Page 96 - Introduction to Computational Fluid Dynamics

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4.3 TRANSFORMATION TO (x, ω) COORDINATES
where, for convenience, May 25, 2005 11:7 75
ψ EI ≡ ψ E − ψ I . (4.12)
Thus, substituting Equation 4.11 in Equation 4.10, we can write Equation 4.9 as

∂ ∂ S
∂ ∂ 2
+ (a + b ω) ρ r u
, (4.13)
= +
∂x ω ∂ω x ∂ψ ∂ψ ρ u
where
−1 ∂ψ I
a ≡−ψ , (4.14)
EI
∂x
b ≡−ψ −1 ∂ψ EI . (4.15)
EI
∂x
Now, invoking Equation 4.2 again, we obtain
∂ −1 ∂
= ψ EI . (4.16)
∂ψ ∂ω

Therefore, Equation 4.13 can be written as

∂ ∂ S
∂ ∂
+ (a + b ω) c , (4.17)
= +
∂x ω ∂ω x ∂ω ∂ω x ρ u
where
−2
2
c ≡ ψ EI ρ r u
. (4.18)
Equation 4.17 represents Equation 4.1 in the (x, ω) coordinate system in
nonconservative form. To develop the conservative counterpart, the equation is
written as

∂ ∂ ∂ S
∂
+ (a + b ω) − c − (a + b ω) = , (4.19)
∂x ∂ω ∂ω ∂ω ρ u
ω
where, since a and b are not functions of ω,
∂
(a + b ω) = b . (4.20)
∂ω
Now, consider the identity

∂ ∂ ∂ψ EI ∂
ψ −1 (ψ EI ) = + ψ −1 = − b . (4.21)
EI EI
∂x ∂x ∂x ∂x
Using the last two equations, we can write Equation 4.19 as

∂ ∂ ∂
ψ EI S
[ψ EI ] + ψ EI (a + b ω) − c = . (4.22)
∂x ∂ω ∂ω ρ u
This is the required boundary layer equation in the (x, ω) coordinate system
written in conservative form. It will be useful at this stage to interpret the terms
in Equation 4.22. Thus, from Equations 4.12 and 4.5, it is easy to show that ψ EI