Basics of Fluid Mechanics, 2014a

520 CHAPTER 12. COMPRESSIBLE FLOW 2–DIMENSIONAL
It should be noted that, oblique shock becomes less “oblique” and more parallel when
other effects start to kick in.
12.3 Prandtl-Meyer Function
12.3.1 Introduction
As discussed in Chapter ?? when the deflection
turns to the opposite direction of the flow, the
flow accelerates to match the boundary condition.
The transition, as opposed to the oblique
shock, is smooth, without any jump in properties.
Here because of the tradition, the deflection
angle is denoted as a positive when it
is away from the flow (see Figure 12.21). In
a somewhat a similar concept to oblique shock
there exists a “detachment” point above which
this model breaks and another model has to
be implemented. Yet, when this model breaks
Flow
direction
positive
angle
maximum angle
Fig. -12.21. The definition of the angle for
the Prandtl–Meyer function.
down, the flow becomes complicated, flow separation occurs, and no known simple
model can describe the situation. As opposed to the oblique shock, there is no limitation
for the Prandtl-Meyer function to approach zero. Yet, for very small angles,
because of imperfections of the wall and the boundary layer, it has to be assumed to
be insignificant.
Supersonic expansion and isentropic compression
(Prandtl-Meyer function), are an extension of the Mach line
concept. The Mach line shows that a disturbance in a field
of supersonic flow moves in an angle of μ, which is defined
as (as shown in Figure 12.22)
μ = sin −1 ( 1
M
)
(12.63)
μ
c
M
1
μ
√
M 2 − 1
U
Fig. -12.22. The angles of
the Mach line triangle.
or
μ = tan −1 1
√
M
1
− 1
(12.64)
A Mach line results because of a small disturbance in the wall contour. This Mach line
is assumed to be a result of the positive angle. The reason that a “negative” angle is
not applicable is that the coalescing of the small Mach wave which results in a shock
wave. However, no shock is created from many small positive angles.
The Mach line is the chief line in the analysis because of the wall contour shape
information propagates along this line. Once the contour is changed, the flow direction
will change to fit the wall. This direction change results in a change of the flow
properties, and it is assumed here to be isotropic for a positive angle. This assumption,