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 wave drag is an addY;tional component of the drag. Thus, for a given body, the drag
 m supersonic flow is usually much higher than thatin subsonic flow. The wave drag
depends on the geometrical shape and thickness ratio of the body and freestream
Mach number.
   Now consider the sharp double wedge airfoil to be at a small angle of attack
(at .< 0) as shown in Fig.  1.40b. The oblique shock waves AA' and AA", formed
respectively on the upper and lower surfaces, are not of equal strength. The reason
for this difference is 'S:at the fiow turning angles on the top and bottom surfaces
are not equal. On the top side AB, it is equal to ar _ 0, whereas on the bol.tom
side AD it is equal to a- +0. Thus, the pressure on AD is higher than that on AB.
Similarly, the strengths of the expansion fans originating at B and D are diffe.rent.
because Mach :rrumbers M" and Mi are different. Actually, Mu > Mt. Therefore,
the pressure on BC is lower compared to that on DC. The net result is that we
                                                          I
have an upward force (FN) acting on the airfoil. The component of this upward
force in the direction perpendicular to the freestream is the lift /, and that a~ong
the freestream direction D is the wave drag.
    For a double wedge airfoil,
Cdw -
4
Cl -
4
[,2+( )2]
(1-55)
(1.56)
(1.57)
Here, the wave drag has two components, one caused by thickness distribution
and the other caused by angle of attack. For a thin fiat plate, t/c = 0 and he,nce
Cdw = O at a = 0.
   A finite wing in supersonic :flow behaves in a different manner compared to
that in a subsonic flow. For subsonic fiow, the effects of the wing tips are felt all
over the wing surface. However,in supersonic flow, the effect of the wing tips are
confined to the Mach cones emanaOng from the leading edges of the tip chord as
shown in Fig. 1.41. Influences of the wing tips AD and BC are confined to the
regions ADF and BCE. The rest of the wing ABE F is not aware of the wing tips
and functions as though it were part of a two-dimensional wing.lf the aspect ratio
is sufficiently high, then the region of the wing falling within these Mach cones is
quite small, and the entire wing can be assumed to behave like a two-dimensional.
wing.
_-
M >I
 poo
40                 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
Fig.1.41   Finite wingsin supersonic flow.
 1.11    Critical Mach Number
      So far we have discussed the flow over an airfoil at low subsonic or supersoruc
 speeds, assuming that the flow is either completely subsonic or completely su-
  personic. However,  if the freestream Mach number is in the high subsonic or low
 supersoruc range, then fiow 1ield around the body may consist of mixed subsonic
 and supersonic flow regions. When the freestream Mach number is in the range
 of 0.8-1.2, the fiow is said to be transonic;fo understand this type of complex
 fiow field, let us study the fiow over an airfoil held at a constant positive angle of
 attack when the freestream Mach numberincreases from a low subsonic to a high
 subsonic or transonic value.
      For a given freestream Mach number, we will have some pointlike P on the top
 surface of the airfoil section, where the local velocity is maximum (Fig. 1.42a).
 At this point, the local velocity will continuously increase as the freestream Mach
 number increases. No drastic changes in the nature of the flow take place as long as
 the local fiow everywhere on the body surface is subsoruc. For this range of Mach
 numbers, the pressure coefficient and lift-curve slopes are given by the well-known
Prandtl-Glauert rule,
         Cp,c= CP,i      .  (1.58)
                                                  ,
              ao.,
　
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