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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
;{}
88               PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
We have
m
h
    v*
a) Eqruvalent air speed
          v
             b) True air speed
Fig.2.12 Levelftighterrvelope.
    T-D
                                                 1
                                   = ~p V2S(CDO +kC~.)
                                                 1
                                 = ~p V2SCDO + (jk,_WS
Equation (2.85) can bc written as follows:
                                      A/r4 _ Tlr2+ B = O
Where
                                   = ~pSCDO
                              .  B = (2kpVg.)
(2.85)
(2.86)
(2.87)
(2.88)
AIRCRAFT PERFORMANCE
SoMng Eq, (2.86), we obtain
Vmax =
Vmin =
~]
(2.89)
(2.90)
Hcywever, a more elegant form of analytical solution can be obtained using the
nondimensional. form ofEqs. (2.27) and (2.28). With y = 0, Eqs. (2.27)'and (2.28)
take the following form,
Then,
U4 _ 2zu2 +1 = 0
 n:1
ILmax = ~
Umin = V
       Vmax - llmax VR
         Vmin -. ILmin VR
(2.91)
(2.92)
(2.93)
(2.94)
(2.95)
(2.96)
(2.97)
Given the variation of thrust available with altitude, using this approach, we can
construct the level fiight envelope for a jet aircraft whose thrust is independent of
flight velocity. At the absolute ceiling, Unun - Umax, which gives z = 1. Physically,
when z - 1, T - Dmn. As said before, at absolute ceiling, level fiight is possible
only at only one speed where the drag is minimum.
   Whether the flight is possible at Vmin depends on the magnitude of the level
flight stalling velocity, which is given by
Vstall -
(2.98)
If Vmm < VstaU, then the airplane cannot fly at Vmin in steady level flight because
it asks for a lift coefficient in excess of CL,max. For such cases, the level flight
solution  Vnun has no physical significance, and  VstaU effectively 'oecomes the low-
speed solution.
      To flyinlevelflight at any speed at aspeed  V  such that V             V  <  'V~x, the pilot
has to throttle the engine so that the drag at this speed equals the thrust available
as schematically shown in Fig. 2.13.
90                  PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
　
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