-
8/13/2019 Behavior of Isentropic Flow in Quasi-1D
1/5
Behavior of Isentropic Flow in Quasi-1D
Recall cons. of mass:.constuA=
Consider a perturbation in the area
0
...
))()((
=++
++++=
+++=
udAAduuAd
TOHudAAduuAduA
dAAduuduA
Or
(1)
Similar manipulations to x-momentum gives:
(2)
Also, since the flow is assumed isentropic, we can use the
definition of the speedof sound to relate d and dp :
=
= .
2
consts
pa
AA
uu
pp
L
L
L
L
=
=
=
=
Lx
Rx
dAAA
duuu
dppp
d
LR
LR
LR
LR
+=
+=
+=
+=
0=+ ududp
dadp2=
0=++A
dA
u
dud
Since we have assumed .consts= in the flow (3)
-
8/13/2019 Behavior of Isentropic Flow in Quasi-1D
2/5
Behavior of Isentropic Flow in Quasi-1D
16.10 2002 2
We are interested in how the flow properties change when the
area changes. So,we use (2) and (3) to eliminate terms from (1).
For example, lets determine how
u changes with A :
0)1(
0
0
0
2
2
2
=+
=++
=++
=++
A
dA
u
duM
A
dA
u
du
a
udu
A
dA
u
du
a
dp
A
dA
u
dud
When 1u
duwhen 0M : 0> u
du
when 0>A
dA
u increases when A increases!
0M ? Clearly, u must behave the opposite of A
regardless of the Mach number. So, what must be happening is
that changes
more rapidly than A for 1>M and, thus u behaves opposite of
what we expect
from subsonic flow behavior. Lets check this:
A
dA
Mu
dp
A
dA
Mu
du
22
2
1
1
1
1
=
=
A
dA
Mu
du21
1
=
A
dA
Mu
dp22 1
1
=
-
8/13/2019 Behavior of Isentropic Flow in Quasi-1D
3/5
Behavior of Isentropic Flow in Quasi-1D
16.10 2002 3
A
dA
Mu
da22
2
1
1
=
These results show that, regardless of whether 1>M or 1 ,:1
pM when A
,p when A
Which are the opposite of how u behaves. It is also useful to
consider the
magnitudes of the different changes:
A
dA
M
Md2
2
1
=
,p when A
,p when A
A
dA
Mu
dp22 1
1
=
A
dA
M
Md2
2
1=
-
8/13/2019 Behavior of Isentropic Flow in Quasi-1D
4/5
Behavior of Isentropic Flow in Quasi-1D
16.10 2002 4
0=++A
dA
u
dud
Flow low ,M changes are small compared to u changes. But for
,1>M
changes more rapidly.
Also, of interest is that ,,dpd and du become extremely large as
1M .
In fact, this requires that 1=M must occur at a minimum in the
area where
0=dA .
Lets consider a converging-diverging duct:
Whenbachp is only a little less than 0p , the flow will be
subsonic:
As bachp is lowered from op , we will eventually hit the bachp
at which 1=M at
the throat.
What happens if bachp is lowered further?
There is one other isentropic flow through this geometry which
occurs when
bachp is very low. In this situation, the flow will become
supersonic in the divergent
section:
0p bachp
M
1
0
decreasing
bachp
01
1
1 22
2
=+
A
dA
A
dA
MA
dA
M
M
-
8/13/2019 Behavior of Isentropic Flow in Quasi-1D
5/5
Behavior of Isentropic Flow in Quasi-1D
16.10 2002 5
M
1
0
Supersonicisentropic flow(very low
bachp )
Subsonicisentropic flows
