Page 161 - 48Fundamentals of Compressible Fluid Mechanics

8.2. DIMENSIONLESS REPRESENTATION 123

the solution is attainable. One can noticed that there are two possible solutions
(because the square power). These different solutions are super sonic and sub-
sonic solution.
, is selected as the choice for the independent

variable. Thus, the equations need to be obtained in the form variable as a function
The distance friction,
. The density is eliminated from equation (8.15) when combined with the

equation (8.6) to became
of
(8.22)
Substituting the velocity (8.22) into equation (8.10) and one can obtain

(8.23)

Equation (8.23) can be rearranged into

(8.24)

Similarly or by other path the stagnation pressure can be expressed as a function

of
(8.25)

%

%

(8.26)

The variables in equation (8.24) can be separated to obtain integrable form as

%

follows

(8.27)

%

(the initial velocity

is positive for any , thus, the term on other

for which
. Thus, the
It can be noticed that at the entrance

to tube isn’t zero.). The term

is the limiting case where from a mathematical point of view. Mach

side has to be positive as well. To obtain this restriction
makes the right hand side integrate negative. The

value
Physical meaning of this value similar to
chocked flow which were discussed

number larger from
in a variable area flow Chapter 4.
the
%
value of right hand side approached infinity ( ). Since the stagnation temperature
%

Further it can be noticed from equation (8.26) that when

%