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this happens when
CDr " 3CDO (2.50)
CL :
(2.51)
AIRCRAFT PERFORMANCE
Let the value oflift coefficient when (C2/2] CD) iS maximum be denoted by CL,m.
Then,
CL,m =
=nCz
With tlus, it can be shown that
-.,), ax = -,
(2.52)
(2-53)
We observe that for glide with minimum sink rate, the induced :is three times
the zero-lift drag. The schematic variations of E and cC;.n~/)gwith angle of
attack are shown in Fig. 2.5.
The velocity Vm for glide with mirumum sink rateis given by
Vm -.
~ 0.76 'VR
(2.54)
From this relation, we observe that the velocity for the glide with minimum sink
rateis about 0*76 VR or 0.76 times the velocity for flattest glide.Whereas the flattest
glide occurs when the drag is minimum, the glide with minimum sink rate occurs
when the power required is a minimum. rfhese velocities are shown in Fig. 2.6.
CL
Jfig. 2.5 Variation of aerodynanuc parameters with lift coefficienL
78 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
., f*.
Minimum
Sirtk Rrte,
t mai
VR
F-I~ttest
GlIde,
R mai
v
mg. 2.6 Optimal rrelocities for gliding flight
The minimum sink rate is given by
hs.nrm = 4
(2.55)
Sailplane pilots fly at V = Vm when they are in the "lift" mode, i.e., when they
encounter an upward gust_ When the "lift ' dies, they accelerate to VR, the velocity
for flattest glide to cover the most ground while searching for another "lift:' An
instrument called "variometet' tells sailplane pilots whether they arein "lift" mode
or not.
The enduranceis the total time the glider remains in the air and can be determined
as follows:
so that
d
.= _l,~r
t = :Lhy
F(-,),h
(2,56)
(2.57)
Assurrung that the angle of attack is held constant during the glide and ignoring
the variations in density because of changes in altitude, we obtain
AIRCRAFT PERFORMANCE
79
If the difference between theinitial and final altitudeis significant, then the variation
in density may have to be considered. In such cases, the following approximate
equation may be used.
p = Poe-0'000114h
(2.59)
whiere Po is the density at sea level and h is the altitude in meters. Usually, Po -
1.225 kg/m3.
For maximum endurar, r has to fiy at that angle of attack or lift coeffi-
cient when the parame{c~n(C,2t/h/~hd) ~. haximum, which occurs when CL = .AcZ
and V -. 0.76 VR. Note that this is also the condition for minimum sink rate. rl'hus,
the endurance is maximum when the sink rate is minimum. Using Eq. (2.53), we
obtain
f-
rmax -
(h4 hf)
(2.60)
The range and endurance are important measures of a glider's performance.
Whereas the maximum range was independent of the weight, the maximum en-
durance depends on the weight. This calls for the designer to make the glider as
light as possible. Furthermore, both the maximum range and endurance improve
if the aerodynamic parameters k and CDO are kept to theirlowest possible values.
Because of this, gliders tend to have an elliptical wing with a high aspect ratio and
an efficient low-drag, laminar-fiow airfoil section.
Example 2.1
A glider having W -2000 N, S = 8.0 fl12, A - 16.0, e - 0.95, and CDO =
0.015 is launched from a height of 300 m. Determine the maximum range, corre-
sponding glide angle, forward velocity, and lift coefficient at sea level.
Solution. At sea level, we have pt, = 1.225 kg/m3. Furthermore,
k- 1
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