Page 84 - 48Fundamentals of Compressible Fluid Mechanics

46 CHAPTER 4. ISENTROPIC VARIABLE AREA FLOW

is positive hence
and
have the same sign. For the subsonic branch
smaller than one

the term

From these observations the trends are, similar to incompressible fluid, an in-

crease in area results in increase of the static pressure (converting the dynamic
pressure to a static pressure). Conversely, if the area decrease (as a function of )
the pressure decreases. Note that the pressure decrease is larger in compressible
flow compared to incompressible flow.

is negative and change the character of the equation.
For the supersonic branch
, the phenomenon is different. For

the term

This behavior is opposite to incompressible flow behavior.

can increase only

(sonic flow) the value of the term
For the special case of

. However, the opposite, not necessarily means that
thus mathematically

or
. Since physically
in a finite amount it must that

that
. In that case, it is possible that

thus in the

.It must also be noted that when

occurs only when
diverging side is in the subsonic branch and the flow isn’t choked.
when
The relationship between the velocity and the pressure can be observed
.
(4.35)
from equation (4.28) by solving it for

(since the

is positive). Hence the pressure increase when the velocity decreases
and vice versa.
has an opposite sign to
From equation (4.35) it is obvious that
From the speed of sound, one can observe that the density, , increases
term
with pressure and visa versa (see equation 4.36).
(4.36)

It can be noted that the derivations of the above equations (4.35 - 4.36), the

equation of state was not used. Thus, the equations are applicable for any gas
(perfect or imperfect gas).

and from
thermodynamics
The second law (isentropic relationship) dictates that