Page 202 - 48Fundamentals of Compressible Fluid Mechanics

164 CHAPTER 9. FANNO FLOW

')(+*-,
0 .0/2143 5062798

Fig. 9.16: The extra tube length as a function of the shock location, supersonic branch

!!!!!!!"$#&%
The proposed procedure is based on figure 9.16.
assuming shock at
the left side (at the max length).
i) Calculated the extra WYXaZ [
and subtract the actual extra W`XaZ [
assuming shock at
the right side (at the entrance).
ii) Calculated the extra WYXaZ [
and subtract the actual extra W`XaZ [
iii) According to the positive or negative utilizes your root finding procedure.
From numerical point of view, the Mach number equal infinity when left side
assume result in infinity length of possible extra (the whole flow in the tube is sub-
sonic). To overcame this numerical problem it is suggested to start the calculation
from distance from the right hand side.
Let denote
(9.51)

. The requirement that has to satisfied
WYXaZ [=<
W`XaZ [HG JI
H > W`XaZ [=?A@ B ?DCFE
A;:
as difference between the maximum possible of length
Note that WYXaZ [KG JI
is smaller than W`XaZ [ML
in which the flow supersonic achieved and the actual length in which the flow is
?PORQ
L
LN
supersonic see figure 9.15. The retreating length is expressed as subsonic but
is that denote W`XaZ [SL
B ?PB
L
LTVU TVU
(9.52)
B ?PB H W`XaZ [ L E W`XaZ [=G JI
W`XaZ [ L
?PORQ
L
L
LTVU TVU
LN