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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
are schematically shown in Fig. 1.45. The graph in which the lift coefficient is
plotted against the drag coe:fficient is called the drag polar. As the Mach number
increases, the drag polars shift to the right and bend forward because at a given
lift coefficient the drag coefficient will be higher as the Mach number increases.
For a conventional Iow-speed airfoil with a well-rounded leading edge, a de-
tached bow shock wave is formed when the freestream Mach number is in the
supersonic range (Fig. 1.42e). Because of symmetry, a detached shock wave is
normal to the fiow direction on the line of symmetry. Because the fiow behind the
normal shock wa've is always subsonic, there is a small patch of subsonic fiow
around the leading edge region. The subsonic fiow quickly accelerates to super-
sonic fiow over the airfoil surface. Somewhere on the upper and lower surfaces, we
have sonic lines as shown, The fiow over this type of airfoil is one of mixed sub-
sonic and supersonic fiow. Furthermore, the wave drag is high because a detached
bow shock wave is formed.
One way of reducing the wave drag of the wing in supersonic fiow and avoiding
the formation of a detached bow shock wave ahead of the airfoil is to use a sharp
leading-edge airfoillike the diamond section ofFig. 1.40. An attached shock wave
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