Page 284 - 48Fundamentals of Compressible Fluid Mechanics

246 CHAPTER 14. PRANDTL-MEYER FUNCTION

or
(14.2)

A Mach line results of a small disturbance of the wall contour is discussed here.

This Mach line is assumed to be results of positive angle. The reasons that “nega-
tive” angle is not applicable is because coalescing of small Mach wave results in a

shock wave. However, no shock is created for many small positive angles.
The reason that Mach line is the chief line in the analysis because this line
is the line on which the information of the shape of contour of the wall propa-
gates. Once, the contour is changed the direction of the flow changes to fit the
wall. This change results in a change of the flow properties and is assumed here
to be isotropic for a positive angle. This assumption turned out to be not far way
from realty. In this chapter a discussion on the relationship between the flow prop-
erties and the flow direction is presented.

14.2 Geometrical Explanation

The change in the flow direction is results
of the change in the tangential component. x dx = dU y cos(90 − µ)
Hence, the total Mach number increases.
Therefore, the Mach angle results is in- ! "#%$'&(
dν
crease and a change in the direction of the ) dy
flow appears. The velocity component at
direction of the Mach line assumed to be Mach line
constant to satisfy the assumption that the
change is results of the contour only. Later,
Fig. 14.3: The schematic of the turning
this assumption will be examined. This
flow
change results in the change in the direc-
tion of the flow. The typical simplifications
for geometrical functions are used
(14.3)

* '*
These simplifications are the core why the change occurs only in the perpendicular
'*

is

direction (*
(14.4)

, is

* " "
" "
). The change of the velocity in the flow direction,
(14.5)
Also in the same manner the velocity in perpendicular to the flow,,+

-+
" " * *
The (see Figure (14.3)) "

(14.6)

"

is the ratio of,+
,+ '*
"