Basics of Fluid Mechanics, 2014a

408 CHAPTER 11. COMPRESSIBLE FLOW ONE DIMENSIONAL
energy equation, the degree of freedom is now energy, i.e., the energy amount “added”
to the shock. This situation is similar to a frictionless flow with the addition of heat,
and this flow is known as Rayleigh flow. This flow is dealt with in greater detail in
Section (11.9).
Since the shock has no heat transfer (a special case of Rayleigh flow) and there
isn’t essentially any momentum transfer (a special case of Fanno flow), the intersection
of these two curves is what really happened in the shock. The entropy increases from
point x to point y.
11.5.1 Solution of the Governing Equations
Equations (11.70), (11.71), and (11.72) can be converted into a dimensionless form.
The reason that dimensionless forms are heavily used in this book is because by doing
so it simplifies and clarifies the solution. It can also be noted that in many cases the
dimensionless equations set is more easily solved.
From the continuity equation (11.70) substituting for density, ρ, the equation of
state yields
Squaring equation (11.76) results in
P x
U x =
P y
U y (11.76)
RT x RT y
2
P x
R 2 2
T U x 2 = P y 2
x R 2 2
T U y 2 (11.77)
y
Multiplying the two sides by the ratio of the specific heat, k, provides a way to obtain
the speed of sound definition/equation for perfect gas, c 2 = kRT to be used for the
Mach number definition, as follows:
P x
2
P y
2
U 2 x =
T x kRT
} {{ x T
}
y kRT y
} {{ }
c 2 x c 2 y
U y
2
Note that the speed of sound is different on the sides of the shock.
definition of Mach number results in
P x
2
T x
M x 2 = P y 2
T y
M y
2
(11.78)
Utilizing the
(11.79)
Rearranging equation (11.79) results in
T y
T x
=
(
Py
P x
) 2 (
My
M x
) 2
(11.80)