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fiow is terminated by a shock wave as shown. As M~ increases further but still
below unity, the region of supersonic flow expands on the upper surface, and a
region of supersonic ffow ma"reven appear onYthe lower surface of the airfoil as
shown in Fig. 1.42d.
      rfhe formation of shock waves on the surface of the airfoilleads to flow separa-
tion, loss oflift  and increase in drag. Downstream of a shock wave, the pressure
is always higher. As a result, an adverse pressure gradient is impressed upon the
boundary lay.er, causing it to separate from the surface. Because the velocity rises
from zero at'the body surface to' the freestream value across the thickness of the
boundary layer, a part of the supersoruc boundary layer is always subsoruc, There-
fore, the formation of the adve"~ pressure gradient is communicated upstream of
a shock wave through the subsonic part of the boundary layer. As a result, the fiow
upstream of the shock wave is aware of the pressure gradient and may separate
  I
even ahead of the shock wave, causing a modrfication in the effecr:ive body shape,
hence a change in the shock wave structure. This process of mutual interaction
between the shock wave and the boundary layer is called shock-boundary layer
interaction.
     As a result of the shock-induced flow separation, the drag coefficient rises very
rapidly, and the lift coefficient drops. Another consequence of the shock-induced
 flow separation i,s the buffeting ofhorizontal tail. Buffeting is said to occur when the
 separated, unsteady, Oirbulent fiow from the wing surface passes over the horizontal
ta17"and causes the tail loads (lift, drag, and pitching moment) to fiuctuate. The
fiuctuating tailloads create problems of stability and control.
    In Figs. 1.43 and 1.44, the schematic variations of lift and drag coefficients
with Mach number at various angles of attack are shown. The lift coefficient
increases in the high subsonic Mach numbers as indicated by Eq. 1.59. This in-
crease continues slightly 'oeyond the critical Mach number because even though
shock waves are formed on the airfoil surface they may not be strong enough to
 cause flow separation. However, as the freestream Mach numberincreases further,
the shock waves becortle stronger and eventually cause flow separation. Around
this Mach number, the lift coefficient reaches its maximum value and then falls
off as shown in Fig.  1.43. This type of stall is sometimes caUed "shock stall:'
The nature and severit}r of the shock stall depends on the camber and thick-
ness ratio of the airfoil. The drop in the lift coefficient is more se'vere for highly
cambered and thicker airfoils because these airfoils have relatively higher local
velocities that lead to stronger shock waves and more severe adverse pressure
gradients.
     The fallin the lift coefficient is usually accompanied by a steep rise in the drag
coefficient as.shown in Fig.  1.44. The drag coefficient reaches a peak value around
Mach 1 and then starts dropping off when a clear supersonic fiow is established
on the entire sru:face of the airfoil as shown in Fig.  1.42e. When this happens, the
REVIEW OF BASIC AERODYNAMIC PRINCIPLES                  43
Cl
                  M
Fig. 1.43    Variation ofsectionallift coefficient at high speeds.
                 M
~g.1.44   Variation ofsectional drag coefficient at high speeds.
44                 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
cl
                Cd
Fig. 1.45     Schematic sketch of sectional drag polars at transonic speeds.
fiow is smooth and attached because shock waves that were formed on the upper
or lower surface of the airfoil and caused flow separation are now pushed towards
the trailing edge.
        To characterize the drag rise in high subsonic/transonic flow,it is usual to define
what is called the drag divergence Mach number Md as that value of freestream
Mach number when the drag coefficient begins to rise sharply as shown schemat-
ically in Fig. 1.44.
     The typical drag polars of an airfoil section in the transonic Mach number range
　
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