Aircraft Stability and Performance
2nd Year, Aerospace Engineering
Dr. M. Turner
Spring Semester
Before anything else: some crucial equations
Drag on a wing (body)
Lift on a wing (body)
L=
1
ρSCL V 2
2
D=
1
ρSCD V 2
2
Drag coefficient:
Lift coefficient:
CD = CD0 + ǫCL2
CL = CL0 + CLα α
ρ
V
S
air density
(True) Airspeed
wing area
kg /m3
m/s
m2
Below ≈ Mach 1: CD0 , CL0 , CLα , ǫ are ≈ constant
Extensive use made of the above in the course!
Lift
◮
◮
◮
◮
◮
Lift makes aircraft fly
Primarily generated by the wings
We consider a “lumped” lift model: L = Lwing + Lbody + Ltail + . . .
In principle can calculate lift using wing geometry etc....
...Often convenient to characterize lift in a simpler, experimental
way, using lift coefficient, CL
CL =
L
1
2
2 ρV S
L
ρ
V
S
L=
1 2
ρV SCL
2
Lift (Newtons)
air density
(true) velocity of aircraft (m/s)
total wing area (m2 )
CL can be considered as the normalised lift: often preferred
Lift coefficient CL
◮
◮
CL depends on several different factors (Mach number) but one of
the most important is the angle of attack, α
α is the angle of incidence to the on-coming airstream.
CL
CL
STALL
Increase in incidence
gives increase in lift
coefficient in
linear region
lift coefficient
decreases
after critical
angle reached
Increase in incidence
gives increase in lift
coefficient in
linear region
STALL
Stall point
higher than
on symmetric
wing
α
α
zero incidence gives
postive lift
zero incidence gives
zero lift
CL -α graph: non-symmetric wing
CL -α graph: symmetric wing
◮ At stall point, lift coefficient is maximum: CL = CL,max
◮ For small α (in linear region):
CL = CL0 +
∂CL
α
∂α
Lift curve slope
“Lift curve slope”
Function of
◮ thickness chord ratio (τ )
dCL
dα
◮
◮
Reynolds number
Mach number (speed)
dCL
dα
Typical subsonic
passenger jet aircraft
1.0
dCL
dα
Mach no.
peaks just below Mach 1.0
One reason many passenger jets fly just below Mach 1
Drag
◮
◮
◮
Drag impedes motion of aircraft
Contributed by wings, body, tailplane, engines...
We consider a “lumped” drag model:
D = Dwing + Dbody + Dtail + . . .
CD =
◮
◮
D=
1 2
ρV SCD
2
Alternatively drag can be expressed as
CD
◮
D
1
2
2 ρV S
CL2
πeA
= CD0 + ǫCL2
= CD0 +
(e
efficiency factor = constant)
CD0 responsible for parasitic drag (form drag)
ǫCL2 responsible for velocity induced (or simply induced) drag
Point-mass approximation of aircraft performance
Straight and level flight
L
V
D
T
mg
Approximating aircraft as point-mass gives
dV
dt
d 2z
m 2
dt
m
=
T −D
=
L − mg
V
T
D
m
L
horizontal velocity
thrust
drag
aircraft mass
lift
Thus for constant height and constant forward velocity, V , we have
T
L
=
=
D
mg
(1)
(2)
Velocity in straight-and-level flight
Lift is given by
L=
1 2
ρV SCL
2
Using equation (2), this means
1
CL ρV 2 S
2
V2
V
= mg
2mg
ρCL S
s
2mg
=
ρCL S
=
As ρ, m, S (and obviously g !) are constant in straight-level flight
V ∝ CL
−1/2
Stall speed: the lowest speed at which the aircraft can maintain steady
flight
r
mg
Vs =
ρCL,max S
Equivalent airspeed
V (VTAS )
VEAS
“True” airspeed i.e. speed of air “hitting“ aircraft
Equivalent airspeed - an airspeed equivalent to that at sea level
◮
VEAS sometimes more convenient
◮
Makes various speeds independent of altitude
e.g. stall speed, minimum drag speed etc.
Example: variation of true airspeed stall with altitude
11
00
00
11
00
11
Vs = 154.42 m/s
0000000000000000000000000000000000000000000
12000m 1111111111111111111111111111111111111111111
6000m
11
00
11
00
0000000000000000000000000000000000000000000
1111111111111111111111111111111111111111111
Vs = 106.06 m/s
000
111
000
111
11111111111111111111111111111111111111111111
Vs = 85.86 m/s
2000m 00000000000000000000000000000000000000000000
Equivalent airspeed
2
ρ0 VEAS
ρ0 2
ρ VEAS
ρ
Main idea: make VEAS altitude independent:
Letting σ = ρ/ρ0
=
ρV 2
=
ρV 2
2
ρσ −1 VEAS
= ρV 2
VEAS
V
Hence
=
=
√
σV
√
σ −1 VEAS
Stall speed
VEAS,S
=
√
σVS =
r
ρ
ρ0
r
mg
=
ρSCL,max
r
mg
ρ0 SCL,max
Drag in straight-and-level flight
“Lumped” model of drag is given by
D=
1
1 2
2
ρV SCD = ρ0 VEAS
SCD
2
2
Approximately:
CD = CD0 + ǫCL2
CD0 , ǫ const
Thus
D
=
1
1
1
2
2
2
ρ0 VEAS
S(CD0 + ǫCL2 ) = ρ0 VEAS
SCD0 + ρ0 VEAS
SǫCL2
2
2
2
Using expression for CL then gives
D
=
=
1
1
2
2
ρ0 VEAS
CD0 + ρ0 VEAS
Sǫ
2
2
1
2
ρ0 VEAS
CD0 +
|2
{z
}
no
lift
drag
L
1
2
2 ρ0 VEAS S
ǫL2
1
2
2 ρ0 VEAS S
lift
|
{z
dependent
}
drag
!2
Drag in straight-and-level flight
No lift drag
Lift dependent drag
independent of lift
dependent on lift
Increases with airspeed
Decreases with airspeed
Lift dependent drag also given by (substituting for L)
ǫL2
=
1
2
2 ρ0 VEAS S
ǫm2 g 2
1
2
2 ρ0 VEAS S
Drag = Required Thrust
AIRCRAFT
DRAG
TOTAL DRAG
NO−LIFT
DRAG
LIFT DEPENDENT DRAG
VS
VE,MD
EQUIVALENT AIRSPEED
VE ,MD
“minimum drag”
speed
Calculating minimum drag
VE ,MD - in some sense an “optimal“ airspeed
◮
We have L = mg : Hence
D=D
◮
L
D
= mg
L
L
2
2
and L = 21 ρ0 CL SVEAS
As D = 12 ρ0 CD SVEAS
D=
◮
CD
mg
CL
⇒ Drag is minimised when CD /CL is minimised
Using CD = CD0 + ǫCL2 necessary condition for minimum:
dCL /CD
dCL
=
=
=
d
CD0 + ǫCL2
dCL
CL
2
d
CD0
ǫCL
d
+
dCL CL
dCL CL
−CD0 CL−2 + ǫ = 0
Calculating minimum drag
◮
Thus we have lift coefficient at minimum drag
r
CD0
CL,MD =
ǫ
◮
Also, as
VE =
s
2mg
ρ0 CL S
substituting for CL = CL,MD gives
s
14
2mg
ǫ
VE ,MD =
ρ0 S CD0
Implications
◮ Ratio CD0 /ǫ is an important consideration in VE ,MD
◮
VE ,MD decreases with S (contributor to lift-inducted drag)
Power and drag
Power used to overcome drag: Pr = DV
√
Pr
=
σPr
=
ǫm2 g 2
1
2
ρ0 VEAS
SCD0 + 1
2
2
2 ρ0 VEAS S
!
VEAS
√
σ
ǫm2 g 2
1
2
ρ0 VEAS
SCD0 + 1
2
2 ρ0 VEAS S
P
TOTAL POWER
NO−LIFT
POWER
VE ,MP
“minimum power”
speed
LIFT DEPENDENT POWER
VE,MP
EQUIVALENT AIRSPEED
Calculating minimum power
As before, using L = mg (at straight & level flight)
Pr = DV =
CD
mgV
CL
p
2mg /(ρSCL) so
s
s
CD
2mg
2
CD
mg
=
(mg )3/2 3/2
Pr =
CL
ρSCL
ρS
C
But at cruise V =
L
Thus, as m, g , S are constants, minimum power when
CD
3/2
CL
is a minimum. Necessary conditions:
d
dCL
CD
3/2
CL
!
=0
...After some differentiation
CL,MP =
r
3CD0
ǫ
Maximum speed
Jet aircraft
Propeller aircraft
TOTAL DRAG
TOTAL DRAG
THRUST
Max. excess
thrust
T,D
T,D
Max. excess
thrust
EQUIVALENT AIRSPEED
VMAX
EQUIVALENT AIRSPEED
V MAX
◮
Maximum speed is when THRUST = DRAG
◮
Acceleration is maximum at point of maximum excessive thrust i.e.
T −D
= V̇
m
◮
Jet typically has larger maximum excessive thrust and higher
intersection of thrust and drag curves