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Fig.2.7 Airplaneinlevelflight
. Line
FdDO
1/ k
W
11 0 02
82 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
can be stated in simple terms as the lift is equal to weight and thrust available is
equal to thrust required. With L - W, the load factor n -. 1 for level flight.
The kinematic equations as given by Eqs. (2.9) and (2.10) assume the form
x - V (2.63)
h-0
(2.64)
A solution ofEqs. (2.61) and (2.62)'gives the velocity V and angle of attack or the
lift coefficient CL for steady, unaccelerated level fiight.
The drag of an airplane in level fiight has an interesting variation with flight
velocity. To understand this,let us proceed as follows:
With L - W, we have
Then,
D = r~p V2S(CDO +kC2)
C~ = p2WS
D = gpl/2S rCDO +
.
4k Wz
p2 \/4S2
2k W2)
= gp V2SCDO + p~S)
(2.65)
(2.66)
(2.67)
The first term on the right-hand side of Eq. (2.67) is the zero-lift drag Do and the
second term is the induced drag Dr. The zero-lift drag varies directly as the square
of the velocity, whereas the induced drag varies inversely with the square of the ve-
locity. The variation of these two drag components and the total dravguwith velocity
are shown in Fig. 2.8. At low speeds, the induced drag is dominant because the lift
coefficient needed to sustain level flight is quite high. As the velocit3r increases,
the zero-lift drag rises very rapidly, but the induced drag becomes insignificant,
At low subsoruc'speeds, the zero-lift drag coefficient CDO can be assumed
constant with respect to the velocity. However, at high subsonic, transonic, and
supersonic speeds, the zero-lift drag coefficient varies with flight speed (Mach
number) because it includes the wave drag component. This variation must be
taken into account.
Because of opposite trends of zero-lift drag and induced drag, the total drag
assumes a minimum value at a certain velocity. The speed at which the total drag
is a minimum can be obtained by differentiating Eq. (2.67) with respect to the
velocity and equating the resulting expression to zero to obtain
4kS% =0
p VSCDO - p S~
V:
(2.68)
(2.69)
It may be observed that this velocityis equal to the reference velocity VR introduced
earlier in Eq. (2.15) for the definition of nondimensional flight speed u. Thus, the
Dmin
AIRCRAFT PERFORMANCE
VFI
Flg. 2.8 Variation of drag components with yelocity.
83
reference velocity happens to be the velocity at which the drag is n:ummum in a
steady, unaccelerated le'vel fiight.
Substituting the expression for velocity from Eq. (2.69) in Eq. (2.67), we obtain
Do - Di
=W -
Dnun - 2Do
- 2Dr
= 2w.j -
(2.70)
(2.71)
-fhus, we observe that when the total drag in level fiight is minimum, the zero-lift
drag equals the induced drag.
The power required in level flight is given by
PR - DV
1( p V3SCDO + (2Lk}_Ws )
= 2p
(2.72)
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