Minimum Drag Speed
Prof. Dr. Mustafa Cavcar
School of Civil Aviation
Anadolu University
Eskisehir, Turkey
[email protected]
Remember that the drag force acting on an aircraft
D = CD
ρ
2
V 2S
(1)
where C D is the drag coefficient, ρ is the air density, V is the airspeed and S is
the wing area of the aircraft. The drag coefficient can be expressed as a simple
parabolic drag polar
C D = C D0 + KC L2
(2)
where C D0 is the parasite drag coefficient, K is the induced drag coefficient, and
C L is the lift coefficient. The lift force acting on the aircraft
L = CL
ρ
2
V 2 S = nW
(3)
where n is the load factor and W is the aircraft weight. Thus, the lift coefficient
becomes
CL =
2nW
ρV 2 S
(4)
In case of steady level flight, n = 1 , and
CL =
2W
ρV 2 S
(5)
From Equations (2) and (5), the drag coefficient of aircraft becomes
C D = C D0 +
4 KW 2
ρ 2V 4 S 2
(6)
and, from Equations (2) and (6), the drag of aircraft becomes
© Prof. Dr. Mustafa Cavcar, 2004
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D = C D0
ρ
2
V 2S +
2 KW 2
ρV 2 S
(7)
Equation (7) results in a drag versus airspeed plot as shown in Figure 1. Because
Equation (7) is also dependent to the aircraft weight, for the different weights of
same type of aircraft, different drag versus airspeed plots are obtained as shown in
Figure 1. As it is seen from Figure 1, for a given weight, the drag of aircraft
reaches to a minimum at an airspeed. This airspeed is called “minimum drag
speed,” Vmd . The minimum drag speed always has a higher value than the stall
speed.
Figure 1 Drag versus airspeed for different aircraft weights.
Because dD / dV = 0 at the minimum drag airspeed, its value can be found from
Equation (7) by
dD
ρ
4 KW 2
= 2C D0 VS −
=0
dV
2
ρV 3 S
(8)
From Equation (8)
Vmd
 K
=
 CD
 0




1/ 4
2W
ρS
© Prof. Dr. Mustafa Cavcar, 2004
(9)
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From Equation (9) it is obvious that Vmd varies with the aircraft weight and
altitude. This is also shown by Figure 1. At higher weights, such as the take-off
weight, minimum drag speed is higher, and at lower weights, such as the landing
weight, minimum drag speed is lower.
As the stall speed is a reference speed for take-off and landing performance of an
aircraft, the minimum drag speed is a reference speed for other phases of the
flight mission such as climb, cruise, and descent. Maximum climb angle,
maximum endurance and best power-off glide angle are achieved at Vmd .
However, due to speed stability reasons aircraft have always flown at airspeeds
slightly above Vmd .
Figure 2 Speed stability.
Speed Stability [1]
Vmd divides the region of normal command and region of reverse command.
Consider a V2 airspeed, above Vmd , in the normal command region as in Figure 2.
Imagine that, without changing altitude or thrust, the speed increases as indicated
by the dotted line labeled D. Now the drag exceeds the thrust. Hence, if level
flight is maintained, the aircraft slows down (decelerates) to the original speed V2 ,
since T = D for steady level flight. Suppose that the speed decreases, as
represented by C. Here, the thrust exceeds the drag. Thus, if level flight is
maintained, the aircraft speeds up until the thrust again equals the drag as
represented by V2 . The aircraft exhibits speed stability, i.e., if a small change in
© Prof. Dr. Mustafa Cavcar, 2004
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speed either above or below V2 occurs, the change is `damped out' and the aircraft
returns to the original speed.
Now look at the slower speed, V1 , below Vmd , in the reverse command region
Again, consider that the speed increases by a small amount as represented by the
dotted line labeled B. Notice that now, because the thrust exceeds the drag, the
aircraft accelerates away from the original speed. Point 1 is said to exhibit speed
instability. In fact, as the aircraft accelerates, the thrust continues to exceed the
drag and the aircraft continues to accelerate in level flight all the way to V2 ,
where the thrust again just equals the drag.
However, consider the decrease in speed from V1 represented by A in Figure 2.
Here the drag exceeds the thrust and, if level flight is to be maintained with
constant thrust, so that T = D , the aircraft must slow down (decelerate). But,
when the aircraft decelerates the drag to maintain level flight increases and the
thrust decreases, and the aircraft continues to slow down. What this means is that
level flight cannot be maintained at the slower airspeed unless the thrust is
increased. If level flight is to be maintained below Vmd , then the thrust must be
increased, i.e., in the reverse command region, to fly slower thrust must be
increased. Due to this fact, airspeeds below Vmd is called the reverse command
region. If thrust is not increased, the aircraft descends or alternatively may
eventually depart controlled flight.
Air Traffic Control (ATC) and Minimum Drag Speed
Figure 3 shows clean configuration minimum drag speeds of various types of
commercial jet aircraft based on Eurocontrol’s Base of Aircraft Data (BADA)
Revision 3.6. It is seen that Vmd varies between 160 and 300 KIAS for maximum
gross take-off weight, depending on the aircraft type. For the minimum weights of
same aircraft types, Vmd varies between 130 and 200 KIAS. Usually none of the
aircraft descents or approaches with their minimum weight, thus even during
descent aircraft will be heavier than their minimum weight, and consequently
their clean configuration minimum drag speeds will be higher than 130 to 200
KCAS. Therefore, when airspeed reduction is desired by ATC for approach radar
sequencing, special attention should be paid in asking speed reduction.
Flight below the clean configuration minimum drag speed requires extension of
the flaps in order to avoid stalling of the aircraft. However, although flaps help
sufficient lift to be generated, they cause extra drag, so that extra fuel
consumption. Therefore, pilots will avoid using flaps in order to fly with reduced
speed to prevent excessive fuel consumption, if the aircraft is still distant enough
from the aerodrome.
© Prof. Dr. Mustafa Cavcar, 2004
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Dumaitre [2] proposes the use of 220 KIAS as the speed for approach radar
sequencing. If Figure 3 is considered, this speed well fits to almost all types of jet
transport aircraft. In addition to fuel consumption advantages, there are other
advantages of flight at 220 KIAS [2]:
. At this speed, the aircraft can re-accelerate easily, if it is desired. When flaps are
extended, acceleration of the aircraft becomes slower.
. At 220 KIAS, all jet aircraft can lower their flaps. Therefore, speed reduction
can be achieved easily if it is desired.
. The turn radius at this speed is small enough, so that the final phases of
sequencing are easier.
. Many types of aircraft, i.e. all jets and most turboprops, are capable of flying at
this speed. Therefore, an homogenous traffic flow is possible.
Figure 3 Minimum drag speed for various jet transport aircraft.
References
[1]
[2]
Rogers, D.F., Speed Stability, NAR Associates, Annapolis, 2001.
Dumaitre, P., Air Traffic Control and Operating techniques, The air traffic
controller and aircraft performance on approach, Ecole Nationale de
l’Aviation Civile, Toulouse, 1991.
© Prof. Dr. Mustafa Cavcar, 2004
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