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nitude of the pressure coe:fficient at any point on the airfoil surface and the section
lift-curve slope increases steadily with Mach number above their incompressible
values. These formulas are applicable as long as the flow ever)rwhere is subsonic
and shock free.
REVIEW OF BASIC AERODYNAMIC PRINCIPLES                  41
~
 M~ < Mcr
          -
M =Mcr
--
M~ > Mcr
a) Subsoruc flow
<;==::: A--J(-:
b) Local soiuc flow
c) Supersonic flow on upper surface
      -
M >Mcr
Wake
d) Supersonic flow on upper and lower surfaces
e) Airfoilin supersonic ftow
Fig.1.42   Flow over airfoil atlugh speeds.
     When the local Mach number at point P  on the airfoil surface reaches the value
of uruty, the corresponding freestream Mach number is called the critical Mach
number and is denoted by Mu  as schematically illustrated in Fig.  1.42b. Elsewhere
on the surface of the airfoil, the local Mach number is below unity, and the flow
is subsonic. The value of Mcr depends on the geometrical shape of the body and
the angle of attack. For a given airfoil, the critical Mach number usually decreases
with increase in angle of attack.
    The typical values of Mcr for airfoils at zero-lift lie in the range 0.6-0.85. For
example, at Ci  = 0, the values of Mcr for NACA 2412, NACA 23012, and NACA
653-418 airfoils are, respectively, equal to 0.69, 0.672, and 0.656. For a thin airfoil
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42               PERFORMANCE, STABILITY DYNAMICS,AND CONTROL
like NACA 0006, the a:itical Mach numberif 0.805 at Ci - 0. The values of Mc.r
for other airfoils may be found elsewhere.3,11
       As the freestream Mach number increases beyond Mcr, the local velocity on the
surface of the airfoil exceeds the sonic velocity at more than one point  In fact, a
 small region appears where thelocal Mach number is either equal to or greater than
unity as shown in Fig. 1.42c. Because the flow Mach num~er behind the airfoil
has to be equal to the freestream value, which is subsonic, the region of supersonic
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
　
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