Page 95 - Introduction to Computational Fluid Dynamics

P1: IWV
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and where ψ is the stream function defined by 2D BOUNDARY LAYERS
∂ψ
=−ρv r, (4.3)
∂x
∂ψ
= ρ ur. (4.4)
∂y
Thus, at any x

ψ = ρ ur dy + C, (4.5)
where C is a constant. The y coordinate is thus related to ψ and the latter, in turn, is
related to ω via Equation 4.2. Suffixes I and E, of course, refer to inner and external
boundaries.

4.3 Transformation to (x, ω) Coordinates
Our task now is to transform Equation 4.1 from the (x, y) coordinate system to
the (x, ω) coordinate syatem. To do this, we shall follow the sequence (x, y) →
(x,ψ) → (x,ω). Making use of the mass conservation equation ( = 1), we can
write Equation 4.1 in nonconservative form as

∂ ∂ 1 ∂ ∂
ρ u + v = r
+ S. (4.6)
∂x ∂y r ∂y ∂y
Now, the transformation (x, y) → (x,ψ) implies that

∂ψ ∂ ∂
∂
= + , (4.7)
∂x ∂ψ ∂x
∂x y x ψ

∂ ∂ψ ∂ ∂
= = ρ ru . (4.8)
∂y x ∂y ∂ψ y ∂ψ y
Substituting these equations in Equation 4.6, we can show that

∂ ∂ S
∂ 2
= ρ r u
+ . (4.9)

∂x ∂ψ ∂ψ ρ u
ψ
Further, the (x,ψ) → (x,ω) transformation implies that

∂ ∂ ∂ω ∂
= + , (4.10)
∂x ψ ∂x ω ∂x ψ ∂ω x
but, from Equation 4.2,

∂ψ

∂ω −1 ∂ψ I ∂ψ EI
= ψ EI − − ω
∂x ∂x ∂x ∂x
ψ ψ

=−ψ −1 ∂ψ I + ω ∂ψ EI , (4.11)
EI
∂x ∂x