Page 97 - Introduction to Computational Fluid Dynamics

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2D BOUNDARY LAYERS
represents the total streamwise mass flow rate through the boundary layer at any x.
Similarly, making use of the definitions of a, b, and c and using Equation 4.16, we
can show that

∂ ∂
ψ EI (a + b ω) − c = r ˙ m − r
, (4.23)
∂ω ∂y
where
∂ ψ EI ∂
= (4.24)
∂ω ρ ru ∂y
and
r ˙ m = r ρv = (1 − ω)r I ˙ m I + ωr E ˙ m E
with ˙ m E = (ρv) E , ˙ m I = (ρv) I . (4.25)
Thus the total transverse mass flux ˙ m at any y is a weighted sum of mass fluxes at
theinner( ˙ m I )andexternal( ˙ m E )boundariesinthepositive y direction.Equation4.23
therefore represents the total convective–diffusive flux in the y direction. Then by
substituting Equation 4.23, Equation 4.22 can be written as
∂ ∂ ∂ ψ EI S
[ψ EI ] + r ˙ m − r
= . (4.26)
∂x ∂ω ∂y ρ u

4.4 Discretisation

Figure 4.3 shows the (x,ω) grid at streamwise location x. Suffix u refers to upstream
and d refers to downstream. Note that nodes N, P, and S are not equidistant because
ω, in general, will not be uniform. This will become apparent in a later section.
To derive the discretised version of Equation 4.26, each term in the equation will
be integrated over the control volume. Thus, assuming source term S to be constant
over the control volume, we have

ψ EI S S
n n n
x d x d x d
dx dω = dx dψ = Sr dx dy
ρ u ρ u
x u s x u s x u s
= Sr P x y = S V, (4.27)
where

V = r P x y. (4.28)

Similarly, the streamwise convection term integrates to

x d
n
∂ d u
[ψ EI ] dx dω = (ψ EI ) − (ψ EI ) P ω. (4.29)
x u s ∂x