Basics of Fluid Mechanics, 2014a

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12.7. FLAT BODY WITH AN ANGLE OF ATTACK 527
In ideal inviscid incompressible flows, the
movement of body does not encounter any resistance.
This result is known as d’Alembert’s
Paradox, and this paradox is examined here.
Supposed that a two–dimensional diamond–
shape body is stationed in a supersonic flow as
shown in Figure (12.27). Again, it is assumed
that the fluid is inviscid. The net force in flow
direction, the drag, is
w
)
D =2(
2 (P 2 − P 4 ) = w (P 2 − P 4 )
(12.100)
1 2
θ 1 Wθ 2
θ 1 θ 2
Fig. -12.27. A simplified diamond shape
to illustrate the supersonic d’Alembert’s
Paradox.
It can be observed that only the area that “seems” to be by the flow was used
in expressing equation (12.100). The relation between P 2 and P 4 is such that the flow
depends on the upstream Mach number, M 1 , and the specific heat, k. Regardless in the
equation of the state of the gas, the pressure at zone 2, P 2 , is larger than the pressure
at zone 4, P 4 . Thus, there is always drag when the flow is supersonic which depends on
the upstream Mach number, M 1 , specific heat, k, and the “visible” area of the object.
This drag is known in the literature as (shock) wave drag.
12.7 Flat Body with an Angle of Attack
Previously, the thickness of a body was shown to
have a drag. Now, a body with zero thickness but
with an angle of attack will be examined. As opposed
to the thickness of the body, in addition to
the drag, the body also obtains lift. Again, the
slip condition is such that the pressure in region 5
and 7 are the same, and additionally the direction
of the velocity must be the same. As before, the
magnitude of the velocity will be different between
the two regions.
12.8 Examples For Prandtl–Meyer Function
1
4
2
3
l
α
e
3
3
4
4
Slip plane
Fig. -12.28. The definition of attack
angle for the Prandtl–Meyer function.
6
7
5