Basics of Fluid Mechanics, 2014a

386 CHAPTER 11. COMPRESSIBLE FLOW ONE DIMENSIONAL
In the same manner the relationship for the density ratio is
Isentropic Density
( ) 1
ρ 0
ρ = T0
k−1
=
(1+ k − 1 ) 1
M 2 k−1
T
2
(11.28)
New useful definitions are introduced for the case when M =1and denoted by superscript
“∗.” The special cases of ratio of the star values to stagnation values are
dependent only on the heat ratio as the following:
Star Relationship
T ∗
T 0
=
V 2
V 1
=
P ∗
P 0
=
ρ ∗
ρ 0
=
c ∗2
c
2 0
( ) 1
T1
T 2
( 2
k +1
( 2
k +1
k−1
=
) k
k−1
) 1
k−1
(
ρ1
ρ 2
)
=
(
P1
P 2
) 1 k
(11.29)
Using all the definitions above relationship between the stagnation properties to star
speed of sound are
√
c ∗ = kR 2 T 0
(11.30)
k +2
11.4.2 Isentropic Converging-Diverging Flow in Cross Section
The important sub case in this chapter
is the flow in a converging–diverging nozzle.
The control volume is shown in Figure
(11.7). There are two models that assume
variable area flow: First is isentropic
and adiabatic model. Second is isentropic
and isothermal model. Here only the
first model will be described. Clearly, the
stagnation temperature, T 0 , is constant
through the adiabatic flow because there
isn’t heat transfer. Therefore, the stagnation
pressure is also constant through the
A ∗ C.V. ρ P P+dP
ρ+dρ
T T+dT
U U+dU
Fig. -11.7. Control volume inside a convergingdiverging
nozzle.
flow because the flow isentropic. Conversely, in mathematical terms, equation (11.25)
and equation (11.27) are the same. If the right hand side is constant for one variable, it