Basics of Fluid Mechanics, 2014a

12.3. PRANDTL-MEYER FUNCTION 523
The momentum equations now obtain the form of
U θ
r
U θ
r
∂U r
∂θ − U θ 2
r
U θ
( ∂Ur
∂θ − U θ
∂U θ
∂θ − U θ U r
r
)
U θ
( ∂Uθ
∂θ − U r
=0
)
=0
= − c2 ∂ρ
rρ ∂θ
= − c2 ρ
∂ρ
∂θ
(12.77)
(12.78)
Substituting the term 1 ∂ρ
ρ ∂θ
from equation (12.76) into equation (12.78) results in
( ) (
∂Uθ
U θ
∂θ − U r = c2
U r + ∂U )
θ
(12.79)
U θ ∂θ
or
(
2
U θ U r + ∂U ) (
θ
= c 2 U r + ∂U )
θ
(12.80)
∂θ
∂θ
And an additional rearrangement results in
(
c 2 2
− U ) ( θ U r + ∂U )
θ
=0 (12.81)
∂θ
From equation (12.81) it follows that
U θ = c (12.82)
It is remarkable that the tangential velocity at every turn is at the speed of sound!
It must be pointed out that the total velocity isn’t at the speed of sound, but only
the tangential component. In fact, based on the definition of the Mach angle, the
component shown in Figure (12.23) under U y is equal to the speed of sound, M =1.
After some additional rearrangement, equation (12.77) becomes
( )
U θ ∂Ur
r ∂θ − U θ =0 (12.83)
If r isn’t approaching infinity, ∞ and since U θ ≠0leads to
∂U r
∂θ = U θ (12.84)
In the literature, these results are associated with the characteristic line. This analysis
can be also applied to the same equation when they are normalized by Mach number.
However, the non–dimensionalization can be applied at this stage as well.