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 that the power developed by the reciprocating engine and propulsive efficiency are
 independent of the fiight velocity. Therefore, for propeller aircraft,itis a common
practice to obtain graphical or numerical solutions to deternune the two speeds in
levelflight 14    and 14na
    To ,o.struct a level flight envelope, we have to obtain this type of graphical
 or numerical solutions at several altitudes. However, this task can be simplified if
 we use the equivalent air speed Ve. which is related to the true air speed by the
following relation:
Ve = V.~
where o  = P/Po is the density ratio and Po is the air density at, sea level.
     7Vith this we can rewrite Eq. (2.67) as follows:
D = ~Po Ve2SCDO +
2k W2
,OO \re2 S
(2.83)
(2.84)
The advantage of introducing the equivalent air speed is obvious. We have only
one drag curye given by Eq. (2.84) that holds for all altitudes. Now, on this drag
curve, let us superpose the thrust~available curves at various altitudes as shown
in Fig. 2.11. The thrust available drops as the altitude increases because of a falJ
in air density. The thrust-available curve intersects the drag curve at two points
designated as Ve,nun cl11(1 Ve,max. At these two intersection points, the level fiight
AIRCRAFT PERFORMANCE
Ve,mi ,,h = 0
          Ve
olutt
ing
Vt ,nnai,h= O
Fig. 2.11     Variahon of maxinmm and mirumum speeds in level flight.
87
Eqs. (2.61) and (2.62) are identically satisfied, and, therefore, each point is a level
fiight solution.
      The schematic variation of Ve.rm i and  Ve,miax with altitude is shown in Fig. 2.12a.
Ifwe plot the true air speeds corresponding to Ve,   n and Ve,max, we obtain the level
fiight envelope as shown in Fig. 2.12b. It is interesting to observe that the true air
speed corresponding to high-speed solution increases initially with altitude and
then begins to decrease, whereas the true air speed corresponding to the low-
speed solution increases monotonically with altitude. At a certain altitude, the two
solutions merge, and we have only one level fiight solution. This altitude is c&lled
the absolute ceiling ofthe airplane. Note that the absolute ceiling is also the altitude
where the thrust-available curve is tangential to the thrust-required (drag) curve.
In other words, at absolute ceiling, the thrust available has dropped so much that
the level flight is possible only at one speed. This speed happens to be the speed at
which the drag is minimum. Also, at absolute ceiling, the rate of climb will be zero
as we will see later. Beyond the absolute ceiling, steady level flight is not possible
because thrust available is not sufficient to balance the aerodynamic drag.
    The velocity corresponding to the high speed solution increases initially with
altitude because at altitudes close to the sea level, the drop in thrust available is very
gradual. As a result,  Vemax, even though decreasing, is still quite close to its value
at sea level. On the other hand, the density ratio a drops at a higher rate so that the
true air speed, which is the ratio of two decreasing quantities with the denominator
decreasing faster than the numerator, increases initially with altitude. At some
altitude, the true air speed attains a maximum value and then starts dropping when
this trend reverses.
2-4.2   AnalyticaJ Solutions for Jet Aircraft
   If we assume that at a given altitude, the thrust developed by a jet engine is
independent of flight velocity, then we can obtain analytical solution for level
flight as discussed in the following.
:g
'f:V
-/:,
'/z.i
:.~c
' -.
I::fi
;{}
136            PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
Far shallow glide angles, sin y .~ Y and cos y N 1, so that
          1
y = E~os~
1V _
 n  rr,CD
=E= CL
          1
n-~
             cos /r
   '{/2
tan br =  Rg
   \/2
R--
         g tan y
(2.291)
(2.292)
(2.293)
(2.294)
(2.295)
  V  gtanp, gr  -f
co = R = ~ = -  V                          (2.296)
　
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