Page 285 - 48Fundamentals of Compressible Fluid Mechanics

14.2. GEOMETRICAL EXPLANATION 247

was shown to be

The ratio " "
(14.7)
"
Combining equation (14.6) and (14.7) transform to
"

(14.8)

After integration of the equation (14.8) results in

*

(14.9)
*

!
!
at .
$'&)( (

+*-, *

The constant can be chosen in a such a way that *
14.2.1 Alternative Approach to Governing equations
In the previous sec-
tion, a simplified ver- back
Mach
sion was derived based line r U r
on geometrical argu- Front
Mach
ments. In this sec- U θ line
θ
tion more rigorous ex-
planation is provided.
It must be recognized
that here the cylindrical
coordinates are advan-
tageous because the
flow turned around a
Fig. 14.4: The schematic of the coordinate for the mathematical
single point.
description
For this coordinate system, the mass conservation can be written as
!
!
(14.10)

"
The momentum equations are expressed as " !
(14.11)

"
"
"
"
!
!
"

$

!
(14.12)
""
"
"
"
!
!

"

$