Page 79 - 48Fundamentals of Compressible Fluid Mechanics

4.1. STAGNATION STATE FOR IDEAL GAS MODEL 41

The usefulness of Mach num- A B
ber and equation (4.9) can be demon-
T 0 T 0
strated by this following simple exam- P 0 velocity P 0
ρ 0 ρ 0
ple. In this example a gas flows through
a tube (see Figure 4.2) of any shape
can be expressed as a function of only
the stagnation temperature as oppose Fig. 4.2: Perfect gas flows through a tube
to the function of the temperatures and
velocities.
The definition of the stagnation provides the advantage of a compact writ-
ing. For example, writing the energy equation for the tube when by an external
forces or energy exchange shown in Figure (4.2). In that case, the energy ques-
tion is reduced to

(4.10)

The ratio of stagnation pressure to the static pressure can be expressed
as a function of the temperature ratio because of the isentropic relationship as

(4.11)

In the same manner the relationship for the density

(4.12)

A new useful definition is introduced for the case when
and denoted by

superscript . The special case of ratio of the star values to stagnation values are
depend only on the heat ratio as following:
(4.13)

(4.14)

(4.15)