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have an upward force (FN) acting on the airfoil. The component of this upward
force in the direction perpendicular to the freestream is the lift /, and that a~ong
the freestream direction D is the wave drag.
For a double wedge airfoil,
Cdw -
4
Cl -
4
[,2+( )2]
(1-55)
(1.56)
(1.57)
Here, the wave drag has two components, one caused by thickness distribution
and the other caused by angle of attack. For a thin fiat plate, t/c = 0 and he,nce
Cdw = O at a = 0.
A finite wing in supersonic :flow behaves in a different manner compared to
that in a subsonic flow. For subsonic fiow, the effects of the wing tips are felt all
over the wing surface. However,in supersonic flow, the effect of the wing tips are
confined to the Mach cones emanaOng from the leading edges of the tip chord as
shown in Fig. 1.41. Influences of the wing tips AD and BC are confined to the
regions ADF and BCE. The rest of the wing ABE F is not aware of the wing tips
and functions as though it were part of a two-dimensional wing.lf the aspect ratio
is sufficiently high, then the region of the wing falling within these Mach cones is
quite small, and the entire wing can be assumed to behave like a two-dimensional.
wing.
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40 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
Fig.1.41 Finite wingsin supersonic flow.
1.11 Critical Mach Number
So far we have discussed the flow over an airfoil at low subsonic or supersoruc
speeds, assuming that the flow is either completely subsonic or completely su-
personic. However, if the freestream Mach number is in the high subsonic or low
supersoruc range, then fiow 1ield around the body may consist of mixed subsonic
and supersonic flow regions. When the freestream Mach number is in the range
of 0.8-1.2, the fiow is said to be transonic;fo understand this type of complex
fiow field, let us study the fiow over an airfoil held at a constant positive angle of
attack when the freestream Mach numberincreases from a low subsonic to a high
subsonic or transonic value.
For a given freestream Mach number, we will have some pointlike P on the top
surface of the airfoil section, where the local velocity is maximum (Fig. 1.42a).
At this point, the local velocity will continuously increase as the freestream Mach
number increases. No drastic changes in the nature of the flow take place as long as
the local fiow everywhere on the body surface is subsoruc. For this range of Mach
numbers, the pressure coefficient and lift-curve slopes are given by the well-known
Prandtl-Glauert rule,
Cp,c= CP,i . (1.58)
,
ao.,
ao,c - -- (1.59)
M~
where Cp is the pressure coefficient and ao is the lift-curve slope of the airfoil.
The suffixes r and c denote incompressible and compressible values, respectively.
It may be noted that the pressure coefficient is defined as
p - Pea
Cp = 1 2p.tl-2 (1.60)
According to Eqs. (1.58-1,60), which are based on potential fiow theory, the mag-
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