Page 286 - 48Fundamentals of Compressible Fluid Mechanics

248 CHAPTER 14. PRANDTL-MEYER FUNCTION

If it is assumed that the flow isn’t a function of the radios, , then all the derivatives
with respect to radios vanish. One has to remember that when enter to the
function, like the first term in mass equation, the derivative isn’t zero. Hence, the
mass equation reduced to !

(14.13)
!

"

" %
After rearrangement equation (14.13) transforms into !
(14.14)
"
!
% "

"
The momentum equations are obtained the form of
*
" "

(14.15)
"
"
"

!
" %
"
!
""
"
*

!
(14.16)

$
"
!

"

Substituting the term 6 from equation (14.14) into equation (14.16) results in

$

" %
*
(14.17)
6
"
"
"
% "

or " % * " *
$
(14.18)
"
% "
" % "
And additional rearrangement results in $ *

"
(14.19)
*
"
" % "
From equation (14.19) it follows that * %
(14.20)
$
"
It is remarkable that tangential velocity at every turn is the speed of sound! It must
be point out that the total velocity isn’t at the speed of sound but only the tangential
$