AE 429 - Aircraft Performance and
Flight Mechanics
Level Turn, Pull Up
and Pull Down
Turning Performance
What is a turn?
∆Ψ
R
r
Center of curvature
–
–
a turn is a change in flight path direction
turn rate is the time rate of change in heading
Ψ ≡ Lim
∆Ψ
∆t → 0 ∆t
1
Turning Performance
More definitions
–
Turn radius, R, is the distance between the flight
path and the instantaneous center of curvature
Lcos φ
Load factor and turn radius
–
Lsin φ
L
φ
Load factor n is defined as
n≡
L
W
mV∞2 / R
R
In a level, un-accelerated turn
W = L cos φ
L
1
=
n≡
W cos φ
– N is a function of φ (bank angle)
only in a steady, level turn
–
cos φ = W / L = 1/( L / W ) = 1/ n
W
φ = arccos(1/ n)
for level turn,
constant altitude
L cos φ = W
Perpendicular to flight path in the
horizontal plane
r2 = R
1
m
(V∞ cosθ )
2
r2
= L sin φ + T sin ε sin φ
0
Performance parameters:
Turn radius R
Turn rate ω =
ψ
dψ / dt
local angular velocity along the curved
flight path
Larger the magnitude of Fr :
tighter and faster will be the turn
Note: L and φ are not independent in level turn
2
Turn Radius
(V )
m ∞
2
= L sin φ
R
R=m
V∞2
W V∞2
1 V∞2
=
=
L sin φ L g sin φ n g sin φ
cos φ = 1/ n
cos2 φ + sin 2 φ = 1
1/ n2 + sin 2 φ = 1
sin 2 φ = 1 − 1/ n2
sin φ = 1 − 1/ n2
V∞2
V∞2
1
=
R=
n g 1 − 1/ n2 g n2 − 1
Small R
Turn Rate
R=
ω = dψ / dt = V∞ / R
V∞2
ω=
g n2 − 1
high ω
high n (large L/W)
low Velocity
dψ V∞ g n2 − 1
=
=
dt
R
V∞
high n (large L/W)
low Velocity
High Performance: smallest R and largest ω for largest n ; lowers speed V
what is the higher possible n ?
R and ω are function of n and V
φ
L
n=
1
cos φ
Level turn:
nmax
D
TR
but T < Tmax A implying that for Tmax A
nmax =
1
1
=
cos φmax cos φTA max
L = n W = ½ ρ V2 S CL
D = T;
1
ρV 2 S
T
2
=
K (W / S ) W
Do not depend on W/S, T/W, k, Cd0, ρ
L/D = n W/T
−
max
CD0
1
ρV 2
2
W /S
1/ 2
=
L T
DW
T=
1
2nW
ρV 2 S CD0 + K
2
ρV 2 S
1 ≤ n ≤ nmax
max
φTmax A
nmax =
2
C
1
ρV 2 L max
2
W /S
3
Minimum Turn Radius
Minimum turn radius
–
Stall speed in straight and level flight (L = W) is
Vs =
–
2W
ρ∞SCLmax
In a level turn, stall speed becomes (L=nW)
2 nW
ρ∞SCLmax
Vsturn =
Vsturn = Vs n
–
Which suggests that
–
Replacing Vs with Vsturn in the turn radius equation gives
the aerodynamic limit on minimum turn radius
Rmin =
Vs2
turn
g n2 − 1
=
Vs2 n
g n2 − 1
=
Vs2
1
g 1− 2
n
Level Turn Chart
4
Pull-Up
Consider a turn in the vertical plane: winglevel Pull-Up (instantaneous turn)
–
–
Different from level turn (constant flight properties)
The radial forces are:
at t = 0; θ = 0
Fr = L − W = W ( n − 1)
R
V 2 W V∞2
=m ∞ =
R
g R
–
Solving for R:
R=
–
θ
L>W
V∞2
g (n − 1)
θ=0
L=W
And for turn rate:
W
g ( n − 1)
V
ω= ∞ =
R
V∞
W
Pull-Down
Now, look at another instantaneous turning maneuver in the
vertical plane -- a “split s”
–
Using the same approach as
for a Pull-Up
at t = 0; θ = 0
m
V∞2
= L +W
R
R=
ω=
–
R=m
V∞2
L +W
W
V∞2
g (n + 1)
g (n + 1)
V∞
The rate is improved
and the radius is enlarged
over pull-ups
L
R
θ
5
Limiting cases: n large
Effect of W/S (wing loading) and Clmax
–
When n is large,
n +1 ≈ n −1 ≈ n
V∞2 =
V2
gn
R ≈ ∞ , ω≈
gn
V∞
2L
ρ∞SCL
–
Recalling that
–
Substituting, we obtain radius and rate of turn
R=
2L
ρ∞SCL g (L / W )
ω=
gn
2L
ρ∞SCL
R=
2 W
ρ∞CL g S
ω=g
ρ∞CL n
2W
S
( )
For minimum turn radius and maximum turn rate
–
Maximize both CL and load factor
Rmin =
–
2
W
ρ∞ gCLmax S
ωmax = g
ρ∞CLmax nmax
2W
S
( )
Practical constraints on load factor
–
nmax is a function of CLmax;
–
at low speeds it will be limited by the aerodynamic lifting capability
(stall) of the lifting surfaces
nmax =
–
–
CL
1
ρ∞V∞2 max
W
2
S
at high speeds, structural loads on the airframe may also limit nmax
for many airplanes, the other force balance (T = D) governs the
minimum turn radius and the maximum turn rate -- turn
performance is limited by available thrust
6
constraints on V
V as small as possible for Rmin and ωmax
–
L = n W = ½ ρ V2 S CL
V∞ =
–
R=
2nW
VStall =
Rmin does not necessarily correspond to nmax
V∞2
g n2 − 1
Rmin =
CL = CL max
ρ∞ SCL
=
2q∞
g ρ∞ n 2 − 1
∂R
=0
∂q∞
2nW
ρ∞ SCL max
1 ≤ n ≤ nmax
4k (W / S )
g ρ ∞ (T / W ) 1 − 4kCDo /(T / W )2
ωmax = q∞ ρ ∞ /(W / S )[(T / W ) /(2k ) − (CDo / k )1/ 2 ]
nR min = 2 − 4kCDo /(T / W )2
V∞ R min = 4k (W / S ) /( ρ∞ (T / W ))
V-n diagram
the V-n diagram illustrates 2 of these constraints
CL max
Corner Velocity or
maneuver velocity
V* =
2nmax W
ρ∞ CL max S
7
Aerodynamic and structural limits on turn
performance
Aerodynamic and thrust limits on turn
performance
Aerodynamics
Wing
Design
nmax =
CL
1
ρ∞V∞2 max
W
2
S
Thrust Available
Drag
Structural
(Materials/
Wing Size)
8