Page 101 - Introduction to Computational Fluid Dynamics

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Plane Flow 0 521 85326 5 2D BOUNDARY LAYERS
From Equations 4.4 and 4.2, it can be shown that
ω
d ω
y = ψ EI = I(say). (4.48)
0 ρ u
Thus, knowing the initially set values of ω j and ω c, j , y j and y c, j can be estimated.
Note that ψ EI and ρ u will change with x. Therefore, y will also change with x.
Axisymmetric Flow
In this case, from Equation 4.45, it follows that

ψ EI
d ω = (r I + y cos α)dy (4.49)
ρ u

and, therefore, from Equation 4.48

y 2
I = r I y + cos α . (4.50)
2
The solution to this quadratic equation suitable for computer implementation is

2 I
y = , (4.51)

2 0.5
r I + r + 2 I cos α
I
where I is given by Equation 4.48. Now, knowing y j and y c, j in this manner, r j
and r c, j can be evaluated using Equation 4.45.

4.6 Boundary Conditions
At the E and I boundaries, three types of boundary conditions are possible: sym-
metry, wall, or free stream. We discuss them in turn.

4.6.1 Symmetry

There can be no mass flux across the symmetry plane. Also, ∂ /∂n| b = 0, where
suffix b denotes the E or I boundary node. This implies that

˙
b = nb and m b = 0, (4.52)

where suffix nb stands for near-boundary node. A further consequence of the
˙ m b = 0 condition is that ∂ψ b /∂x = 0or ψ b = constant. The boundary condition
can be effected by setting AS 2 = 0 at the I boundary or AN JN−1 = 0 at the E
boundary.