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bend well ahead of the body and smoothly flow past it as shown in Fig. 1.38a.
However, in supersonic flow, a shock wave is formed at the leading edge'of the
body as shown in Fig. 1.38b, and the fluid particles experience a sudden rise in
pressure, temperature, and density on crossing the shock wave. The Mach number
downstream of the shock wave is always smaller than the upstream Mach number.
The shock wave is attached to the leading edge if the lead:;g edge is sufficiently
sharp and the Mach number is much greater than unity.lf the leading edge is blunt
or the Mach number is close to unity or both, then a detached bow shock wave is
formed ahead of the body as shown in Fig. 1.38c.
When a supersonic flow goes around a corner that turns away from the fiow
direction, then an expansion fan originates at the corner as shown in Fig. 1.39a.
In contrast to a shock wave, the temperature, pressure, and density change gradu-
ally across an expansion fan. On crossing an expansion wave, the Mach number
increases, and pressure and temperature decrease. Such a corner is called an ex-
pansion corner. On the other hand, if the corner tums into the flow (Fig. 1.39b),
a shock wave is formed at the corner, and the pressure, temperature, and density
increase on crossing the shock wave. Such a corner is called a compression corner.
Consider supersonic flow past a sharp symmetrical double wedge airfoil at zero
angle of attack as shown in Fig. 1.40a. At the leading edge, an attached shock
wave is formed. As a result, pressures on surfaces AB and AD are higher than
freestream pressure P{x, because of a compression produced by the shock wave. At
B and D, expansion fans are formed. The fiow undergoes a gradual expansion and
pressures on surfaces BC and DC fall below the freestream value p~. Because
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38 PERFORMANCE, STABfLJTC DYNAMJCS, AND CONTROL
Oblique Shock
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a) Expansion corner
b) Compression corner
Fan
mg 139 Supersonicllow around corners.
Expansion Wavc
A
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Oblique Shock Wave
b)a>0
Fig.1.40 Superscuuc flow over double wedge airfoil.
REVIEW OF BASIC AERODYNAMIC PRINCIPLES
39
pressures on AB and AD are equal, the net upward force or lift is zero. But the
axial components do not cancel, and we have a nonzero force in the axial direction.
Similarly, the component of force in the direction normal to the flow on surfraces
BC and DC cancel, but the axial components add up to give a nonzero force in. the
axial direction. The two nonzero axial force components add up to a net nonzero
force in the flow direction, which is called wave drag. The wave drag is solely
caused by the compressible nature of the fiuid. Therefore, in supersonic flow', the
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
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