Page 287 - 48Fundamentals of Compressible Fluid Mechanics

14.2. GEOMETRICAL EXPLANATION 249

component. In fact, based on definition of the Mach angle, the component shown
is equal to speed of sound, .
After some additional rearrangement equation (14.15) becomes
in Figure (14.3) under "
(14.21)
"
"
"
If isn’t approaching infinity, and since " % leads to
(14.22)
%
*
"

"
1
In the literature, these results associated with line of characteristic line . This

analysis can be also applied to the same equation when they normalized by Mach
number. However, the dimensionlization can be applied at this stage as well.
The energy equation for any point on stream line is
(14.23)
"

"

For enthalpy in ideal gas with a constant specific heat, , is

(14.24)

and substituting this equality, (equation (14.24)) into equation (14.23) results

$
"

"

(14.25)
"

"

"
Utilizing equation (14.20) for the speed of sound and substituting the radial velocity

equation (14.22) transformed equation (14.25) into
$

(14.26)

"

After some rearrangement equation (14.27) becomes

(14.27)

"

"

Note, " must be positive. The solution of the differential equation (14.27) incor-

porating the constant into it becomes

%
*
(14.28)

"

1 This topic is under construction.

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