Page 153 - Introduction to Computational Fluid Dynamics

P1: IWV
12:28
May 20, 2005
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and, using Equation 5.91, 2D CONVECTION – CARTESIAN GRIDS
2
1 y P τ w y P ∂u 1 u u 1P
τ
P = dy = dy = (5.98)
y P 0 ρ y P 0 ∂y y P
or, using Equations 5.90 and 5.93,
3/4 3/2
C µ e
+
P = P ln(Ey ). (5.99)
P
κ y P
It is now easy to effect the boundary condition via
2
µ eff u V P
1P
Su e = Su e + 2 , (5.100)
y
P
3/4 1/2
ρ C µ e P
+
Sp e = Sp e + ln(Ey ) V P . (5.101)
P
κ y P
=
To evaluate P , we combine Equations 5.91 and 5.97 so that
τ w ∂u 1 2 ∂u 1
P = = u τ . (5.102)
ρ ∂y ∂y
But, from Equation 5.86, ∂u 1 /∂y = u τ /(κ y). Therefore,
e
u 3 C 3/4 3/2
P = τ = P . (5.103)
κ y P κ y P
To effect this condition, we set

30
30
Su = 10 P , Sp = 10 . (5.104)
= T
In this case, AS =
eff x 1 /y P , where
eff = k eff /C p . Again, we set AS = 0 and
absorb the boundary condition via an augmented source. Thus

eff x 1 q w
Su T = Su T + (T b − T P ) = Su T + x 1 . (5.105)
y P C p
Substituting for (T b − T P ) from Equation 5.87, it follows that

eff ρ u τ
= . (5.106)
y P Pr t (u + + PF)
1P
Thus, if q w is specified, we set
q w
Su T = Su T + x 1 , Sp T = Sp T + 0, (5.107)
C p