Page 113 - 48Fundamentals of Compressible Fluid Mechanics

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Note that the greater–equal signs were not used. The reason is that the process
is irreversible, and therefore no equality can exist. Mathematically the parameters
are
and , which are needed to be solved. For ideal gas, equation (5.5) is

(5.6)

It can be also noticed that entropy, , can be expressed as a function of the
other parameters. Now one can view these equations as two different subsets of
equations. The first set is the energy, continuity and state equations, and the sec-
ond set is the momentum, continuity and state equations. The solution of every set
of these equations produces one additional degree of freedom, which will produce
a range of possible solutions. Thus, one can instead have a whole range of solu-
tions. In the first case, the energy equation is used, producing various resistance
to the flow. This case is called Fanno flow, and the Chapter (9) deals extensively
with this topic. The mathematical explanation is given there in greater detail. In-
stead of solving all the equations that have been presented, one can solve only 4
equations (including the second law), which will require additional parameters. If
the energy, continuity and state equations will be solved for the arbitrary value of
, a parabola in the
– diagram will be obtained. On the other hand, when
the momentum equation is solved instead of the energy equation, the degree of
freedom now is energy i.e., the energy amount “added” to the shock. This situation
the
is similar to frictionless flow with the addition of heat, and this flow is known as
Rayleigh flow. This flow is dealt with in greater detail in chapter 10.

Since the shock has
no heat transfer (a special subsonic
case of Rayleigh flow) and "$#&% ' ( flow
supersonic
there isn’t essentially mo-

flow
mentum transfer (a special T
case of Fanno flow), the shock jump
intersection of these two !
curves is what really hap- Rayleigh
Fanno
pened in the shock. In line line
Figure (5.2) the intersec-
tion is shown and two solu-
tions are obtained. Clearly
the increase of the entropy
determines the direction of s
flow. The entropy increases
from point to point . It Fig. 5.2: The intersection of Fanno flow and Rayleigh flow
is also worth noting that the produces two solutions for the shock wave

temperature at
on Rayleigh flow is larger than that on the Fanno line.