1. And from energy equation dh = -VdV … (2)
A course in Gas Dynamics…………………………………..….…Lecturer: Dr.Naseer Al-Janabi
CHAPTER TWO
ISENTROPIC FLOW
2-1 Isentropic Flow in varying Area ducts
For isentropic flow, from continuity
dp dV dA
p V A
And from energy equation
dh = -VdV … (2)
We now introduce the property relation d
Since flow is isentropic (ds = 0).
Thus equation (3) becomes dp
And from (2), so Or pV
We introduce (6) into the differential form of the continuity equation (1)
becomes
dp dA dp
Solve this for dp/ρ and show that + +
= … (1)
d = d … (3) p
dh = … (4) p
-VdV =
… (5) p
dV = -
… (6)
p A - pV2 +
= … (7)

2. Recall the definition of sonic velocity: ap ap s
A course in Gas Dynamics…………………………………..….…Lecturer: Dr.Naseer Al-Janabi
d dA dp
p A p
Recall the definition of sonic velocity: ap
ap s
Since our flow is isentropic, we may drop the subscript and change the
partial derivative to an ordinary derivative: dp Or
dp = a2 dp … (11)
Substituting this expression for dp into equation (8) yields
dp V2 dA dp
Introduce the definition of Mach number, V a
dp dA dp
p A p
and combine the terms in dρ/ρ to obtain the following relation between
density and area changes:
dp M dA
If we now substitute equation (14) into the differential form of the
continuity equation (1), we can obtain a relation between velocity and area
changes. Show that:
dV dA
1 - M A
Now equation (6) can be divided by V to yield
= V2 ( + )
… (8) a2
= ( )
… (9) a2 =
… (10)
p a2 ( A = +
p )
… (12)
M =
= M2 (
+ )
… (13)
p = *1 - M2) A
… (14)
= - ( )
… (15) V

3. For convenience, we collect the three important relations that will be
A course in Gas Dynamics…………………………………..….…Lecturer: Dr.Naseer Al-Janabi
dV dp
V pV2
If we equate (15) and (16), we can obtain a relation between pressure and
area
changes. Show that dA
1 - M A
Dividing by p ,
dp pV dA 1 So
dp kV dA
Finally
dp kM dA
p M A
For convenience, we collect the three important relations that will be
referred to in the analysis that follows:
dp kM dA
p M A
dp M dA
dV dA
V M A
= -
… (16)
dp = pV2 ( )
… (17) = (
- M2 )
… (18) p p A = (
- M2 )
… (19) p a2 A
= ( )
… (20)
= ( )
= - ( )

4. For subsonic flow, M < 1, then (1 - M2 ) is + ve.
A course in Gas Dynamics…………………………………..….…Lecturer: Dr.Naseer Al-Janabi
Let us consider what is happening to fluid properties as it flows through
a variable-area duct.
For subsonic flow, M < 1, then (1 - M2 ) is + ve.
When dA is negative (area is decreasing), then dp is negative (pressure
decreases) and dp is negative (density decreases) and dV is positive
(velocity increases) and vice versa.
For supersonic flow, M > 1, then (1 - M2 ) is - ve.
When dA is negative (area is decreasing), then dp is positive (pressure
increases) and dp is positive (density increases) and is dV negative
(velocity decreases) and vice versa.
We summarize the above by saying that as the pressure decreases, the
following variations occur:
A similar chart could easily be made for the situation where pressure
increases, but it is probably more convenient to express the above in an
alternative graphical form, as shown in Figure2.1

5. From this equation we see that:
A course in Gas Dynamics…………………………………..….…Lecturer: Dr.Naseer Al-Janabi
Combines equations (13) and (15) to eliminate the term dA/A with the
following result:
dp dV
p V
From this equation we see that:
= -M2
… (21)

6. Figure 2.3 The cross section of a nozzle at the smallest flow area is
A course in Gas Dynamics…………………………………..….…Lecturer: Dr.Naseer Al-Janabi
At low Mach numbers, density variations will be quite small. This means
that the density is nearly constant (dp � 0) in the low subsonic regime
(M � 0. 3 ) and the velocity changes compensate for area changes.
At a Mach number equal to unity, we reach a situation where density
changes and velocity changes compensate for each another and thus no
change in area is required (dA=0 ).
At supersonic flow, the density decreases so rapidly that the
accompanying velocity change cannot accommodate the flow and thus the
area must increase.
Consider the operation of devices such as nozzles and diffusers. A
nozzle is a device that converts enthalpy (or pressure energy for the case of
an incompressible fluid) into kinetic energy. From Figure 2.1 we see that an
increase in velocity is accompanied by either an increase or decrease in
area, depending on the Mach number. Figures 2.3 and 2.4 show what these
devices look like in the subsonic and supersonic flow regimes.
Figure 2.3 The cross section of a nozzle at the smallest flow area is
called the throat.