue take-off if engine fails
after this point
Stop take-off if engine fails before
this point
Acceleration at full power
γ c
Total take-off if distance
VCRVMCGVTVS
hc
L
W
R
μRD
T
V
T (Jet)
Lift, Drag
, Thrust, Resistance
–lb
L, D
, T, R
T (Prop)
D +μ R
Ground Speed – ft/s
Texcess(Prop)=T(Prop) -(D+μ R)Texcess(Jet)=T(Jet) -(D+μ R)
Vground
( )
ReactionGroundLWR
RDTg
WV
−=
+−= µ
Ground Run (continue -1)
13
20 VCVBTT ++=
cVbVaVd
td
cVbVa
V
Vd
sd xs
++=
++==
2
2
1
Takeoff (continue – 4)
Ground Run (continue – 2)
To obtain an Analytic Solution assume that during the Ground Run the Thrust can be approximated by
Using
=
=
L
D
CSVL
CSVD
2
2
2
12
1
ρ
ρ
( )
−=
=
+−−=
µ
µρ
W
Tgc
W
gBb
W
gCCC
W
Sga LD
0:
2:
2:
where
SOLO Aircraft Flight Performance
Ground run sg
Transition distance st
Climb distance sc
Stall safety
Take-off possible with one engine
Continue take-off if engine fails
after this point
Stop take-off if engine fails before
this point
Acceleration at full power
γ c
Total take-off if distance
VCRVMCGVTVS
hc
L
W
R
μRD
T
V
14
cVbVaVd
td
cVbVa
V
Vd
sd xs
++=
++==
2
2
1
Takeoff (continue – 5)
Ground Run (continue – 3)
Integrating those equations between two velocities V1 and V2 gives
−−⋅
++
−+
++++=
2
1
1
2
2
121
222
1
1
1
1ln
42
ln2
1
a
a
a
a
caba
b
cVbVa
cVbVa
asg
−−⋅
++
−=
1
2
2
1
2 1
1
1
1ln
4
1
a
a
a
a
cabtg
where
cab
bVaa
cab
bVaa
4
2:
4
2:
2
22
2
11
−+=
−+=
SOLO Aircraft Flight Performance
Ground run sg
Transition distance st
Climb distance sc
Stall safety
Take-off possible with one engine
Continue take-off if engine fails
after this point
Stop take-off if engine fails before
this point
Acceleration at full power
γ c
Total take-off if distance
VCRVMCGVTVS
hc
L
W
R
μRD
T
V
20 VCVBTT ++=
15
Takeoff (continue – 6)
Ground Run (continue – 4)
then
( )( )
−−−−
=
+=
TL
LDLD
g
CWTCCCCg
SW
c
cVa
as
µµµρ
/1
1ln
/
ln2
1
0
22
0,00 01 ==⇐== CBTTV
Assume
where
22:&
/2: VV
V
SWC T
TLT
==ρ
A further simplification, using , givesZZ
Z 1
1
1ln
<<≈
−
−
=µρ
W
TCg
SWs
TL
g0
/
SOLO Aircraft Flight Performance
gL sCg
SW
W
T
Tρ
/0 >
Ground run sg
Transition distance st
Climb distance sc
Stall safety
Take-off possible with one engine
Continue take-off if engine fails
after this point
Stop take-off if engine fails before
this point
Acceleration at full power
γ c
Total take-off if distance
VCRVMCGVTVS
hc
L
W
R
μRD
T
V
16
Takeoff (continue – 7)
Rotation Distance
At the ground roll and just prior to going into transition phase, most aircraft areRotated to achieve an Angle of Attack to obtain the desired Takeoff Lift CoefficientCL. Since the rotation consumes a finite amount of time (1 – 4 seconds), the distancetraveled during rotation sr, must be accounted for by using
where Δt is usually taken as 3 seconds.
SOLO Aircraft Flight Performance
tVs tr ∆=
Ground RunV = 0
sg
sTO
sr str
V TORotation
Transition
sCL
θ CL
htr
hobs
R
L
W
R
μRD
T
V
17
Takeoff (continue – 8)
Transition Distance
In the Transition Phase the Aircraft is in the Air (μ = 0) and turn to the Climb Angle.The Equation of Motion are:
SOLO Aircraft Flight Performance
Ta
Tat
Tat
VV
DT
VV
g
Wt
DT
VV
g
Ws
>
−−=
−−=
2
2
22
DT
gW
Vd
td
DT
gWV
Vd
sd xs
−=
−==
/
/
Assuming T – D = const., we canIntegrate the Equations of Motion(assuming Va > VT)
Ground RunV = 0
sg
sTO
sr str
V TORotation
Transition
sCL
θ CL
htr
hobs
R
18
Takeoff (continue – 9)
Climb Distance
The Climb Distance is evaluated from the following (see Figure):
SOLO Aircraft Flight Performance
c
c
c
cc
hhs
c
γγγ 1
tan
<<
≈=
For small angles of Climb L = W.We can write
Ground RunV = 0
sg
sTO
sr str
V TORotation
Transition
sCL
θ CL
htr
hobs
R
cL
cLD
c
cc C
CkC
W
T
L
D
W
T
,
2,0 +
−=−=γ
We have
cLcLD
cc CkCCWT
hs
,,0 // −−≈
19
Takeoff (continue – 10)
SOLO Aircraft Flight Performance
19
ctrgTO sssss +++=
sec41−=∆∆= ttVs tr
Ta
Tat
Tat
VV
DT
VV
g
Wt
DT
VV
g
Ws
>
−−=
−−=
2
2
22
−−⋅
++
−+
++++=
2
1
1
2
2
121
222
1
1
1
1ln
42
ln2
1
a
a
a
a
caba
b
cVbVa
cVbVa
asg
cab
bVaa
cab
bVaa
4
2:
4
2:
2
22
2
11
−+=
−+=
−−⋅
++
−=
1
2
2
1
2 1
1
1
1ln
4
1
a
a
a
a
cabtg
Ground RunV = 0
sg
sTO
sr str
V TORotation
Transition
sCL
θ CL
htr
hobs
R
Takeoff Summary
Rotation Phase
Climb Phase
Transition Phase
Ground Run
cLcLD
cc CkCCWT
hs
,,0 // −−≈
20Minimum required takeoff runway lengths.
Summary of takeoff requirementsIn order to establish the allowable takeoff weight for a transport category airplane, at any airfield, the following must be considered:•Airfield pressure altitude•Temperature•Headwind component•Runway length•Runway gradient or slope•Obstacles in the flight path
Return to Table of Content
21
LandingLanding is similar to Takeoff, but in reverse.We assume again that the Aircraft doesn’t haveVTOL capabilities.The Landing Phase can be divide in the followingPhases:
1. The Final approach when the Aircraft Glides toward the runway at a steady speed and rate of descent.
2. The Flare, or Transition phase. The Pilot attempts to rotate the Aircraft nose up and reduce the Rate of Sink to zero and the forward speed to a minimum, that is larger than Vstall. When entering this phase the velocity is less than 1.3Vstall and 1.15 Vstall at touchdown.
3. The Floating Phase, which is necessary if at the end of Flare phase, when the rate of descent is zero, an additional speed reduction is necessary. The Float occurs when the
Aircraft is subjected to ground effect which requires speed reduction for touchdown.
4. The Ground Run after the Touchdown the Aircraft must reduce the speed to reach a sufficient low one to be able to turn off the runway. For this it can use Thrust Reverse (if available), spoilers or drag parachutes (like F-15 or MIG-21) and brakes are applied.
SOLO Aircraft Flight Performance
Ground Run sgr
Transition
Airborne Phase
Total Landing Distance
Float
sfFlare stGlide sg
γ
hg
hf
Touchdown
22
Landing (continue – 1)
Descending Phase
SOLO Aircraft Flight Performance
The Aircraft is aligned with the landing runaway at an altitude hg and a gliding angle γ.The Aircraft Glides toward the runway at a steady speed and rate of descent, until it reachesThe altitude ht at which it goes to Transition Phase, turning with a Radius of Turn R. The Descending Range on the ground is :
γγγ
γγ RhRhhh
s ggtgg
−≈
−=
−=
<<1
tan
cos
tan
Ground Run sgr
Transition
Airborne Phase
Total Landing Distance
Float
sfFlare stGlide sg
γ
hg
hf
Touchdown
23
Landing (continue – 2)
Transition Phase
SOLO Aircraft Flight Performance
If γ is the descent angle and R is the turn radiusthen the Aircraft must start the Transition Phase at an altitude ht, above the ground, given by:
( )γcos1−=RhtThe Transition Range on the ground is
γγ RRst ≈= sin
To calculate the turn radius we must use the flight velocity which varies between 1.3 Vstall
at the beginning to 1.1 Vstall at Touchdown. Let use an average velocity
3.11.1 −∈= tstalltt mVmV
If the Transition Turn Acceleration is nt = 1.15 – 1.25 g than the Turn Radius is
( ) gn
VR
t
t
1
2
−=
The Transition Turn time is ( ) gn
V
RVt
t
t
tt 1/ −
== γγ
Ground Run sgr
Transition
Airborne Phase
Total Landing Distance
Float
sfFlare stGlide sg
γ
hg
hf
Touchdown
24
Landing (continue – 3)
Float Phase
SOLO Aircraft Flight Performance
In this phase the Pilot brings the nose wheel to the ground at the touchdown velocity Vt:
tVs tf ∆=
where Δt is between 2 to 3 seconds.
Ground Run sgr
Transition
Airborne Phase
Total Landing Distance
Float
sfFlare stGlide sg
γ
hg
hf
Touchdown
25
Landing (continue – 4)
Ground Run Phase
SOLO Aircraft Flight Performance
The equations of motion are the same as those developed for Takeoff, but with different parameters, adapted for Landing. Those equations are:
cVbVaVd
td
cVbVa
V
Vd
sd xs
++=
++==
2
2
1
( )
−=
=
+−−=
µ
µρ
W
Tgc
W
gBb
W
gCCC
W
Sga grLgrD
0
,,
:
2:
2:
where
−−⋅
++
−+
++++=
2
1
1
2
21
21
222
1
1
1
1ln
42ln
2
1
a
a
a
a
caba
b
cVbVa
cVbVa
asg
−−⋅
++
−=
1
2
2
1
2 1
1
1
1ln
4
1
a
a
a
a
cabtg
wherecab
bVaa
cab
bVaa
4
2:
4
2:
2
22
2
11
−+=
−+=
20 VCVBTT ++=
Assume a constant Thrust T = T0: B = 0, C = 0. V1 = Vtouchdown, V2 = final velocity
cVa
cVa
asg +
+−=22
21ln
2
1
−−⋅
++
−=
1
2
2
1
1
1
1
1ln
4
1
a
a
a
a
catg ( )
−==−−= µµρW
TgcbCC
W
Sga LD
0:,0,2
:
Ground Run sgr
Transition
Airborne Phase
Total Landing Distance
Float
sfFlare stGlide sg
γ
hg
hf
Touchdown
26
Landing (continue – 5)
Ground Run Phase (continue – 1)
SOLO Aircraft Flight Performance
where( )
−=
−−=
µ
µρ
W
Tgc
CCW
Sga grLgrD
0
,,
:
2:
cab
Vaa
ca
Vaa
touchdown
4
2:
4
2:
2
11
−=
−=
Assume a constant Thrust T = T0: B = 0, C = 0. V1 = Vtouchdown, V2 = final velocity
cVa
cVa
asg +
+−=22
21ln
2
1
−−⋅
++
−=
1
2
2
1
1
1
1
1ln
4
1
a
a
a
a
catg
For the Landing Ground Run Phase the following must included:• if Thrust Reversal exists we must change T0 to – T0_reversal .•The Drag Coefficient CD0,gr must consider: - the landing gear fully extended. - spoilers or drag parachutes (if exist)•μ – the friction coefficient must be increased to describe the brakes effect.
Ground Run sgr
Transition
Airborne Phase
Total Landing Distance
Float
sfFlare stGlide sg
γ
hg
hf
Touchdown
27
Landing (continue – 6)
Summary
SOLO Aircraft Flight Performance
where( )
−=
−−=
µ
µρ
W
Tgc
CCW
Sga grLgrD
0
,,
:
2:
cab
Vaa
ca
Vaa
touchdown
4
2:
4
2:
2
11
−=
−=cVa
cVa
asg +
+−=22
21ln
2
1
−−⋅
++
−=
1
2
2
1
1
1
1
1ln
4
1
a
a
a
a
catg
Ground Run Phase
tVs tf ∆=
Float Phase
( ) gn
VRs
t
tt 1
2
−== γγ ( ) gn
V
RVt
t
t
tt 1/ −
== γγTransition Phase
Descent PhaseGround Run sgr
Transition
Airborne Phase
Total Landing Distance
Float
sfFlare stGlide sg
γ
hg
hf
Touchdown
( )γγ
γγ
1/
tan
cos
tan
2 −−=
−=
−= ttggfg
g
nVhRhhhs
28
H.H. Hurt, Jr., “Aerodynamics for Naval Aviators “,NAVAIR 00=80T-80 1-1-1965, pg. 35
Return to Table of Content
29
Fixed Wing Fighter Aircraft Flight PerformanceSOLO
Level Flight
The forces acting on an airplane in Level Flight are shown in Figure
0=
=
h
Vx
Lift and Drag Forces:
( ) TCkCSVCSVD
WCSVL
LDD
L
=+==
==
20
22
2
2
1
2
12
1
ρρ
ρ2
2
VS
WCL ρ
=
+=
+=
SV
WkCSV
SV
WkCSVD DD 2
2
02
242
2
02 2
2
14
2
1
ρρ
ρρ
Lift
DragThrust
Weight
Equations of motion:
0
0
=−=−
DT
WLQuasi-Static
30
Fixed Wing Fighter Aircraft Flight PerformanceSOLO
Level Flight
DragInducedDragParasite
D SV
WkCSVD
2
2
02 2
2
1
ρρ +=
Because of opposite trends in Parasite Drag and Induced Drag, with changes in velocity, the Total Drag assumes a minimum at a certain velocity. If we ignore the change in velocity of CD0 and k with velocity we obtain
04
3
2
0 =−=SV
WkCSV
Vd
DdD ρ
ρ
The velocity of minimum TotalDrag is
*4
0
2V
C
k
S
WV
D
==ρ
We see that the velocity of minimum Total Drag is equal to the Reference Velocity.
02
2
1DCSVρ
SV
Wk2
22
ρ*V
Lift
DragThrust
Weight
31
Fixed Wing Fighter Aircraft Flight PerformanceSOLO
Level Flight
For the velocity, V*, of minimum Total Drag we have
02*
22Di CkW
SV
WkD ==
ρ
DragInducedDragParasite
D SV
WkCSVD
2
2
02 2
2
1
ρρ +=
000min 2 DDD CkWCkWCkWD =+=
and
02
2
1DCSVρ
SV
Wk2
22
ρ*V
Lift
DragThrust
Weight
32M. Corcoran, T. Matthewson, N. W. Lee, S. H. Wong, “Thrust Vectoring”
Comparison of Takeoff Weight and Empty Weight of different Aircraft
33
Fixed Wing Fighter Aircraft Flight PerformanceSOLO
Level Flight
The Power Required, PR, for Level Flight is
SV
WkCSVVDP DR ρ
ρ2
03 2
2
1 +=⋅=
The Power Required for Level Flight assumes a minimum at a certain velocity Vmp. If we ignore the change in velocity of CD0 and k with velocity we obtain
02
2
32
2
02 =−=
SV
WkCSV
Vd
PdD
R
ρρ
or
*4
0 3
1
3
2V
C
k
S
WV
Dmp ==
ρ
*02, 3
32L
D
mpmpL C
k
C
VS
WC ===
ρ
( )*
000
02
,0
, 866.01
4
3
/3
/3e
CkkCkC
kC
CkC
Ce
DDD
D
mpLD
mpLmp ==
+=
+= *
2
min, 866.03
8
e
VW
SV
WkP mp
mpR ==
ρ
03
2
1DCSVρ
SV
Wk
ρ
22
*
3
1V
min,RP
Lift
DragThrust
Weight
34
Fixed Wing Fighter Aircraft Flight PerformanceSOLO
Available Aircraft Power and Thrust
• Throttle Effect
10 ≤≤= ηη ATT
• Propeller
airspeedwithvariationsmallVTP propellerA ≈⋅=,V
Pa, propeller
Power
Propeller Aircraft Available Powerat Altitude (h)
At a given Altitude h
• Turbojet
airspeedwithvariationsmallT jetA ≈,
V
Ta, jet
Thrust
Jet Aircraft Available Powerat Altitude h
At a given Altitude h
Lift
DragThrust
Weight
Lift
DragThrust
Weight
Level Flight
35
Fixed Wing Fighter Aircraft Flight PerformanceSOLO
Vmin Vmax
Pa, propeller
PRPmin
BA
ηaPa, propeller
Propeller Aircraft
Vmin Vmax
Ta, jet
TR
Dmin
η Ta, jet
AB
Jet Aircraft
Level Flight
To have a Level Flight the requirement must be satisfied by the available propulsion performance.•For a Propeller Aircraft, the available power Pa,propeller , at a given altitude h, is almost insensitive with changes in velocity. The Velocity in Level Flight is steady when the graph of Required Power PR intersects the graph of Pa,propeller at points A and B. We get two velocities Vmin (h) at A and Vmax (h) at B. By controlling the Propeller Power ηa Pa,propeller (0< ηa <1) we can reach any velocity between Vmin (h) and Vmax (h).
•For a Jet Aircraft, the available Thrust Ta,jet , at a given altitude h, is almost insensitive with changes in velocity. The Velocity in Level Flight is steady when the graph of Required Thrust TR intersects the graph of Ta,jet at points A and B. We get two velocities Vmin (h) at A and Vmax (h) at B. By controlling the Jet Thrust η Ta,jet (0< η <1) we can reach any velocity between Vmin (h) and Vmax (h).
36
Fixed Wing Fighter Aircraft Flight PerformanceSOLO
Vmin Vmax
Ta, jet
TR
Dmin
η Ta, jet
AB
Jet Aircraft
Level Flight
We have
Analytical Solution for Jet Aircraft
( )SV
WkCSVCkCSVDT DLD 2
2
022
02 2
2
1
2
1
ρρρ +=+==
Define
0
*
00
*
**
4
0
2:
2*,*,:
2:*,
*:
D
DDD
L
D
L
D
CkW
T
W
eTz
CCk
CC
C
Ce
C
k
S
WV
V
Vu
==
===
==ρ
2
2
/1
20
0
2
2
0
2
2
2
u
D
u
Dz
D V
Ck
SW
Ck
SW
VT
CkW
ρ
ρ
+=012 24 =+− uzu
Lift
DragThrust
Weight
37
Fixed Wing Fighter Aircraft Flight PerformanceSOLO
Vmin Vmax
Ta, jet
TR
Dmin
η Ta, jet
AB
Jet Aircraft
Level Flight
Analytical Solution for Jet Aircraft
012 24 =+− uzu Solving we obtain
1
1
2max
2min
−+=
−−=
zzu
zzu
4
0maxmaxmax
4
0minminmin
2*
2*
D
D
C
k
S
WuVuV
C
k
S
WuVuV
ρ
ρ
==
==
Lift
DragThrust
Weight
0
*
00
*
**
4
0
2:
2*,*,:
2:*,
*:
D
DDD
L
D
L
D
CkW
T
W
eTz
CCk
CC
C
Ce
C
k
S
WV
V
Vu
==
===
==ρ
38
Fixed Wing Fighter Aircraft Flight PerformanceSOLO
Level Flight
Analytical Solution for Jet Aircraft
1
1
2max
2min
−+=
−−=
zzu
zzu
12min −−= zzu
12max −+= zzu
At the absolute Ceiling (when is only one possible velocity) we have umax = umin, thereforez = 1.
max,
2
Lstall CS
WV
ρ=
Lift
DragThrust
Weight
0
*
00
*
**
4
0
2:
2*,*,:
2:*,
*:
D
DDD
L
D
L
D
CkW
T
W
eTz
CCk
CC
C
Ce
C
k
S
WV
V
Vu
==
===
==ρ
39
Drag Characteristics
Fixed Wing Fighter Aircraft Flight PerformanceSOLO
40
Fixed Wing Fighter Aircraft Flight PerformanceSOLO
Level Flight
Aircraft Range in Level Flight
Lift
DragThrust
Weight
Range in Level Flight of Jet Aircraft
Equations of motion:
0
0
=−=−
DT
WL
0=
=
h
Vx
We add the equation of fuel consumption
TcW −= c – specific fuel consumption
We assume that fuel consumption is constant for a given altitude.
Vtd
Wd
Wd
xd
td
xd ==
Dc
V
Tc
V
W
V
Wd
xd DT
−=−===
41
Fixed Wing Fighter Aircraft Flight PerformanceSOLO
Level Flight
Aircraft Range in Level Flight
Lift
DragThrust
Weight
Range in Level Flight of Jet Aircraft
Dc
V
Wd
xd −=
The quantity dx/dW is called the “Instantaneous Range” and is equal to the Horizontal Range traveled per unit load of fuel or the “Specific Range”. Multiply and divide by L = W
Wc
V
C
C
Wc
V
D
L
Wd
xd
D
L
−=
−=
Integrating we obtain
∫
−=−= f
i
W
WD
Lif W
WdV
cC
CxxR
1:
42
Fixed Wing Fighter Aircraft Flight PerformanceSOLO
Level Flight
Aircraft Range in Level Flight
Lift
DragThrust
Weight
Range in Level Flight of Jet Aircraft
To perform the integration we must specify the variation of CL, CD and V. Let consider two cases:
∫
−=−= f
i
W
WD
Lif W
WdV
cC
CxxR
1:
a. Range at Constant Altitude of Jet Aircraft
We have LCVSLW 2
2
1 ρ==LCS
WV
ρ2=
The velocity changes (decreases) since the weight W decreases due to fuel consumption.
[ ]ifD
LW
WD
L WWC
C
cW
Wd
ScC
CR
f
i
−
=
−= ∫
221
ρ
43
Fixed Wing Fighter Aircraft Flight PerformanceSOLO
Level Flight
Aircraft Range in Level Flight
Lift
DragThrust
Weight
a. Range at Constant Altitude of Jet Aircraft
[ ]ifD
LW
WD
L WWC
C
cW
Wd
ScC
CR
f
i
−
=
−= ∫
221
ρ
The maximum range is obtained when
[ ]ifD
L WWC
C
cR −
=
max
max
2
max
20max
+
=
LD
L
D
L
CkC
C
C
C
( ) 030
221
2022
0
20
20
=−⇒=+
−+
=
+ LD
LD
LL
L
LD
LD
L
L
CkCCkC
CkCC
CkC
CkC
C
Cd
d
The maximum range is obtained when*0
3
1
3
1L
DL C
k
CC ==
The Velocity at maximum range is ( ) ( ) ( ) ( )tVCS
tW
CS
tWtV
LL
*4*
4*
32
33/
2 ===ρρ
44
Fixed Wing Fighter Aircraft Flight PerformanceSOLO
Level Flight
Aircraft Range in Level Flight
Lift
DragThrust
Weight
b. Range at Constant Velocity of Jet Aircraft
=
−= ∫
f
i
D
LW
WD
L
W
W
C
C
c
V
W
WdV
cC
CR
f
i
ln1
The Velocity V is constant and equal to V* corresponding to initial weight Wi.
4
0*
* 22
D
i
L
i
C
k
S
W
CS
WV
ρρ==
The maximum range is obtained when
=
=
f
i
f
i
D
L
W
We
c
V
W
W
C
CV
cR lnln
1 **
max
max
To keep Velocity V constant when weight W decreases, the air density ρ must also decrease, hence the Aircraft will gain (qvasistatic) altitude
( )Pc
td
Wd
td
hde
td
hdp
hh −==−= − 0/0ρρ
45
Fixed Wing Fighter Aircraft Flight PerformanceSOLO
Level Flight
Aircraft Range in Level Flight
Range in Level Flight of Propeller Aircraft
Lift
DragThrust
Weight
The equation of fuel consumption
PcW P−=
cp – specific fuel consumption (consumed per unit power developed by the engine per unit time
We assume that fuel consumption is constant for a given altitude.
Vtd
Wd
Wd
xd
td
xd ==
Pc
V
W
V
Wd
xd
p
−==
- Required PowerVDPR ⋅=
PP pA ⋅=η - Available Power
ηp – propulsive efficiency
AR PP =p
VDP
η⋅=
46
Fixed Wing Fighter Aircraft Flight PerformanceSOLO
Level Flight
Aircraft Range in Level Flight
Range in Level Flight of Propeller Aircraft
Lift
DragThrust
Weight
WcC
C
WcD
L
DcPc
V
Wd
xd
p
p
D
L
p
pWL
p
p
p
ηηη
−=−=−=−=
=
Integration gives ∫
−=−= f
i
W
Wp
p
D
Lff W
Wd
cC
CxxR
η:
We assume• Angle of Attack is kept constant throughout cruise, therefore e = CL/CD is constant•ηp is independent on flight velocity
f
i
p
p
W
We
cR ln
η= Bréguet Range Equation
The maximum range of Propeller Aircraft in Level Flight is
f
i
Dp
p
f
i
p
p
W
W
CkcW
We
cR ln
2
1ln
0
*max
ηη==
47
Louis Charles Bréguet(1880 – 1955)
The Bréguet Range Equation
The Bréguet range equation determines the maximum flight distance. The key assumptions are that SFC, L/D, and flight speed, V are constant, and therefore take-off, climb, and descend portions of flights are not well modeled (McCormick, 1979; Houghton, 1982). ( )
⋅
=final
initial
W
W
SFCg
DLVRange ln
/
Winitial = Wfuel + Wpayload + Wstructure + Wreserve
Wfinal = Wpayload + Wstructure + Wreserve
where
( )
++
+⋅
=reservestructurepayload
fuel
WWW
W
SFCg
DLVRange 1ln
/
where SFC, L/D, and Wstructure are technology parameters while Wfuel, Wpayload, and Wreserve are operability parameters.
Fixed Wing Fighter Aircraft Flight PerformanceSOLO
48
Fixed Wing Fighter Aircraft Flight PerformanceSOLO
Level Flight
Aircraft Range in Level Flight
Range in Level Flight of Propeller Aircraft
Lift
DragThrust
Weight
Let assume that the flight to maximum range is performed in one of two ways
1. Propeller Aircraft Flight at Constant Altitude
In Constant Altitude Flight the velocity changes with the decrease of weight such that
( ) ( )4
0
* 2
DC
k
S
tWVtV
ρ==
2. Propeller Aircraft Flight with Constant Velocity
In Constant Velocity Flight the velocity is the V* velocity based on the initial weight of the Aircraft
.2
4
0
* constC
k
S
WVV
D
i ===ρ
49
Fixed Wing Fighter Aircraft Flight PerformanceSOLO
Level Flight
Aircraft Endurance in Level Flight
Lift
DragThrust
Weight
The Endurance of an Airplane remains in the air and is usually expressed in hours.
Endurance of Jet Aircraft in Level Flight
We have TcW −=
c – specific fuel consumption
W
Wd
c
e
W
Wd
D
L
cDc
Wd
Tc
Wdtd
WLDT
−=−=−=−=== 1
Integrating we obtain ∫−= f
i
W
W W
Wd
c
et
Assuming that the Angle of Attack is held constant throughout the flight, e =CL/CD is constant
f
i
W
W
c
et ln=
f
i
Df
i
W
W
CkcW
W
c
et ln
2
1ln
0
*
max ==
The Maximum Endurance for Jet Aircraft occurs for e = e*, CL = CL*, V = V*, D = Dmin.
50
Fixed Wing Fighter Aircraft Flight PerformanceSOLO
Level Flight
Aircraft Endurance in Level Flight
The Endurance of an Airplane remains in the air and is usually expressed in hours.
Endurance of Propeller Aircraft in Level Flight
We have ppp VDcPcW η/⋅−=−=
W
Wd
Ve
cW
Wd
VD
L
cVD
Wd
ctd
p
p
p
pWL
p
p 11 ηηη−=−=
⋅−=
=
Assuming that the Angle of Attack is held constant throughout the flight, e =CL/CD is constant
Lift
DragThrust
Weight cp – specific fuel consumption (consumed per unit power developed by the engine per unit time.ηp – propulsive efficiency
Integrating we obtain
∫−= f
i
W
Wp
p
W
Wd
Ve
ct
1η
The Endurance of Propeller Aircraft depends on Velocity, therefore we will assume two cases1.Flight at Constant Altitude2.Flight with Constant Velocity
51
Fixed Wing Fighter Aircraft Flight PerformanceSOLO
Level Flight
Endurance of Propeller Aircraft in Level Flight
Lift
DragThrust
WeightThe velocity will change to compensate for the decrease in weight
∫ =−= f
i
W
WD
L
p
p
C
Ce
W
Wd
Ve
ct
1η
1. Propeller Aircraft Flight at Constant Altitude
We haveLCVSLW 2
2
1 ρ==LCS
WV
ρ2=
−
=
ifD
L
p
p
WW
S
C
C
ct
11
2
2 2/3 ρη
For Maximum Endurance Propeller Aircraft has to fly at that Angle of Attack such that (CL
3/2/CD) is maximum, which occurs when CL=√3 CL* and V = 0.76 V*.
−
=
ifDp
p
WW
S
Ckct
11
2
27
4
12
03max
ρη
52
Fixed Wing Fighter Aircraft Flight PerformanceSOLO
Level Flight
Endurance of Propeller Aircraft in Level Flight
Lift
DragThrust
Weight
∫ =−= f
i
W
WD
L
p
p
C
Ce
W
Wd
Ve
ct
1η
2. Propeller Aircraft Flight with Constant Velocity
f
i
p
p
W
W
Ve
ct ln
1η=
For Maximum Endurance Propeller Aircraft has to fly at a velocity such that e=(CL/CD) is maximum, which occurs when CL=CL
* and V = V*, which is based on initial weight Wi
4
0*
* 22
D
i
L
i
C
k
S
W
CS
WV
ρρ==
0
*
2
1
DCke =
f
i
D
i
p
p
f
i
D
i
Dp
p
f
i
p
p
W
W
CkS
W
cW
W
C
k
S
W
CkcW
W
Ve
ct ln
1
2ln
2
2
1ln
14 3
0
4
00*
*max ρ
ηρ
ηη===
53
D=TR
V
V*tmax
Slope min(PR/V)
Bréguet
Velocities for Maximum Range and Maximum Endurance of Propeller Aircraft
Fixed Wing Fighter Aircraft Flight PerformanceSOLO
Graphical Finding of Maximum Range and Endurance of Jet Aircraft in Level Flight
=
⇔
V
D
D
VR
VVminmaxmax
Maximum Range
From Figure we can see that min (D/V) is obtained by taking the tangent to D graph that passes through origin.The point of tangency will give D and V for (D)min.
Maximum Endurance
∫∫<
=
<
−=−=00
111Wd
DcWd
Tct
DT
( )DD
tVVmin
1maxmax =
⇔
From Figure we can see that min (PR) is obtained by taking the PR and V for (PR)min.
Lift
DragThrust
Weight
∫∫<
−==0
WdDc
VxdR
54
PR
V
V*Rmax
0.866 V*tmax
Slope min(PR/V)
Velocities for Maximum Range and Maximum Endurance of Propeller Aircraft
Lift
DragThrust
Weight
Fixed Wing Fighter Aircraft Flight PerformanceSOLO
Graphical Finding of Maximum Range and Endurance of Propeller Aircraft in Level Flight
∫∫∫∫>>>
⋅−=−=−==
000
WdVD
V
cWd
P
V
cWd
Pc
VxdR
p
p
Rp
p
p
ηη
DV
P
P
VR
V
R
VR
Vminminmaxmax =
=
⇔
Maximum Range
From Figure we can see that min (PR/V) is obtained by taking the tangent to PR graph that passes through origin.The point of tangency will give PR and V for (PR/V)min.
Maximum Endurance
∫∫<<
−=−=00
11Wd
PcWd
Pct
Rp
p
p
η
( ) ( )VDPP
tV
RV
RV
⋅==
⇔ minmin
1maxmax
From Figure we can see that min (PR) is obtained by taking the PR and V for (PR)min.
55
H.H. Hurt, Jr., “Aerodynamics for Naval Aviators “,NAVAIR 00-80T-80 1-1-1965, pg. 35
Fixed Wing Fighter Aircraft Flight Performance
56
H.H. Hurt, Jr., “Aerodynamics for Naval Aviators “,NAVAIR 00-80T-80 1-1-1965, pg. 35
Fixed Wing Fighter Aircraft Flight Performance
57
H.H. Hurt, Jr., “Aerodynamics for Naval Aviators “,NAVAIR 00-80T-80 1-1-1965, pg. 35
Fixed Wing Fighter Aircraft Flight Performance
58
Flight Ceiling by the available Climb Rate- Absolute 0 ft/min- Service 100 ft/min- Performance 200 ft/min
True Airspeed
Altitude
Absolute CeilingService CeilingPerformance Ceiling
Excess Thrust provides the ability to accelerate or climb
True Airspeed
Thrust AvailableThrust
RequiredThrust
True Airspeed
Thrust
AvailableThrust
RequiredThrust
A AB B
C D
E
E
Thrust
True Airspeed
AvailableThrust
RequiredThrust
C D
Jet Aircraft Flight Envelope Determined by Available ThrustFlight Envelope: Encompasses all Altitudes
and Airspeeds at which Aircraft can Fly
Stengel, MAE331, Lecture 7, Gliding, Climbing and Turning Performance
Fixed Wing Fighter Aircraft Flight PerformanceSOLO
Lift
DragThrust
Weight
Changes in Jet AircraftThrust with Altitude
59
Propeller Aircraft Ceiling Determined by Available Power
To find graphically the maximum Flight Altitude (Ceiling) for a Propeller Aircraft we use the PR (Power Required) versus V (Velocity) graph. The maximum Flight Altitude corresponds to maximum Range Rmax.
We have shown that to find Rmax we draw the Tangent Line to PR Graph, passing trough the origin.
Fixed Wing Fighter Aircraft Flight PerformanceSOLO
Lift
DragThrust
Weight
Changes in Propeller Aircraft Powerand Thrust with Altitude
VC
Pa, propeller
PR
hcruiseA
h2
h1
h0
h0 < h1 <h2 < hcruise
The intersection point A with PR Graph defines the Ceiling Velocity VC, and the Pa (Available Power – function of Altitude) with this point defines the Ceiling Altitude.
Return to Table of Content
60
Fixed Wing Fighter Aircraft Flight PerformanceSOLO
Gliding Flight
A Glider is an unpowered airplane.
0
sin
cos
==
=
W
Vh
Vx
γγ
1<<γ0=+
=γWD
WL
.constW
Vh
Vx
==
=
γ
Lift and Drag Forces:
( ) γρρ
ρ
WCkCSVCSVD
WCSVL
LDD
L
−=+==
==
20
22
2
2
1
2
12
1
LCS
WV
ρ2=
eC
C
L
D
W
D
L
DLW 1−=−=−=−=
=γ
Equations of motion:
0sin
0cos
=+=−
γγ
WD
WLQuasi-SteadyFlight
61
Fixed Wing Fighter Aircraft Flight PerformanceSOLO
Gliding Flight
We found
LCS
WV
ρ2= eC
C
L
D
W
D
L
DLW 1−=−=−=−=
=γ
Flattest Glide” (γ = γmin)
The Flattest Glide (γ = γmin) is given by:
0*
max
minmin 22
1DL CkCk
eW
D −=−=−=−=γ
e
LC*LC
*2
1
LCk
CL/CD as a function of CL
k
CC DL
0* =
The flight velocity for the Flattest Glide is given by:
4
0*..
2*
2
DL
GF C
k
S
WV
CS
WV
ρρ===
The flight velocity for the Flattest Glide is equal to the reference velocity V* or u = 1.The Flattest Glide is conducted at constant dynamic pressure.
.2
1
0*
2.. const
C
kW
C
WVSq
DL
GFG ==== ρ
62
H.H. Hurt, Jr., “Aerodynamics for Naval Aviators “,NAVAIR 00=80T-80 1-1-1965, pg. 35
Gliding Flight
CLEAR CONFIGURATION
LANDING CONFIGURATION
LIFT to DRAGRATIO
L/D(L/D)max
LIFT COEFFICIENT, CL
CLEAR CONFIGURATION
LANDING CONFIGURATION
RATE OFSINK
VELOCITY
(L/D)max
TANGENT TO RATE OF SINK GRAPH AT THE ORIGIN
Gliding Performance
Fixed Wing Fighter Aircraft Flight PerformanceSOLO
63
Fixed Wing Fighter Aircraft Flight PerformanceSOLO
Gliding Flight
We have:
Distance Covered with respect to Ground
The maximum Ground Range is covered for the Flattest Glide at the reference velocity V* or u = 1.
γVtd
hd
Vtd
xd
=
=
D
Le
V
V
hd
xd −=−===γγ1
Assuming a constant Angle of Attack during Glide, e is constant and the Ground Range R, to descend from altitude hi to altitude hf is given by:
( ) hehhehdexxR fi
h
hif
f
i
∆=−=−=−= ∫:
and
0
maxmax2 DCk
hheR
∆=∆=
e
LC*LC
*2
1
LCk
CL/CD as a function of CL
64
Fixed Wing Fighter Aircraft Flight PerformanceSOLO
Gliding Flight
Rate of sink is defined as:
Rate of Sink
==⋅=−=−=
=
=2/32
22
L
D
L
D
L
L
D
W
D
CS
WV
sC
C
S
W
C
C
CS
W
W
VDV
td
hdh
L
ρργ
ρ
The term DV = PR represents the Power Required to sustain the Gliding Flight.Therefore the Rate of Sink is minimum when the Power Required is minimum, or(CD/CL
3/2) is minimum
( ) ( )0
2
3
2
3423
2
2/50
2
2/5
20
2
3
20
2/12/3
2/3
20
2/3 =−=+−=+−
=
+=
L
DL
L
LDL
L
LDLLL
L
LD
LL
D
L C
CCk
C
CkCCk
C
CkCCCCk
C
CkC
Cd
d
C
C
Cd
d
Denote by CL,m the value of Lift Coefficient CL for which (CD/CL3/2) is minimum
*0, 3
30*
L
k
CC
DmL C
k
CC
DL =
== 274
3
3
03
2/3
0
00
min
2/3D
D
DD
L
D Ck
kC
kC
kC
C
C =
+=
65
Fixed Wing Fighter Aircraft Flight PerformanceSOLO
Gliding Flight
Rate of Sink
*0, 3
30*
L
k
CC
DmL C
k
CC
DL =
==0
3
max
2/3 27
4
1
DD
L
CkC
C =
We found:
The velocity Vm for glide with minimum sink rate is given by:
*4
0
76.0~
4
4
0,
76.02
3
1
3
22
*
VC
k
S
W
C
k
S
W
CS
WV
V
D
DmLm
≈
=
==
ρ
ρρ
S
CkW
C
C
S
Wh D
L
Ds ρρ 27
22 03
min
2/3min, =
=
The minimum sink rate is given by:
66
Fixed Wing Fighter Aircraft Flight PerformanceSOLO
Gliding Flight
Endurance
The Endurance is the total time the glider remains in the air.
MinimumSink Rate
tmax
FlatestGlideRmax
−== 2/3
2
L
D
C
C
S
WV
td
hd
ργ
−==
D
L
C
C
W
S
V
hdtd
2/3
2
ργ
( )fiD
Lh
hD
L hhC
C
W
Shd
C
C
W
St
f
i
−
=
−= ∫
2/32/3
22
ρρ
Assuming that the Angle of Attack is held constant during the glide and ignoring the variation in density as function of altitude, we have
For Maximum Endurance the Glider has to fly at that Angle of Attack such that (CL
3/2/CD) is maximum, which occurs when CL=√3 CL
* and V = 0.76 V*.
−=
4
27
24
03max
fi
D
hh
CkW
St
ρReturn to Table of Content
67
Performance of an Aircraft with Parabolic PolarSOLO
W
LTn
+= αsin:'
W
Ln =:
20
:LD
L
D
L
CkC
C
CSq
CSq
D
Le
+===
We assume a Parabolic Drag Polar:2
0 LDD CkCC +=Let find the maximum of e as a function of CL
( ) ( ) 02
220
20
220
220 =
+
−=+
−+=∂∂
LD
LD
LD
LLD
L CkC
CkC
CkC
CkCkC
C
e e
LC*LC
*2
1
LCk
CL/CD as a function of CL
The maximum of e is obtained for
k
CC D
L0* =
( ) 00
02
0 2** DD
DLDD Ck
CkCCkCC =+=+=
Start with
Load Factor
Total Load Number
Lift to Drag Ratio
Climbing Aircraft Performance
68
Performance of an Aircraft with Parabolic PolarSOLO
e
LC*LC
*2
1
LCk
CL/CD as a function of CL
The maximum of e is obtained for
k
CC D
L0* =
( ) 00
02
0 2** DD
DLDD Ck
CkCCkCC =+=+=
*2
1
*2
1
2
1
2*
**
2200
0
LLDD
D
D
L
CkCkCkCk
C
C
Ce =====
We have WnCSVCSqL LL === 2
2
1 ρ
Let define for n = 1
=
=
==
2
4
0
*2
1:*
*:
2
*21
:*
Vq
V
Vu
C
k
S
W
CS
WV
DL
ρ
ρρ
20
:LD
L
D
L
CkC
C
CSq
CSq
D
Le
+===
Climbing Aircraft Performance
69
Performance of an Aircraft with Parabolic PolarSOLO
Using those definitions we obtain
L
L
L
L
C
Cnqq
WCSq
WnCSqL **
**=→
===
22
2
1
21
*21
*
uV
Vn
q
q ==ρ
ρ
2
**
*
u
CnC
q
qnC L
LL ==
( )
+=
+=
+=+=
=
2
22
0402
02
*
4
22
022
0
**
**
02
u
nuCSq
u
CnCuSq
u
CnkCuSqCkCSqD
DD
D
CCk
LDLD
DL
=
=
=
2
0
*2
1:*
*:
2:*
Vq
V
Vu
CS
kWV
D
ρ
ρ
*2
1
**** 0
0 eW
C
CCSqCSq
L
DLD ==
+=
2
22
*2 u
nu
e
WD
Therefore
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Climbing Aircraft Performance
70
Performance of an Aircraft with Parabolic PolarSOLO
We obtained
+=
2
22
*2 u
nu
e
WD
u 0
- - - - 0 + + + + +
D ↓ min ↑
n
u
D
∂∂
Let find the minimum of D as function of u.
nu
u
nu
e
W
u
nu
e
W
u
D
=→
=−=
−=
∂∂
2
3
24
3
2
0*
22*2
*2min e
WnDD
nu==
=
Aircraft Drag
Climbing Aircraft Performance
=
=
=
2
0
*2
1:*
*:
2:*
Vq
V
Vu
CS
kWV
D
ρ
ρ
71
Performance of an Aircraft with Parabolic PolarSOLO
Aircraft Drag
( )MAXn
W
VhLn ≤= ,
+=
= 2
22
*2 u
nu
e
WD MAX
nn MAX
Maximum Lift Coefficient or Maximum Angle of Attack( ) ( ) ( )VhorMCMC STALLMAXLL ,, _ ααα ≤≤
We haveu
C
C
u
n
u
CnC
q
qnC
L
MAXL
CC
LLL
MAXLL*
**
* _
2
_
=→===
2
2
_
2
2
_2
*1
*2
**2_
uC
C
e
W
uC
Cu
e
WD
L
MAXL
L
MAXL
CC MAXLL
+=
+=
=
Maximum dynamic pressure limit
( ) ( ) MAXMAX
MAXMAX uV
VuhVVorqVhq =<→≤≤= :
*2
1 2ρ
*eW
D
MAXLC _
2
2
_12
1u
C
C
L
MAXL
+
+=
2
22
2
1*
u
nue
W
D MAX
LIMIT
nn MAX=
2min * ueW
D =
+=
2
22
2
1*
u
nue
W
D
MAXuu =MAX
MAXL
LCORNER n
C
Cu
_
*=
n
LIMIT
uMAXnu =
as a function of u*eW
D
Maximum Load Factor
Climbing Aircraft Performance
=
=
=
2
0
*2
1:*
*:
2:*
Vq
V
Vu
CS
kWV
D
ρ
ρ
72
Performance of an Aircraft with Parabolic PolarSOLO
Energy per unit mass ELet define Energy per unit mass E:
g
VhE
2:
2
+=Let differentiate this equation:
( ) ( )W
VDT
W
VDT
W
DTg
g
VV
g
VVhEps
−≈−=
−−+=+== αγαγ cos
sincos
sin:
*&*2 2
22 VuV
u
nu
e
WD =
+=
Define *: eW
Tz
=
We obtain ( )
+−=
+−
=−=2
22
2
22
2
1
*
**
2
1*
*
u
nuzu
e
V
W
Vuu
nue
W
T
e
W
W
VDTps
or ( )u
nuzu
e
Vps
224 2
*2
* −+−=
020 224 =+−→==
nuzupconstns
( ) ( )2
224
2
2243 23
*
*244
*
*
u
nuzu
e
V
u
nuzuuuzu
e
V
u
p
constn
s ++−=−+−−+−=∂
∂
=
0=∂∂
=constn
s
u
p 221
2uu
uuMAX <<+
nz >
Climbing Aircraft Performance
nznzzu
nzzu>
−+=
−−=22
2
221
3
3 22 nzzuMAX
++=
=
=
=
2
0
*2
1:*
*:
2:*
Vq
V
Vu
CS
kWV
D
ρ
ρ
73
Performance of an Aircraft with Parabolic PolarSOLO
Energy per unit mass Esp
2u1u MAXu2
21 uu + u
MAXn
n
1=n
( )u
nuzu
e
Vps
224 2
*
* −+−=ps as a function of u
uV
peuzunnuzuu
V
pe ss
*
*222
*
*2 242224 −+−=→−+−=
From which uV
peuzun s
*
*22 24 −+−=
( )*
*244 3
2
V
peuzu
u
n s
constps
−+−=∂
∂
=
( )3
0412 22
22 zuzu
u
n
constps
=→=+−=∂
∂
=
( )u
nuzu
e
Vps
224 2
*2
* −+−=
Climbing Aircraft Performance
=
=
=
2
0
*2
1:*
*:
2:*
Vq
V
Vu
CS
kWV
D
ρ
ρ
74
Performance of an Aircraft with Parabolic PolarSOLO
Load Factor n
u
3
z z
z2z3
z
u
2n
0=sp
0>sp
0<sp
0<sp
0=sp
0>sp
( )u
n
∂∂ 2
( )2
22
u
n
∂∂
3
zu
( ) ( ) 22
2
22
,, nu
n
u
n
∂∂
∂∂ as a function of u
( )3
0412 22
22 zuzu
u
n
constps
=→=+−=∂
∂
=
( )*
*244 3
2
V
peuzu
u
n s
constps
−+−=∂
∂
=
Integrating once
uV
peuzun s
*
*22 24 −+−=
Integrating twice
Climbing Aircraft Performance
=
=
=
2
0
*2
1:*
*:
2:*
Vq
V
Vu
CS
kWV
D
ρ
ρ
75
Performance of an Aircraft with Parabolic PolarSOLO
Load Factor n
For ps = 0 we have
zuuzun 202 24 ≤≤+−=
Let find the maximum of n as function of u.
022
4424
3
=+−+−=
∂∂
uzu
uzu
u
n
Therefore the maximum value for n is achieved for zu =
( ) znMAXps
==0
u 0 √z √2z
∂ n/∂u | + + + 0 - - - - | - -
n ↑ Max ↓
z2z
u
n
0=sp
0>sp
0<sp
MAXn
z
MAXMAXL
L nC
C
_
*
n as a function of u
Climbing Aircraft Performance
=
=
=
2
0
*2
1:*
*:
2:*
Vq
V
Vu
CS
kWV
D
ρ
ρ
Performance of an Aircraft with Parabolic PolarSOLO
Energy per unit mass E
g
VhE
2:
2
+=
Climbing Aircraft Performance
Energy Height versus Mach NumberEnergy Height versus True Airspeed
( )hV
VM
sound
=:( )00
:T
TV
T
TMhVTAS sound ==
Return to Table of Content
77
Performance of an Aircraft with Parabolic PolarSOLO
Steady Climb (V, γ = constant)
Climbing Aircraft Performance
0sin
0cos
==−−
==−
td
Vd
g
WWDT
td
dV
g
WWL
γ
γγ
Equation of Motion for Steady Climb:
γγ
sin
cos
Vh
Vx
=
=
Define the Rate of Climb:( )
sRa
C pW
PP
W
DTVVh =−=−⋅== γsin
wherePa = V T - available powerPR = V D - required powerps - excess power per unit weight
Weight
ThrustExcess
W
DT =−=γsin
C
C
WL
const
γγγ
cos
.
===
Lift
Drag
Thrust
Weight
78
Performance of an Aircraft with Parabolic PolarSOLOClimbing Aircraft Performance
LC CSVW 2
2
1cos ργ =
( )s
CD
LD
C pSV
WkCSVVT
WW
CkCSVVTh =
−−=+−
=ρ
γρρ
21
cos
2
1121
22
03
20
3
Let find the velocity V for which the Rate of Climb is maximum, for the Propeller Aircraft:
0cos2
2
312
22
02 =
+−==
SV
WkCSV
Wtd
pd
td
hdC
DsC,Prop
ργρ
Steady Climb (V, γ = constant)
For a Propeller Aircraft we assume that Pa=T V= constant.
or **
44
04
76.03
12
3
1VV
C
k
S
WV
DClimb.Prop ===
ρ
sC
DaPropC pSV
WkCSVP
Wh =
−−=
ργρ
22
03
,
cos2
2
11
We can see that the velocity at which the Rate of Climb of Propeller Aircraft is maximum is the same as the velocity at which the Required Power in Level Flight is maximum.
Lift
Drag
Thrust
Weight
79
Performance of an Aircraft with Parabolic PolarSOLOClimbing Aircraft Performance
LC CSVW 2
2
1cos ργ =
( )
−−=+−
=SV
WkCSVVT
WW
CkCSVVTh C
D
LD
C
ρ
γρρ
21
cos
2
1121
22
03
20
3
Let find the velocity V for which the Rate of Climb is maximum, for the Jet Aircraft:
0cos2
2
312
22
02 =
+−=
SV
WkCSVT
Wtd
hd CD
C
ργρ
Steady Climb (V, γ = constant)
For a Jet Aircraft we assume that T = constant.
Define
0
*
00
*
**
4
0
2:2*,*,:
2:*,
*:
D
DDD
L
D
L
D CkW
T
W
eTzCC
k
CC
C
Ce
C
k
S
WV
V
Vu =======
ρ
0cos
2
23
2 2
/1
20
0
2
2
0
2
2
=+− C
u
D
u
Dz
D V
Ck
SW
Ck
SW
VT
CkWγρ
ρ
0cos23 224 =−− Cuzu γ
Czzu γ22 cos3++=
80
Performance of an Aircraft with Parabolic PolarSOLOClimbing Aircraft Performance
Steady Climb (V, γ = constant)
ps versus the nondimensional velocity u
ps versus the velocity V
0sin ==−−td
Vd
g
WWDT γ
1
2
22
*2 =
+=
nu
nu
e
WD
Define
0
*
00
*
**
4
0
2:2*,*
,:2
:*,*
:
D
DDD
L
D
L
D
CkW
T
W
eTzCC
k
CC
C
Ce
C
k
S
WV
V
Vu
====
===ρ
+−==−=
22
*
12
2
1sin
uuz
eV
p
W
DT sγ
To find the maximum γ we must have
02
22
1sin3*
=
−−=
uu
eud
d γ
4
0
2*
max
DC
k
S
WVV
ργ ==
( ) ( )1*
*2
*2
*
11
224
, max−=−+−=
==
ze
V
u
nuzu
e
Vp
un
s γ *
,max
1sin
max
max
e
z
V
ps −==γ
γγ
1max
=γu
SOLO
81
Aircraft Flight Performance
Construction of the Specific Excess Power contours ps in the Altitude-Mach Number map for a Subsonic Aircraft below the Drag-divergence Mach Number. These contour are constructed for a fixed load factor W/S and Thrust factor T/S, if the load or thrust factor change, the ps contours will shift.
( ) ( )W
VDT
W
VDT
W
DTg
g
VV
g
VVhEps
−≈−=
−−+=+== αγαγ cos
sincos
sin:
In Figure (a) is a graph of Specific Excess Power contours ps versus Mach Number. Each curve is for a specific altitude h. In Figure (b) each curve is for a given Specific Excess Power ps in Altitude versus Mach Number coordinates. The points a, b, c, d, e, f for ps = 0 in Figure (a) are plotted on the curve for ps = 0 in Figure (b). Similarly all points ps = 200 ft/sec in Figure (a) on the line AB are projected on the curve ps = 200 ft/sec in Figure (b).
Specific Excess Power contours ps for a Subsonic Aircraft
Specific Excess Power contours ps
SOLO
82
Aircraft Flight Performance
Specific Excess Power contours ps for a Supersonic Aircraft
In the graphs of Specific Excess Power ps versus Mach Number Figure (a) for a Supersonic Aircraft we see a “dent” in h contour in the Transonic Region. This is due to the increase in Drag in this region.
2
In Figure (b) the graphs of Altitude versus Mach Number we see a “closed” ps = 400 ft/sec contour due to the increase in Drag in this Transonic Region.
Specific Excess Power contours ps
( ) ( )W
VDT
W
VDT
W
DTg
g
VV
g
VVhEps
−≈−=
−−+=+== αγαγ cos
sincos
sin:
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83
Performance of an Aircraft with Parabolic PolarSOLOClimbing Aircraft Performance
Optimum Climbing Trajectories using Energy State Approximation (ESA)
We defined the Energy per unit mass E (Specific Energy):g
VhE
2:
2
+=Differentiate this equation:
( ) ( )W
VDT
W
VDT
W
DTg
g
VV
g
VVh
td
Edps
−≈−=
−−+=+== αγαγ cos
sincos
sin:
Minimum Time-to-Climb
The time to reach a given Energy Height Ef is computed as follows
E
Edtd
= ∫= fE
Ef E
Edt
0
The minimum time to reach the given Energy Height Ef is obtained by using at each level.
( )∫= fE
Ef E
Edt
0max
max,
( )maxE
84
Performance of an Aircraft with Parabolic PolarSOLOClimbing Aircraft Performance
Optimum Climbing Trajectories using Energy State Approximation (ESA)
Minimum Time Climb Profiles for Subsonic Speed
( ) ( )W
VDT
W
VDT
W
DTg
g
VV
g
VVhEps
−≈−=
−−+=+== αγαγ cos
sincos
sin:
Stengel, MAE331, Lecture 7, Gliding, Climbing and Turning Performance
The minimum time to reach the given Energy Height Ef is obtained by using at each level.
( )maxE
Energy can be converted from potential to kinetic or vice versa along lines of constant energy in zero time with zero fuel expended. This is physically not possible so the method gives only an approximation of real paths.
SOLO
85
Aircraft Flight Performance
Stengel, MAE331, Lecture 7, Gliding, Climbing and Turning Performance
Minimum Time Climb Profiles for Supersonic Speed
( ) ( )W
VDT
W
VDT
W
DTg
g
VV
g
VVhEps
−≈−=
−−+=+== αγαγ cos
sincos
sin:
The minimum time to reach the given Energy Height Ef is obtained by using at each level.
( )maxE
The optimum flight profile for the fastest time to altitude or time to speed involves climbing to maximal altitude at subsonic speed, then diving in order to get through the transonic speed range as quickly as possible, and than climbing at supersonic speeds again using .( )maxE
86
SOLO
Climbing Aircraft Performance
Optimum Climbing Trajectories using Energy State Approximation (ESA)
Shaw, “Fighter Combats – Tactics and Maneuvering”
Minimum Time Climb Profiles
Aircraft Flight Performance
The minimum time to reach the given Energy Height Ef is obtained by using at each level .
( )maxE
87
SOLOClimbing Aircraft Performance
Optimum Climbing Trajectories using Energy State Approximation (ESA)
A.E. Bryson, Course “Performance Analysis of Flight Vehicles”, AA200, Stanford University, Winter 1977-1978
Aircraft Flight Performance
A.E. Bryson, Jr., “Energy-State Approximation in Performance Optimization of Supersonic Aircraft”, Journal of Aircraft, Vol.6, No. 5, Nov-Dec 1969, pp. 481-488
Approximate (ESA) Solutions.
Implicit to ESA Approximation is the possibility of instantaneous jump between kinetic to potential energy (from A to B ).This non physical situation is called a “zoom climb” or “zoom dive”.
A
B
The minimum time to reach the given Energy Height Ef is obtained by using at each level.
( )maxE
SOLOClimbing Aircraft Performance
Optimum Climbing Trajectories using Energy State Approximation (ESA)
“Exact” calculated using Optimization MethodsComputations
Aircraft Flight Performance
Comparison between “Exact” and Approximate(ESA) Solutions.
Implicit to ESA Approximation is the possibility of instantaneous jump between kinetic to potential energy (fromA to B , and from C to D).This non physical situation is called a “zoom climb” or “zoom dive”. We can see the “exact” solution in those cases.
A
B
C
D
The minimum time to reach the given Energy Height Ef is obtained by using at each level.
( )maxE
88
A.E. Bryson, Course “Performance Analysis of Flight Vehicles”, AA200, Stanford University, Winter 1977-1978
A.E. Bryson, Jr., “Energy-State Approximation in Performance Optimization of Supersonic Aircraft”, Journal of Aircraft, Vol.6, No. 5, Nov-Dec 1969, pp. 481-488
89http://msflights.net/forum/showthread.php?1184-Supersonic-Level-Flight-Envelopes-in-FSX
F-15 Streak Eagle Time to Climb Records, which follow the ideal path to reach set altitudes in a minimal amount of time. The Streak Eagle could break the sound barrier in a vertical climb, so the ideal flightpath to 30000m involved a large Immelmann.
https://www.youtube.com/watch?v=HLka4GoUbLo https://www.youtube.com/watch?v=S7YAN9--3MAF-15 Streak Eagle Record Flights part 2F-15 Streak Eagle Record Flights part 1
SOLOClimbing Aircraft Performance
Optimum Climbing Trajectories using Energy State Approximation (ESA)
Aircraft Flight Performance
90
How to climb as fast as possible
Takeoff and pull up: You want to build energy (kinetic or potential) as quickly as you can. Peak acceleration is at mach 0.9, which is the speed that energy is gained the fastest. You should first accelerate to near that speed. Avoid bleeding off energy in a high-g pull up. Start a smooth pull up before at mach 0.7-0.8 and accelerate to mach 0.9 during the pull.
http://msflights.net/forum/showthread.php?1184-Supersonic-Level-Flight-Envelopes-in-FSX
SOLOClimbing Aircraft Performance
Optimum Climbing Trajectories using Energy State Approximation (ESA)
F-15 Streak Eagle Time to Climb Records, which follow the ideal path to reach set altitudes in a minimal amount of time. The Streak Eagle could break the sound barrier in a vertical climb, so the ideal flightpath to 30000m involved a large Immelmann.
Aircraft Flight Performance
Climb again: to 36000ft for maximum speed, or higher as to not exceed design limits or to save fuel for a longer run
Climb: Adjust your climb angle to maintain mach 0.9. In a modern fighter, the climb angle may be 45-60 degrees. If you need a heading change, during the pull and climb is a good time to make it.
Level off: between 25000 and 36000ft by rolling inverted. Maximum speed is reached at 36000, but remember the engines produce more thrust at higher KIAS, so slightly denser air may not hurt acceleration through the sound barrier.
Break the mach barrier: Accelerate to mach 1.25 with minimal wing loading (don't turn, try to set 0AoA)
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91
SOLOClimbing Aircraft Performance
Optimum Climbing Trajectories using Energy State Approximation (ESA)
Aircraft Flight Performance
Minimum Fuel-to- Climb Trajectories using Energy State Approximation (ESA)
The Rate of Fuel consumed by the Aircraft is given by:
=−=AircraftJetTc
AircraftPropellerPc
td
Wd
td
fd
T
p
We can write ( )DTV
EdW
E
Edtd
−==
The fuel consumed in a flight time , tf for a Jet Aircraft is:
( )∫∫∫ −=== fff t
Tt
T
t
f EdTDV
Wc
E
EdTctd
td
fdf
000 /1
The minimum fuel consumed in a flight time tf is obtained when using Maximum Thrust and the Mach Number that minimize the integrand:
( )∫ −= ft T
Mf Ed
TDV
Wcf
0max
min, /1minarg
for each level of E.
92
SOLOClimbing Aircraft Performance
Optimum Climbing Trajectories using Energy State Approximation (ESA)
Aircraft Flight Performance
Minimum Fuel-to- Climb Trajectories using Energy State Approximation (ESA)
Assuming W nearly constant, during the climb period, contours of constant( )
max
max
Tc
DTV
T
− can be computed, as we see in the Figure
A.E. Bryson, Course “Performance Analysis of Flight Vehicles”, AA200, Stanford University, Winter 1977-1978A.E. Bryson, Jr., “Energy-State Approximation in Performance Optimization of Supersonic Aircraft”, Journal of Aircraft, Vol.6, No. 5, Nov-Dec 1969, pp. 481-488
The Minimum Fuel-to- Climb Trajectory is obtained by choosing at each state.
( )max
max
Tc
DTV
T
−
The Minimum Time-to- ClimbPath is also displayed.
Implicit to ESA Approximation is the possibility of instantaneous jump between kinetic to potential energy(from A to B) where the Total Energy is constant.
A
B
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93
SOLOClimbing Aircraft Performance
Optimum Climbing Trajectories using Energy State Approximation (ESA)
Aircraft Flight Performance
Maximum Range during Glide using Energy State Approximation (ESA)
Equations of motion
A.E. Bryson, Jr., “Energy-State Approximation in Performance Optimization of Supersonic Aircraft”, Journal of Aircraft, Vol.6, No. 5, Nov-Dec 1969, pp. 481-488
γ
γγ
sin
0cos
WDTVg
W
WLtd
dV
g
W
−−=
≈−=
( )W
VDTEps
−== :
g
V
W
DT −−=γsin
γγ
sin
cos
Vh
Vx
=
=
−−=g
V
W
DTVh
γγ cos
1
cos
−−===g
V
W
DT
V
h
x
h
xd
hd
During Glide we have: T = 0, W = constant, dE≤0, |γ| <<1, therefore
+−=
g
V
W
D
xd
hd
( )γcos
1
VW
DT
xd
Ed −=
2
2
1: VhE +=
( ) ( )( )EL
VED
W
VED
td
Ed −≈−=
Vtd
xd = ( )( )( )ED
EL
ED
W
Ed
xd −≈−= ( )( )( )∫∫∫ −≈−== EdED
ELEd
ED
WxdR
94
SOLOClimbing Aircraft Performance
Optimum Climbing Trajectories using Energy State Approximation (ESA)
Aircraft Flight Performance
Maximum Range during Glide using Energy State Approximation (ESA)
We found
A.E. Bryson, Jr., “Energy-State Approximation in Performance Optimization of Supersonic Aircraft”, Journal of Aircraft, Vol.6, No. 5, Nov-Dec 1969, pp. 481-488
( )( )( )∫∫ −≈−= EdED
ELEd
ED
WR
Using the first integral we see that to maximize R we must choose the path that minimizes the drag D (E). The approximate optimal trajectory can be divided in:1.If the initial conditions are not on the maximum range glide path the Aircraft shall either “zoom dive” or “zoom climb” at constant E0, A to B path in Figure .2.The Aircraft will dive on the min D (E) until it reaches the altitude h = 0 at a velocity V and Specific Energy E1=V2/2, B to C in the Figure.3.Since h=0 no optimization is possible and to stay airborne one must keep the drag such that L = W, by increasing the Angle of Attack and decreasing velocity until it reaches Vstall and Es=Vstall
2/2, C to D in Figure Since h=0, d E=V dV.
( ) ( ) ( )[ ]∫ ∫∫=
=
−−−= 1
0 1
0
00min
10
max
E
E
E
Eh
pathon
E
E
s
VdVD
VWEd
ED
WEd
ED
WR
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95
Performance of an Aircraft with Parabolic PolarSOLO
−+=
=
γσα
γσ
coscossin
cossin
V
g
Vm
LTq
V
gr
W
W
nW
L
W
LTn =≈+= αsin
:'
Therefore
( )
−=
=
γσ
γσ
coscos'
cossin
nV
gq
V
gr
W
W
γσγσγσω 2222222 coscoscoscos'2'cossin +−+=+= nnV
gqr WW
or
γγσω 22 coscoscos'2' +−= nnV
g
γγσω 22
2
coscoscos'2'
1
+−==
nng
VVR
Aircraft Turn Performance
96
Performance of an Aircraft with Parabolic PolarSOLO
( ) ( )
( )γσσ
γαχ
γσγσαγ
cos
sinsin
cos
sin
coscos'coscossin
V
gLT
nV
g
V
g
Vm
LT
=+=
−=−+=
2. Horizontal Plan Trajectory ( )0,0 == γγ
( )
1'
1
1''
11'sin'
cos
1'01cos'
2
2
22
−=
−=
−==
=→=−=
ng
VR
nV
g
nn
V
gn
V
g
nnV
g
σχ
σσγ
Aircraft Turn Performance
1. Vertical Plan Trajectory (σ = 0)
( )
γ
γγ
χ
cos'
1
cos'
0
2
−=
−=
=
ng
VR
nV
g
97
98
Vertical Plan Trajectory (σ = 0) SOLO
Prof. Earll Murman, “Introduction to Aircraft Performance and Static Stability”, September 18, 2003
99
R
V=:χ1'2 −= nV
gχ
Contours of Constant n and Contours of Constant Turn Radiusin Turn-Rate in Horizontal Plan versus Mach coordinates
Horizontal Plan Trajectory SOLO
100Maneuverability Diagram
R
V=:χ1'2 −= n
V
gχ
Horizontal Plan Trajectory
101F-5E Turn Performance
Horizontal Plan Trajectory
102
Performance of an Aircraft with Parabolic PolarSOLO
2. Horizontal Plan Trajectory ( )0,0 == γγ
We can see that for n > 1
We found that2
2 *
*u
C
Cn
u
CnC
L
LLL =→=
n
1n
2n
MAXn
u u
LC
MAXLC _
1_
nC
C
MAXL
LMAX
MAXL
Lcorner n
C
Cu
_
*=
*2 L
MAXL C
u
nC =
MAXMAXL
Lcorner n
C
Cu
_
*= MAXL
L nC
C
1
*
MAXLC _
2LC
1LC
2
*1 u
C
Cn
L
L=
MAXn
n, CL as a function of u
Aircraft Turn Performance
=
=
=
2
0
*2
1:*
*:
2:*
Vq
V
Vu
CS
kWV
D
ρ
ρ1
1
1'
1
11'
2
2
2
2
22
−≈
−=
−≈−=
ng
V
ng
VR
nV
gn
V
gχ Horizontal Turn Rate
Horizontal Turn Radius
103
Performance of an Aircraft with Parabolic PolarSOLO
MAX
MAX
L
MAXL
n
n
C
C
V
g 1
**
2_ −
MAXMAXL
Lcorner n
C
C
V
gu
_
*
*=
MAXL
L
C
C
V
gu
_1
*
*=
MAXn
2n1n
MAXLC _
2LC
1LC
u
χ
MAXu
Horizontal Turn Rate as function of u, with n and CL as parameters χ
We defined 2
*&
*: u
C
Cn
V
Vu
L
L==
We found 22
2
22 1
**1
*1
uu
C
C
V
gn
Vu
gn
V
g
L
L −
=−=−=χ
This is defined for 1:**
1__
<=≥≥= uC
Cun
C
Cu
MAXL
LMAX
MAXL
Lcorner
2. Horizontal Plan Trajectory ( )0,0 == γγ
Aircraft Turn Performance
=
=
=
2
0
*2
1:*
*:
2:*
Vq
V
Vu
CS
kWV
D
ρ
ρ
104
Performance of an Aircraft with Parabolic PolarSOLO
From2
2
2
22 1
**1
*1
uu
C
C
V
gn
Vu
gn
V
g
L
L −
=−=−=χ
4
2
2
2
22
1
*
1*
1
*:
uC
Cg
V
n
u
g
VVR
L
L −
=
−==
χ
Therefore
cornerMAXMAXL
L
MAXL
L
L
MAXL
Cun
C
Cu
C
Cu
uC
Cg
VR
MAXL=≤≤=
−
=
__1
4
2
_
2 **
1
*
1*_
cornerMAXMAXL
L
MAX
nun
C
Cu
n
u
g
VR
MAX=≥
−=
_2
22 *
1
*
MAXL
L
L
L
L
L
Cn
C
Cu
C
Cu
uC
Cg
VR
L
**
1
*
1*1
4
2
2
≤≤=
−
=
nC
Cu
n
u
g
VR
MAXL
Ln
_2
22 *
1
* ≥−
=
2. Horizontal Plan Trajectory ( )0,0 == γγ
Aircraft Turn Performance
=
=
=
2
0
*2
1:*
*:
2:*
Vq
V
Vu
CS
kWV
D
ρ
ρ
105
Performance of an Aircraft with Parabolic PolarSOLO
R (Radius of Turn) a function of u, with n and CL as parameters
1
**2
_
2
−MAX
MAX
MAXL
L
n
n
C
C
g
V
MAXMAXL
Lcorner n
C
C
V
gu
_
*
*=
MAXL
L
C
C
V
gu
_1
*
*=
MAXn
2n
1nMAXLC _
2LC 1L
C
u
R
4
2
2
2
22
1
*
1*
1
*:
uC
Cg
V
n
u
g
VVR
L
L −
=
−==
χ
2. Horizontal Plan Trajectory ( )0,0 == γγ
Return to Table of Content
Aircraft Turn Performance
=
=
=
2
0
*2
1:*
*:
2:*
Vq
V
Vu
CS
kWV
D
ρ
ρ
106
Performance of an Aircraft with Parabolic PolarSOLO
Horizontal Turn Rate as Function of ps, n ( )
u
nuzu
e
Vps
224 2
*2
* −+−=
upV
euzun s*
*22 242 −+−=
2
24
2
2 1*
*22
*
1
* u
upV
euzu
V
g
u
n
V
g s −−+−=−=χ
2
24
4
2423
1*
*22
2
1*
*222
*
*244
*
u
upV
euzu
u
upV
euzuuup
V
euzu
V
g
us
ss
−−+−
−−+−−
−+−
=∂∂ χ
Therefore
−−+−
++−=
∂∂
1*
*22
1*
*
*244
4
upV
euzuu
upV
eu
V
g
us
sχ
For ps = 0
222
12
24
011
12
*uzzuzzu
u
uzu
V
gsp
=−+<<−−=−+−==
χ
( ) 222
1244
4
0
1112
1
*uzzuzzu
uzuu
u
V
g
usp
=−+<<−−=−+−
+−=∂∂
=
χ
Aircraft Turn Performance
=
=
=
2
0
*2
1:*
*:
2:*
Vq
V
Vu
CS
kWV
D
ρ
ρ
107
Performance of an Aircraft with Parabolic PolarSOLO
Horizontal Turn Rate as Function of ps, n For ps = 0
222
12
24
011
12
*uzzuzzu
u
uzu
V
gsp
=−+<<−−=−+−==
χ
( ) 222
1244
4
0
1112
1
*uzzuzzu
uzuu
u
V
g
usp
=−+<<−−=−+−
+−=∂∂
=
χ
Let find the maximum of as a function of u χ
( )12
1
* 244
4
0 −+−
+−=∂∂
= uzuu
u
V
g
usp
χ
( ) ( )12*
100
−=====
zV
gu
ss ppMAX χχ
u 0 u1 1 (u1+u2)/2 u2
∞ + + 0 - - - - - - -∞
↑ Max ↓
u∂∂ χ
χ
From
2
24 1**2
2
* u
upVe
uzu
V
g s −−+−=χ
−−+−
++−=
∂∂
1**2
2
1**
* 244
4
upVe
uzuu
upVe
u
V
g
us
sχ
Aircraft Turn Performance
=
=
=
2
0
*2
1:*
*:
2:*
Vq
V
Vu
CS
kWV
D
ρ
ρ
108
Performance of an Aircraft with Parabolic PolarSOLO
Horizontal Turn Rate as Function of ps, n
u
u
0<sp
0<sp
0=sp
0=sp 0>sp
0>sp
χ
u∂∂ χ
( )12*
−zV
g
1=u1u2u
as a function of u with ps asparameter
u∂∂ χχ
,
−−+−
++−=
∂∂
1**2
2
1**
* 244
4
upVe
uzuu
upVe
u
V
g
us
sχ
2
24 1**2
2
* u
upVe
uzu
V
g s −−+−=χ
Because ,we have0*
* >uV
e
000 >=<>>
sss pppχχχ
01
01
01
0>
==
=<
= ∂∂<=
∂∂<
∂∂
sss pu
pu
pu uuu
χχχ
Aircraft Turn Performance
=
=
=
2
0
*2
1:*
*:
2:*
Vq
V
Vu
CS
kWV
D
ρ
ρ
109
Performance of an Aircraft with Parabolic PolarSOLO
Horizontal Turn Rate as Function of ps, n
a function of u, with ps as parameter
χ
2
24 1**2
2
* u
upVe
uzu
V
g s −−+−=χ
Aircraft Turn Performance
=
=
=
2
0
*2
1:*
*:
2:*
Vq
V
Vu
CS
kWV
D
ρ
ρ
SustainedTurn
InstantaneousTurn
110
Performance of an Aircraft with Parabolic PolarSOLO
Horizontal Turn Rate as Function of ps, n
2
24 1*
*22
* u
upV
euzu
V
g s −−+−=χ
( ) ( )ss
s
puupuup
V
euzu
u
g
VVR 21
24
42
1*
*22
* <<−−+−
==χ
3242
232
24
4
224
34243
2
1*
*222
2*
*322
*
1*
*22
2
1*
*22
*
*2441
*
*224
*
−−+−
−−
=
−−+−
−−+−
−+−−
−−+−
=∂∂
upV
euzuu
upV
euzu
g
V
upV
euzu
u
upV
euzu
pV
euzuuup
V
euzuu
g
V
u
R
s
s
s
s
ss
Aircraft Turn Performance
=
=
=
2
0
*2
1:*
*:
2:*
Vq
V
Vu
CS
kWV
D
ρ
ρ
111
Performance of an Aircraft with Parabolic PolarSOLO
Horizontal Turn Rate as Function of ps, n
324
22
1*
*22
2*
*32
*
−−+−
−−
=∂∂
upV
euzu
upV
euzu
g
V
u
R
s
s
or
We have
>+
+
=
<+
−
=→=
∂∂
04
16*
*9
*
*3
04
16*
*9
*
*3
02
2
2
1
z
zpV
eup
V
e
u
z
zpV
eup
V
e
u
u
R
ss
R
ss
R u 0 u1 uR2 u2
∞ - - - 0 + + ∞
↓ min ↑u
R
∂∂
R
222
124
42
011
12
*uzzuzzu
uzu
u
g
VR
sp=−+<<−−=
−+−=
=
( )( ) 2
221324
22
0
1112
1*2uzzuzzu
uzu
uzu
g
V
u
R
sp
=−+<<−−=−+−
−=∂∂
=
Aircraft Turn Performance
=
=
=
2
0
*2
1:*
*:
2:*
Vq
V
Vu
CS
kWV
D
ρ
ρ
112
Performance of an Aircraft with Parabolic PolarSOLO
Horizontal Turn Rate as Function of ps, n
u
R
0>sp
0=sp
0<sp
MAXL
L
C
C
_
*
1
**2
_ −MAX
MAX
MAXL
L
n
n
C
C
g
V
1
1*2 −zg
V
4
2
_
1*
1*
uC
Cg
V
MAXL
L −
1
*2
22
−MAXn
u
g
V
MAXMAXL
L nC
C
_
*
LIMIT
C MAXL_
LIMIT
nMAX
z
1
12 −− zz 12 −+ zz
1**2
2
*
24
42
−−+−=
upVe
uzu
u
g
VR
s
Minimum Radius of Turn R is obtained for zu /1=
1
1*2
2
0 −=
=zg
VR
sp
R (Radius of Turn) a function of u, with ps as parameter
( ) ( )ss
s
puupu
upVe
uzu
u
g
VVR
21
24
42
1**2
2
*
<<
−−+−==
χ
Return to Table of Content
Because ,we have0*
* >uV
e000 >=<
<<sss ppp
RRR000 minminmin >=<
<<sss pRpRpR uuu
Aircraft Turn Performance
=
=
=
2
0
*2
1:*
*:
2:*
Vq
V
Vu
CS
kWV
D
ρ
ρ
113
Performance of an Aircraft with Parabolic PolarSOLO
Horizontal Turn Rate as Function of nV ,
( )W
VDT
g
VVhEps
−≈+==
:
For an horizontal turn 0=h
Vg
Vu
g
VVps
*==
We found2
24 1*
*22
* u
upV
euzu
V
g s −−+−=χ
from which2
24 1*2
* u
uegV
zu
V
g−
−+−
=
χ
defined for
2
22
1 :1**1**: ueg
Vze
g
Vzue
g
Vze
g
Vzu =−
−+
−≤≤−
−−
−=
Aircraft Turn Performance
=
=
=
2
0
*2
1:*
*:
2:*
Vq
V
Vu
CS
kWV
D
ρ
ρ
114
Performance of an Aircraft with Parabolic PolarSOLO
Horizontal Turn Rate as Function of nV ,
Let compute
2
24
4
2423
1*2
2
1*22*44
*
u
ueg
Vzu
u
ueg
Vzuuuue
g
Vzu
V
g
u−
−+−
−
−+−−
−+−
=∂∂
χ
−
−+−
+−=∂∂
1*2
1
*244
4
ueg
Vzuu
u
V
g
u
χ
or
u 0 u1 1 (u1+u2)/2 u2
∞ + + 0 - - - - - - -∞
↑ Max ↓u∂
∂ χ
χ
−−= 1*2
*e
g
Vz
V
gMAX
χ
Aircraft Turn Performance
=
=
=
2
0
*2
1:*
*:
2:*
Vq
V
Vu
CS
kWV
D
ρ
ρ
115
Performance of an Aircraft with Parabolic PolarSOLO
Horizontal Turn Rate as Function of nV ,
u
0<V
0=V
0>V
χ
( )12*
−zV
g
1=u1u2u
2
24 1*2
* u
uegV
zu
V
g−
−+−
=
χ
1*
2 −MAXnuV
g
22
2
_ 1
** uu
C
C
V
g
L
MAXL −
MAXL
L
C
C
_
*
MAXMAXL
L nC
C
_
*
LIMIT
nMAXLIMIT
C MAXL _
MAX
MAX
L
MAXL
n
n
C
C
V
g 1
**
2_ −
as function of uand as parameterχ
V
Return to Table of Content
Aircraft Turn Performance
=
=
=
2
0
*2
1:*
*:
2:*
Vq
V
Vu
CS
kWV
D
ρ
ρ
116http://forum.keypublishing.com/showthread.php?69698-Canards-and-the-4-Gen-aircraft/page11
Example of Horizontal Turn, versus Mach, Performance of an Aircraft
SOLO Aircraft Flight Performance
117
Mirage 2000 at 15000ft.http://forums.eagle.ru/showthread.php?t=98497
Max sustained rate (at around 6.5G on the 0 Ps line) occurring at around 0.9M/450KCASlooking at around 12.5 deg sec
9G Vc (Max instant. Rate) is around 0.65M/320KCAS looking at 23.5 deg sec
SOLO Aircraft Flight Performance
118http://n631s.blogspot.co.il/2011/03/book-review-boyd-fighter-pilot-who.html
Example of Horizontal Turn, versus Mach, Performance of MiG-21
SOLO Aircraft Flight Performance
SOLO
119
Aircraft Flight Performance
Comparison of Sustained ( ) Turn Performance of three Fightry AircraftsF-16, F-4 and MiG-21 at Altitude h = 11 km = 36000 ft
0=V
120
SOLO Aircraft Flight Performance
121
The black lines are the F-4D, the dark orange lines are the heavy F-4E, and the blue lines are the lightweight F-4E (same weight as F-4D). Up to low transonic mach numbers and up to medium altitudes, the F-4E is about 7% better than the F-4D (15% better with the same weight). At higher mach numbers, the F-4 doesn't have to pull as much AoA to get the same lift, so the slats actually cause a drag penalty that allows the F-4D to perform better. For reference, the F-14 is known to turn about 20% better than the unslatted F-4J. So, if the slats made the F-4S turn about 15% better, sustained turn rates would almost be pretty close between the F-14 and F-4S. The F-4E, being heavier, would still be significantly under the F-14. However, with numbers this close, pilot quality is everything rather than precise performance figures.
http://combatace.com/topic/71161-beating-a-dead-horse-us-fighter-turn-performance/
F-4
SOLO Aircraft Flight Performance
122http://www.worldaffairsboard.com/military-aviation/62863-comparing-fighter-performance-same-generations-important-factor-war-2.html
F-15F-4
SOLO Aircraft Flight Performance
123
http://www.airliners.net/aviation-forums/military/print.main?id=153429
SOLO Aircraft Flight Performance
Return to Table of Content
124
Corner Speed
MaximumPositive
Capability(CL) max
MaximumNegative
Capability(CL) min
Load Factor -
n
Structural Limit
Structural Limit
Limit Airspeed
Area ofStructural Damage of
Failure
Vmin V
n
OperationalLoad Limit
OperationalLoad Limit
StructuralLoad Limit
StructuralLoad Limit
Typical Maneuvering EnvelopeV – n Diagram
Maneuvering Envelope:Limits on Normal Load Factor andAllowable Equivalent Airspeed-Structural Factor-Maximum and Minimum allowable Lift Coefficient-Maximum and Minimum Airspeeds-Corner Velocity: Intersection of Maximum Lift Coefficient and Maximum Load Factor
SOLO Aircraft Flight Performance
125
Typical Maneuvering EnvelopeV – n Diagram
Performance of an Aircraft with Parabolic PolarSOLO
126R.W. Pratt, Ed., “Flight Control Systems, Practical issues in design and implementation”,AIAA Publication, 2000
SOLO Aircraft Flight Performance
Return to Table of Content
127
Air-to-Air Combat
Destroy Enemy Aircraft to achieve Air Supremacy in order to prevent the enemyto perform their missions and enable to achieve tactical goals.
SOLO
See S. Hermelin, “Air Combat”, Presentation, http://www.solohermelin.com
128
http://forum.warthunder.com/index.php?/topic/110779-taktik-ve-manevralar-hakk%C4%B1ndaki-e%C4%9Fitim-g%C3%B6rselleri-oz/page-2
Air-to-Air Combat
Before the introduction of all-aspect Air-to-Air Missiles destroying an Enemy Aircraft was effective only from the tail zone of the Enemy Aircraft, so the pilots had to maneuver to reach this position, for the minimum time necessary to activate the guns or launch a Missile.
Return to Table of Content
SOLO
129
Energy–Maneuverability Theory
Aircraft Flight Performance
Energy–maneuverability theory is a model of aircraft performance. It was promulgated by Col. John Boyd, and is useful in describing an aircraft's performance as the total of kinetic and potential energies or aircraft specific energy. It relates the thrust, weight, drag, wing area, and other flight characteristics of an aircraft into a quantitative model. This allows combat capabilities of various aircraft or prospective design trade-offs to be predicted and compared.
Colonel John Richard Boyd (1927 –1997)
Boyd, a skilled U.S. jet fighter pilot in the Korean War, began developing the theory in the early 1960s. He teamed with mathematician Thomas Christie at Eglin Air Force Base to use the base's high-speed computer to compare the performance envelopes of U.S. and Soviet aircraft from the Korean and Vietnam Wars. They completed a two-volume report on their studies in 1964. Energy Maneuverability came to be accepted within the U.S. Air Force and brought about improvements in the requirements for the F-15 Eagle and later the F-16 Fighting Falcon fighters
130
131
Turning Capability Comparison of F4E and MiG21 at Sea Level
http://forum.keypublishing.com/showthread.php?96201-fighter-maneuverability-comparison
F-4E
MiG-21
Aircraft Flight Performance
132
http://www.aviationforum.org/military-aviation/16335-fighter-maneuverability-comparison.html
F4 _Phantom versus MIG 21
MiG-21
SOLO Aircraft Flight Performance
133
Aircraft Flight Performance
134
135
SOLO
136
Aircraft Flight Performance
In combat, a pilot is faced with a variety of limiting factors. Some limitations are constant, such as gravity, drag, and thrust-to-weight ratio. Other limitations vary with speed and altitude, such as turn radius, turn rate, and the specific energy of the aircraft. The fighter pilot uses Basic Fighter Maneuvers (BFM) to turn these limitations into tactical advantages. A faster, heavier aircraft may not be able to evade a more maneuverable aircraft in a turning battle, but can often choose to break off the fight and escape by diving or using its thrust to provide a speed advantage. A lighter, more maneuverable aircraft can not usually choose to escape, but must use its smaller turning radius at higher speeds to evade the attacker's guns, and to try to circle around behind the attacker.[13]
BFM are a constant series of trade-offs between these limitations to conserve the specific energy state of the aircraft. Even if there is no great difference between the energy states of combating aircraft, there will be as soon as the attacker accelerates to catch up with the defender. Instead of applying thrust, a pilot may use gravity to provide a sudden increase in kinetic energy (speed), by diving, at a cost in the potential energy that was stored in the form of altitude. Similarly, by climbing the pilot can use gravity to provide a decrease in speed, conserving the aircraft's kinetic energy by changing it into altitude. This can help an attacker to prevent an overshoot, while keeping the energy available in case one does occur
Energy Management
SOLO
137
Aircraft Flight Performance
Energy Management
Colonel J. R. Boyd:
In an air-to-air battle offensive maneuvering advantage will belong to the pilot who can enter an engagement at a higher energy level and maintain more energy than his opponent while locked into a maneuver and counter-maneuver duel. Maneuvering advantage will also belong to the pilot who enters an air-to-air battle at a lower energy level, but can gain more energy than his opponent during the course of the battle, From a performance standpoint, such an advantage is clear because the pilot with the most energy has a better opportunity to engage or disengage at his own choosing. On the other hand, energy-loss maneuvers can be employed defensively to nullify an attack or to gain a temporary offensive maneuvering position.
http://www.ausairpower.net/JRB/fast_transients.pdf
“New Conception for Air-to-Air Combat”, J. Boyd, 4 Aug. 1976
138http://www.alr-aerospace.ch/Performance_Mission_Analysis.php
F-16
SOLO Aircraft Flight Performance
139Comparative Ps Diagram for Aircraft A and Aircraft B. Two Multi-Role Jet Fighters
SOLO Aircraft Flight Performance
140
http://www.simhq.com/_air/air_065a.html
http://en.wikipedia.org/wiki/Lavochkin_La-5
Comparison of Turn Performance of two WWII Fighter Aircraft: Russian Lavockin La5 vs German Messershmitt Bf 109
http://en.wikipedia.org/wiki/Messerschmitt_Bf_109
SOLO Aircraft Flight Performance
141
Comparison of Turn Performance of two WWII Fighter Aircraft: Russian Lavockin La5 vs German Messershmitt Bf 109
http://en.wikipedia.org/wiki/Lavochkin_La-5http://en.wikipedia.org/wiki/Messerschmitt_Bf_109
http://www.simhq.com/_air/air_065a.html
SOLO Aircraft Flight Performance
142F-86F Sabre and MiG-15 performance comparison
North American F-86 Sabre
MiG-15
SOLO Aircraft Flight Performance
143Falcon F-16C versus Fulcrum MIG 29,
left is w/o afterburner, right is with it, fuel reserves 50%
http://forum.keypublishing.com/showthread.php?47529-MiG-29-kontra-F-16-(aerodynamics-)
FulcrumMiG-29F-16
SOLO Aircraft Flight Performance
144
Comparison of Turn Performance of two Modern Fighter Aircraft: Russian MiG-29 vs USA F-16
FulcrumMiG-29
F-16
http://www.simhq.com/_air/air_012a.html
http://www.evac-fr.net/forums/lofiversion/index.php?t984.html
145
Comparison of Turn Performance of two Modern Fighter Aircraft: Russian MiG-29 vs USA F-16
FulcrumMiG-29
F-16
http://www.simhq.com/_air/air_012a.html
http://www.evac-fr.net/forums/lofiversion/index.php?t984.html
146
http://www.simhq.com/_air/air_012a.html
Comparison of Turn Performance of two Modern Fighter Aircraft: Russian MiG-29 vs USA F-16
Fulcrum MiG-29
F-16
http://www.evac-fr.net/forums/lofiversion/index.php?t984.html
147http://www.simhq.com/_air3/air_117e.html
While the turn radius of both aircraft is very similar, the MiG-29 has gained a significant angular advantage.
Comparison of Turn Performance of two Modern Fighter Aircraft: Russian MiG-29 vs USA F-16
MiG-29F-16
SOLO Aircraft Flight Performance
148
http://www.evac-fr.net/forums/lofiversion/index.php?t984.html
Comparison of Turn Performance of two Modern Fighter Aircraft: Russian MiG-29 vs USA F-16
With afterburner, fuel reserves 50%Without afterburner, fuel reserves 50%
MiG-29
F-16
SOLO Aircraft Flight Performance
149
http://forums.eagle.ru/showthread.php?t=30263
SOLO Aircraft Flight Performance
150
An assessment is made of the applicability of Energy Maneuverability techniques (EM)to flight path optimization. A series of minimum time and fuel maneuvers using the F-4Caircraft were established to progressively violate the assumptions inherent in the EM programand comparisons were made with the Air Force Flight Dynamics Laboratory's (AFFDL)Three-Degree-of-Freedom Trajectory Optimization Program and a point mass option of theSix-Degree-of-Freedom flight path program. It was found the EM results were always optimisticin the value of the payoff functions with the optimism increasing as the percentageof the maneuver involving constant energy transitions Increases. For the minimum timepaths the resulting optimism was less than 27%f1o r the maneuvers where the constant energypercentage was less than 35.',", followed by a rather steeply rising curve approaching in thelimit 100% error for paths which are comprised entirely of constant energy transitions. Twonew extensions are developed in the report; the first is a varying throttle technique for useon minimum fuel paths and the second a turning analysis that can be applied in conjunctionwith a Rutowski path. Both extensions were applied to F-4C maneuvers in conjunction with'Rutowski’s paths generated from the Air Force Armament Laboratory's Energy Maneuverabilityprogram. The study findings are that energy methods offer a tool especially useful inthe early stages of preliminary design and functional performance studies where rapidresults with reasonable accuracy are adequate. If the analyst uses good judgment in its applicationsto maneuvers the results provide a good qualitative insight for comparative purposes.The paths should not, however, be used as a source of maneuver design or flightschedule without verification especially on relatively dynamic maneuvers where the accuracyand optimality of the method decreases.
David T. Johnson, “Evaluation of Energy Maneuverability Procedures in Aircraft Flight Path Optimization and Performance Estimation”, November 1972, AFFDL-TR-72-53
SOLO Aircraft Flight Performance
151
Lockheed F-104 Starfighter
SOLO Aircraft Flight Performance
152Typical Ps Plot for Lockheed F-104 Starfighter
Lockheed F-104 Starfighter
SOLO Aircraft Flight Performance
153
SOLO Aircraft Flight Performance
F-104 Flight Envelope
Lockheed F-104 Starfighter
154F-104A flight envelope
Lockheed F-104 Starfighter
SOLO Aircraft Flight Performance
Return to Table of Content
155
http://defence.pk/threads/design-characteristics-of-canard-non-canard-fighters.178592/
SOLO Aircraft Flight Performance
Aircraft Combat Performance Comparison
156
http://defence.pk/threads/design-characteristics-of-canard-non-canard-fighters.178592/
SOLO Aircraft Flight Performance
157http://img138.imageshack.us/img138/4146/image4u.jpg
SOLO Aircraft Flight Performance
158
https://s3-eu-west-1.amazonaws.com/rbi-blogs/wp-content/uploads/mt/flightglobalweb/blogs/the-dewline/assets_c/2011/05/chart%20combat%20radius-thumb-500x375-125731.jpg
Aircraft Combat Performance Comparison
SOLO Aircraft Flight Performance
Return to Table of Content
159
Supermaneuverability is defined as the ability of an aircraft to perform high alpha maneuvers that are impossible for most aircraft is evidence of the aircraft's supermaneuverability. Such maneuvers include Pugachev's Cobra and the Herbst maneuver (also known as the "J-turn").Some aircraft are capable of performing Pugachev's Cobra without the aid of features that normally provide post-stall maneuvering such as thrust vectoring. Advanced fourth generation fighters such as the Su-27, MiG-29 along with their variants have been documented as capable of performing this maneuver using normal, non-thrust vectoring engines. The ability of these aircraft to perform this maneuver is based in inherent instability like that of the F-16; the MiG-29 and Su-27 families of jets are designed for desirable post-stall behavior. Thus, when performing a maneuver like Pugachev's Cobra the aircraft will stall as the nose pitches up and the airflow over the wing becomes separated, but naturally nose down even from a partially inverted position, allowing the pilot to recover complete control.
http://en.wikipedia.org/wiki/Supermaneuverability
Supermaneuverability
SOLO Aircraft Flight Performance
160
SOLO Aircraft Flight Performance
161
Sukhoi Su-30MKI
SOLO Aircraft Flight Performance
http://vayu-sena.tripod.com/interview-simonov1.html
162
SOLO Aircraft Flight Performance
The Herbst maneuver or "J-Turn" named after Wolfgang Herbst is the only thrust vector post stall maneuver that can be used in actual combat but very few air frames can sustain the stress of this violent maneuver.
Herbst Maneuver
http://en.wikipedia.org/wiki/Herbst_maneuver
Return to Table of Content
163
Constraint AnalysisSOLO Aircraft Flight Performance
The Performance Requirements can be translated into functional relationship between the Thrust-to-Weight or Thrust Loading at Sea Level Takeoff (TSL/WTO) and the Wing Loading at Takeoff (WTO/S). The keys to the development are•Reasonable assumption hor Aircraft Lift-to-Drag Polar.•The low sensibility of Engine Thrust with Flight Altitude and Mach Number.
The minimum of TSL/WTO as functions of WTO/S are required for:•Takeoff from a Runway of a specified length.•Flight at a given Altitude and Required Speed.•Climb at a Required Speed.•Turn at a given Altitude, Speed and a required Rate.•Acceleration capability at constant Altitude.•Landing without reverse Thrust on a Runway of a given length.
164
Energy per unit mass E
Let define Energy per unit mass E:g
VhE
2:
2
+=Let differentiate this equation:
( ) ( )W
VDT
W
VDT
W
DTg
g
VV
g
VVhEps
−≈−=
−−+=+== αγαγ cos
sincos
sin:
define
10 ≤<= ββ TOWW WTO – Take-off Weight
( ) ( ) ( ) SLThThhT αα === 0 TSL – Sea Level Thrust
V
p
W
D
W
T s+=
Load FactorW
CSq
W
Ln L==:
SOLO Aircraft Flight Performance
TOL WSq
nW
Sq
nC β==
+=
V
p
W
D
W
T s
TO
SL
αβ
Constraint Analysis
165
SOLO Aircraft Flight Performance
General Mission Description of a Typical Fighter Aircraft
10: ≤<= ββTOW
W
WTO – Take-off Weight
W – Aircraft Weight during Flight
Constraint Analysis
166
Assume a General Lift-to-Drag Polar Relationship
Total DragRD CSqCSqRD +=+
D, CD - Clean Aircraft Drag and Drag Coefficient
R, CR – Additional Aircraft Drag and Additional Drag Coefficient caused by External Stores, Bracking Parachute, Flaps, External Hardware
02
2
1022
1 DTOTO
DLLD CS
W
q
nK
S
W
q
nKCCKCKC +
+
=++= ββ
TOL WSq
nW
Sq
nC β==
( )
++=V
pCC
W
Sq
W
T sRD
TOTO
SL
βαβ
+
++
+
=
V
pCC
S
W
q
nK
S
W
q
nK
W
Sq
W
T sDRD
TOTO
TOTO
SL02
2
1
βββα
β
SOLO Aircraft Flight Performance
Constraint Analysis
167
( )WLntd
Vd
td
hd ==== ,1,0,0
Case 1: Constant Altitude/Speed Cruise (ps = 0)
Given:
+++
=
SW
q
CCK
S
W
qK
W
T
TO
DRDTO
TO
SL
ββ
αβ 0
21
We obtain:
We can see that TSL/WTO → ∞ for WTO/S → 0 and WTO/S→∞, therefore a minimum exist. By differentiating TSL/WTO with respect to WTO/S and setting the result equal to zero, we obtain:
1
0
/min K
CCq
S
W DRD
WT
TO +=
β
( )[ ]210
min
2 KKCCW
TDRD
TO
SL ++=
αβ
Lift
DragThrust
Weight
SOLO Aircraft Flight Performance
Constraint Analysis
168M. Corcoran, T. Matthewson, N. W. Lee, S. H. Wong, “Thrust Vectoring”
Case 1: Constant Altitude/Speed Cruise (ps = 0)
SOLO Aircraft Flight Performance
Constraint Analysis
169
( )WLntd
hd ≈≈= ,1,0
Case 2: Constant Speed Climb (ps = dh/dt)
Given:
We can see that TSL/WTO → ∞ for WTO/S → 0 and WTO/S→∞, therefore a minimum exist. By differentiating TSL/WTO with respect to WTO/S and setting the result equal to zero, we obtain:
1
0
/min K
CCq
S
W DRD
WT
TO +=
β
( )
+++=
td
hd
VKKCC
W
TDRD
TO
SL 12 210
minαβ
We obtain:
+
+++
=
td
hd
VSW
q
CCK
S
W
qK
W
T
TO
DRDTO
TO
SL 1021 β
βαβ
SOLO Aircraft Flight Performance
170
,1,0,0,,
>== ntd
hd
td
Vd
givenhVgivenhV
Case 3: Constant Altitude/Speed Turn (ps = 0)
Given:
We can see that TSL/WTO → ∞ for WTO/S → 0 and WTO/S→∞, therefore a minimum exist. By differentiating TSL/WTO with respect to WTO/S and setting the result equal to zero, we obtain:
1
0
/min K
CC
n
q
S
W DRD
WT
TO +=
β
( )[ ]210
min
2 KKCCn
W
TDRD
TO
SL ++=
αβ
We obtain:
+
+++
=
td
hd
VSW
q
CCnK
S
W
qnK
W
T
TO
DRDTO
TO
SL 102
21 β
βαβ
2
0
22
0
11
+=
Ω+=cRg
V
g
Vn
SOLO Aircraft Flight Performance
Constraint Analysis
171
( )WLntd
hd
givenh
=== ,1,0
Case 4: Horizontal Acceleration (ps = (V/g0) (dV/dt) )
Given:
We obtain:
+
+++
=
td
Vd
gSW
q
CCK
S
W
qK
W
T
TO
DRDTO
TO
SL
0
021
1β
βαβ
SOLO Aircraft Flight Performance
Lift
DragThrust
Weight
This can be rearranged to give:
+++
=
SW
q
CCK
S
W
qK
W
T
td
Vd
g TO
DRDTO
TO
SL
ββ
βα 0
210
1
Constraint Analysis
172
( )WLntd
hd
givenh
=== ,1,0
Case 4: Horizontal Acceleration (ps = (V/g0) (dV/dt) ) (continue – 1)
Given:
SOLO Aircraft Flight Performance
Lift
DragThrust
Weight
We obtain:
+++
=
SW
q
CCK
S
W
qK
W
T
td
Vd
g TO
DRDTO
TO
SL
ββ
βα 0
210
1
This equation can be integrated from initial velocity V0 to final velocity Vf, from initial t0 to final tf times.
( )∫=− fV
Vs
f Vp
VdV
gtt
00
0
1
where
+++
−=
SW
q
CCK
S
W
qK
W
TVp
TO
DRDTO
TO
SLs β
ββα 0
21
The solutions of TSL/WTO for different WTO/S are obtained iteratively.
Constraint Analysis
173http://elpdefensenews.blogspot.co.il/2013_04_01_archive.html
Constraint AnalysisSOLO Aircraft Flight Performance
174
0=givenh
td
hd
Case 5: Takeoff (sg given and TSL >> (D+R) )
Given:
SOLO Aircraft Flight Performance
Ground RunV = 0
sg
sTO
sr str
V TORotation
Transition
sCL
θ CL
htr
hobs
R
Start from:
( )
TO
T
SL
s W
VRDT
td
Vd
g
V
td
hdp
SL
β
αα
+−
≈+=
≈
0
==
TO
SL
V
W
Tg
td
sd
sd
Vd
td
Vd
βα 0
/1
VdVT
W
gsd
SL
TO
=
0αβ
max,2
2
0max,2
0 2
1
2
1L
TO
TOLstallstallTO CS
k
VCSVLW ρρβ ===
The take-off velocity VTO is VTO = kTO Vstall
Where Vstall is the minimum velocity at at which Lift equals weight and kTO ≈ 1.1 to 1.2:
==S
W
C
kVk
V TO
L
TOstallTO
TO
max,0
222
2
22 ρβ
Integration from:
s = 0 to s = sg
V = 0 to V = VTO
2
2
0
TO
SL
TOg
V
T
W
gs
=
αβ
sg – Ground Run
Constraint Analysis
175
Case 5: Takeoff (sg given and TSL >> (D+R) ) (continue – 1)
SOLO Aircraft Flight Performance
Ground RunV = 0
sg
sTO
sr str
V TORotation
Transition
sCL
θ CL
htr
hobs
R
2
2
0
TO
SL
TOg
V
T
W
gs
=
αβ
==S
W
C
kVk
V TO
L
TOstallTO
TO
max,0
222
2
22 ρβ
=S
W
Cgs
k
W
T TO
Lg
TO
TO
SL
max,00
22
ρβ
αβ
We obtained:
from which:
=
S
W
C
k
T
W
gs TO
L
TO
SL
TOg
max,0
2
0 ρβ
αβ
We have a Linear Relation between TSL/WTO and WTO/S
Constraint Analysis
176
Case 6: Landing
SOLO Aircraft Flight Performance
where ( ) ( )
−=
−−=
µ
µβ
ρ
W
Tgc
CCSW
ga grLgrD
TO
0
,,
:
/2:
cab
Vaa
ca
Vaa
touchdown
4
2:
4
2:
2
11
−=
−=cVa
cVa
asg +
+−=2
2
21ln
2
1
−−⋅
++
−=
1
2
2
1
1
1
1
1ln
4
1
a
a
a
a
catg
Ground Run Phase
We foundGround Run sgr
Transition
Airborne Phase
Total Landing Distance
Float
sfFlare stGlide sg
γ
hg
hf
Touchdown
20 VCVBTT ++=
For a given value of sg , there is only one value of WTO/S that satisfies this equation.
( )gTO sfSW =/
This constraint is represented in the TSL/WTO versus WTO/S plane as a vertical line, at WTO/S corresponding to the required sg.
Constraint Analysis
177Constraint Diagram
SOLO Aircraft Flight Performance
+++
=
S
W
q
CCK
S
W
qK
W
T
TO
DRDTO
TO
SL
ββ
αβ 0
21
+
+++
=
td
hd
VS
W
q
CCnK
S
W
qnK
W
T
TO
DRDTO
TO
SL 102
21 β
βαβ
=S
W
Cgs
k
W
T TO
Lg
TO
TO
SL
max,00
22
ρβ
αβ
( )gTO sfSW =/
Constraint Analysis
178
Comparison of Fighter Aircraft Propulsion SystemsSOLO
179
Comparison of Fighter Aircraft Propulsion SystemsSOLO
180
SOLO Aircraft Flight Performance
Composite Thrust Loading versus Wing Loading – for different Aircraft
Constraint Analysis
181Constraint Diagram for F-16
SOLO Aircraft Flight Performance
Constraint Analysis
Return to Table of Content
182Weapon System Agility
Weapon System Agility
Return to Table of Content
183
References
SOLO
Miele, A., “Flight Mechanics , Theory of Flight Paths, Vol I”, Addison Wesley, 1962
Aircraft Flight Performance
J.D. Anderson, Jr., “Introduction to Flight”, McGraw Hill, 1978, Ch. 6, “Elements of Airplane Performance”
A. Filippone, “Flight Performance of Fixed and Rotary Wing Aircraft”, Elsevier, 2006
M. Saarlas, “Aircraft Performance”, John Wiley & Sons, 2007
Stengel, MAE 331, Aircraft Flight Dynamics, Princeton University
J.D. Anderson, Jr., “Aircraft Performance and Design”, McGraw Hill, 1999
N.X. Vinh, “Flight Mechanics of High-Performance Aircraft”, Cambridge University Press, 1993
F.O. Smetana, “Flight Vehicle Performance and Aerodynamic Control”, AIAA Education Series, 2001
L. George, J.F. Vernet, “La Mécanique du Vol, Performances des Avions et des Engines”, Librairie Polytechnique Ch. Béranger, 1960
L.J. Clancy, “Aerodynamics”, Pitman International Text, 1975
184
Brandt, “Introduction to Aerodynamics – A Design Perspective”, Ch. 5 , Performance and Constraint Analysis
SOLO Aircraft Flight Performance
J.D. Mattingly, W.H. Heiser, D.T. Pratt, “Aircraft Engine Design”, 2nd Ed., AIAA Education Series, 2002
Prof. Earll Murman, “Introduction to Aircraft Performance and Static Stability”, September 18, 2003
Naval Air Training Command, “Air Combat Maneuvering”, CNATRA P-1289 (Rev. 08-09)
Patrick Le Blaye, “Agility: Definitions, Basic Concepts, History”, ONERA
Randal K. Liefer, John Valasek, David P. Eggold, “Fighter Aircraft Metrics, Research , and Test”, Phase I Report, KU-FRL-831-2
References (continue – 1)
B. N. Pamadi, “Performance, Stability, Dynamics, and Control of Airplanes”, AIAA Educational Series, 1998, Ch. 2 , Aircraft Performance
L.E. Miller, P.G. Koch, “Aircraft Flight Performance”, July 1978, AD-A018 547,AFFDL-TR-75-89
185
Courtland_D._Perkins,_Robert_E._Hage, “Airplane Performance Stability and Control”, John Wiley & Sons, 1949
SOLO
Asselin, M., “Introduction to Aircraft Aerodynamics”, AIAA Education Series, 1997
Aircraft Flight PerformanceReferences (continue – 2)
Donald R. Crawford, “A Practical Guide to Airplane Performance and Design”,Crawford Aviation, 1981
Francis J. Hale, “ Introduction to aircraft performance, Selection and Design”, John Wiley & Sons, 1984
J. Russell, ‘Performance and Stability of Aircraft“, Butterworth-Heinemann, 1996
Jan Roskam, C. T. Lan, “Airplane Aerodynamics and Performance”, DARcorporation, 1997
Nono Le Rouje, “Performances of light aircraft”, AIAA, 1999
Peter J. Swatton, “Aircraft performance theory for Pilots”, Blackwell Science, 2000
S. K. Ojha, “Flight Performance of Aircraft “, AIAA, 1995
W. Austyn Mair, David L._Birdsall, “Aircraft Performance”, Cambridge University Press, 1992
186
SOLO
E.S. Rutowski, “Energy Approach to the General Aircraft Performance Problem”, Journal of the Aeronautical Sciences, March 1954, pp. 187-195
Aircraft Flight PerformanceReferences (continue – 3)
A.E. Bryson, Jr., “Applications of Optimal Control Theory in Aerospace Engineering”, Journal of Spacecraft and Rockets, Vol. 4, No.5, May 1967, pp. 553
W.C. Hoffman, A.E. Bryson, Jr., “A Study of Techniques for Real-Time, On-Line Optimum Flight Path Control”, Aerospace System Inc., ASI-TR-73-21, January 1973, AD 758799
A.E. Bryson, Jr., “A Study of Techniques for Real-Time, On-Line Optimum Flight Path Control. Algorithms for Three-Dimensional Minimum-Time Flight Paths with Two State Variables”, AD-A008 985, December 1974
M.G. Parsons, A.E. Bryson, Jr., W.C. Hoffman, “Long-Range Energy-State Maneuvers for Minimum Time to Specified Terminal Conditions”, Journal of Optimization Theory and Applications, Vol.17, No. 5-6, Dec 1975, pp. 447-463
A.E. Bryson, Jr., M.N, Desai, W.C. Hoffman, “Energy-State Approximation in Performance Optimization of Supersonic Aircraft”, Journal of Aircraft, Vol.6, No. 6, Nov-Dec 1969, pp. 481-488
187
SOLO Aircraft Flight Performance
References (continue – 4)
Solo Hermelin Presentations http://www.solohermelin.com
• Aerodynamics Folder
• Propulsion Folder
• Aircraft Systems Folder
Return to Table of Content
188
SOLO
TechnionIsraeli Institute of Technology
1964 – 1968 BSc EE1968 – 1971 MSc EE
Israeli Air Force1970 – 1974
RAFAELIsraeli Armament Development Authority
1974 –
Stanford University1983 – 1986 PhD AA
SOLO
189
OODA loop
Aircraft Flight Performance
Colonel John Richard Boyd (1927 –1997)
The OODA loop (for Observe, Orient, Decide, and Act) is a concept originally applied to the combat operations process, often at the strategic level in military operations. The concept was developed by military strategist and USAF Colonel John Boyd.
191
192M. Corcoran, T. Matthewson, N. W. Lee, S. H. Wong, “Thrust Vectoring”
Comparison Tables
193
SOLO Aircraft Flight Performance
SOLO
194
Aircraft Avionics
195Ray Whitford, “Design for Air Combat”
R.W. Pratt, Ed., “Flight Control Systems, Practical issues in design and implementation”,AIAA Publication, 2000
SOLO
196
Aircraft Flight Performance
SOLO
SOLO
199
Ray Whitford, “Design for Air Combat”Northrop F-5 Freedom Fighter
The Northrop F-20 Tigershark (initially F-5G) was a privately financed light fighter, designed and built by Northrop. Its development began in 1975 as a further evolution of Northrop's F-5E Tiger II, featuring a new engine that greatly improved overall performance, and a modern avionics suite including a powerful and flexible radar. Compared with the F-5E, the F-20 was much faster,
F-20 Powerplant: 1 × General Electric F404-GE-100 turbofan, 17,000 lbf (76 kN)
•F-5 Powerplant: 2 × General Electric J85-GE-21B turbojet • Dry thrust: 3,500 lbf (15.5 kN) each• Thrust with afterburner: 5,000 lbf (22.2 kN) each
The rear fuselage presented a problem, however, since the F-5 is, along with other twin-engined aircraft, characterised by a wide, very flat belly. This also contributes favourably to high-AOA performance. The question of how to reconcile this with a single engine basically circular in section was solved by adding shelves, not unlike those on the F-16, aft of the wing trailing edge to flatten the aft underbody. The increased skin friction drag was a small price to pay to lessen the risks of the radical change represented by the switch from a twin to a single-engined layout. The shelves house the horizontal tail control runs.
The diagram shows flight envelopes for two aircraft, the Northrop F-5E and F-20, at two load factors, 1g and 4g. Several points stand out:1 The F-20 is a Mach 2 aircraft and displays a significantly extended high-speed envelope whereas the F-5E is limited to Mach 1.64 at typical combat weight, albeit at the same 11,000 m altitude.2 Sustained maneuvering (the 4g load factor case is shown) greatly curtails the flight envelope for both aircraft as a result of the large increase in lift-dependent drag.3 The primary air battle zone, shown shaded, is limited to subsonic speeds. This is because the pilots tend to fly their aircraft to maximize turn rate — so as to gain an angular position on their opponents — and this can currently be achieved only at subsonic speed. In Vietnam and subsequent conflicts in which adversaries had Mach 2+ aircraft at their command, the combat speed range was predominantly Mach 0.5–0.9, with very little time being spent above Mach 1.1.
200http://www.simhq.com/_air3/air_117c.html
F-16
201http://www.simhq.com/_air3/air_117c.html
F-16
202
http://forum.warthunder.com/index.php?/topic/174942-wing-loading-and-turning/
The three most important (but far from the only) things to consider about an aircraft's turning performance are shown and explained in relation to the P-51's EM (energy/maneuverability) diagram below;
203
http://forum.keypublishing.com/showthread.php?129077-A-quot-Rough-quot-F-35-Kinematics-Analysis/page2
F-15 FlightF-15
204
http://forum.keypublishing.com/showthread.php?129077-A-quot-Rough-quot-F-35-Kinematics-Analysis/page2
F-15 Drag
F-15
205
206
H.H. Hurt, Jr., “Aerodynamics for Naval Aviators “,NAVAIR 00=80T-80 1-1-1965, pg. 35
207
H.H. Hurt, Jr., “Aerodynamics for Naval Aviators “,NAVAIR 00=80T-80 1-1-1965, pg. 35
208
H.H. Hurt, Jr., “Aerodynamics for Naval Aviators “,NAVAIR 00=80T-80 1-1-1965, pg. 35
209
H.H. Hurt, Jr., “Aerodynamics for Naval Aviators “,NAVAIR 00=80T-80 1-1-1965, pg. 35
210
H.H. Hurt, Jr., “Aerodynamics for Naval Aviators “,NAVAIR 00=80T-80 1-1-1965, pg. 35
211
H.H. Hurt, Jr., “Aerodynamics for Naval Aviators “,NAVAIR 00=80T-80 1-1-1965, pg. 35
212
H.H. Hurt, Jr., “Aerodynamics for Naval Aviators “,NAVAIR 00=80T-80 1-1-1965, pg. 35
213
H.H. Hurt, Jr., “Aerodynamics for Naval Aviators “,NAVAIR 00=80T-80 1-1-1965, pg. 35
214
H.H. Hurt, Jr., “Aerodynamics for Naval Aviators “,NAVAIR 00=80T-80 1-1-1965, pg. 35
215
H.H. Hurt, Jr., “Aerodynamics for Naval Aviators “,NAVAIR 00=80T-80 1-1-1965, pg. 35
216
Fixed Wing Fighter Aircraft Flight PerformanceSOLO
Level Flight
217Stengel, MAE331, Lecture 7, Gliding, Climbing and Turning Performance
218Stengel, MAE331, Lecture 7, Gliding, Climbing and Turning Performance
219Stengel, MAE331, Lecture 7, Gliding, Climbing and Turning Performance
220
Corner Speed
MaximumPositive
Capability(CL) max
MaximumNegative
Capability(CL) min
Load Factor - n
Structural Limit
Structural Limit
Limit Airspeed
Area ofStructural Damage of
Failure
Vmin V
n
OperationalLoad Limit
OperationalLoad Limit
StructuralLoad Limit
StructuralLoad Limit
Typical Maneuvering EnvelopeV – n Diagram
SOLO Aircraft Flight Performance
Corner Velocity Turn
• Corner Velocity
SC
WnV
Lcorner ρ
max
max2=
• For Steady Climbing or Diving Flight
W
DT −= maxsin γ
• Turning Radius
γγ22
max
22
maxcos
cos
−=
ng
VR
• Turning Rate
( )
γ
γγ
χ
cos'
1
cos'
0
2
−=
−=
=
ng
VR
nV
g
( )γ
γcos
cos22max
V
ngcorner
−=Ω
• Time to Complete a Full Circle
γ
γπ
22max
2cos
cos
−=
ng
Vt
221http://www.worldaffairsboard.com/military-aviation/62863-comparing-fighter-performance-same-generations-important-factor-war-2.html
222
http://www.iitk.ac.in/aero/fltlab/cruise.html
http://www.zweefportaal.nl/main/forum/viewthread.php?thread_id=2537&rowstart=0
223
http://selair.selkirk.bc.ca/training/aerodynamics/range_prop.htm
Effect of Altitude on Specific Range and Endurance
Maximum Range at L/Dmax
How Wind Affects Range and Optimum Cruise Speed
SOLO
224
Aircraft Flight Performance
Drag
SOLO
225
Aircraft Flight Performance
Drag
226
SOLO
227
Aircraft Flight Performance
Drag
228
http://elementsofpower.blogspot.co.il/2013_04_01_archive.html
229http://elementsofpower.blogspot.co.il/2013_04_01_archive.html
230
http://elementsofpower.blogspot.co.il/2013_04_01_archive.html
231http://www.f-16.net/forum/viewtopic.php?t=5487
F-16
Comparison of Climb Performance of F-16 and F-$
232http://indiandefence.com/threads/comparing-modern-western-fighters.41124/page-16
F-15
233http://indiandefence.com/threads/comparing-modern-western-fighters.41124/page-16
234
Cobra Turn
http://defence.pk/threads/supermaneuverability.39916/
Immelmann turn
Split S
Roller
Scissors
235
http://www.f-16.net/forum/viewtopic.php?t=13114
F-15
F-22
F-15 versus F-22
236
F-15
F-22
237http://forum.keypublishing.com/showthread.php?72673-Boyd-s-E-M-Theory
238http://defence.pk/threads/cope-india-how-the-iaf-rewrote-the-rules-of-air-combat.300282/page-3
239
http://www.f-16.net/forum/viewtopic.php?t=5487
240
F-16
241
F-16
242
243
F-16
SOLO
244
Aircraft Flight Performance
Performance in Level Flight
SOLO
245
Aircraft Flight Performance
SOLO
246
Aircraft Flight Performance
Determination of Maximum Flight Altitude in Level Flight
247
http://selair.selkirk.bc.ca/training/aerodynamics/range_jet.htm
248
http://forum.warthunder.com/index.php?/topic/110779-taktik-ve-manevralar-hakk%C4%B1ndaki-e%C4%9Fitim-g%C3%B6rselleri-oz/page-2
Air-to-Air Combat
249
http://forum.warthunder.com/index.php?/topic/110779-taktik-ve-manevralar-hakk%C4%B1ndaki-e%C4%9Fitim-g%C3%B6rselleri-oz/page-2
Air-to-Air Combat
250
http://forum.warthunder.com/index.php?/topic/110779-taktik-ve-manevralar-hakk%C4%B1ndaki-e%C4%9Fitim-g%C3%B6rselleri-oz/page-2
Air-to-Air Combat
251Configurations Evolution
252
North American P-51 Mustang
253Ps Diagram for a Multi-Role Fighter Aircraft at n = 1
254Ps Diagram for a Multi-Role Fighter Aircraft at n = 5
255
SOLO Aircraft Flight Performance
256
SOLO Aircraft Flight Performance
257
Generic E/M Diagram