Page 189 - 48Fundamentals of Compressible Fluid Mechanics

9.5. SUPERSONIC BRANCH 151

assume the flow in the entire tube is supersonic:

a) set N HI
Note this Mach number is the high Value.
b
b) calculated N
2) Calculated the entrance Mach assuming shock at the entrance.
’ for subsonic branch
a) set N HI
’
b
b) add WYXaZ [
and calculated N
Note this Mach number is the low Value.
b
for N
c) calculated N
8

3) according your root finding algorithm calculated or guess the shock loca- To check Secant Method.
.
b
tion and then compute as above the new N
’ as on the subsonic branch
a) set N HI
’
b) for the new WYX(Z [
and compute the new N
c) calculated N
’ for the N
4) guess new location for the shock according to your finding root procedure

b
d) Add the leftover of W`XaZ [
and calculated the N
and according the result repeat previous stage until finding the solution.

Q^S
Q^S
IUS
Q0I ( % I % ( I(Q

Q^S

SQ
QUQ
Q
IUS QUQ
Q
Q

(c) The way that numerical procedure of solving this problem is by finding W`XaZ [
must be calculated (see
the chapter on the program with its algorithms.).
b

H
and N
that will produce N
. In the process N

9.5 Supersonic Branch
In chapter ?? it was shown that the isothermal model cannot describe the ade-
quately the situation because the thermal entry length is relatively large compared
to the pipe length and the heat transfer is not sufficient to maintain constant temper-
ature. In the Fanno model there is no heat transfer, and, furthermore, the because
the very limited amount of heat transformed it closer to a adiabatic flow. The only
limitation of the model is uniform of the velocity (assuming parabolic flow for lam-
inar and different profile for turbulent flow.). The information from the wall to the
9
tube center is slower in reality. However, experiments by many starting with 1938
8 You can use any method you which, but be-careful second order methods like Newton-Rapson
method can be unstable.
9 The word information referred into the shear stress transformed from the wall to centerer of the
tube.