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103
uτ
K
F2 = 1.0
F1 =
⎤
1 ⎡ ⎛ uτ δ ⎞
⎟ + 1.0 + C ⎥
ln⎜
⎢
K⎣ ⎝ ν ⎠
⎦
F4 = 1.0
Φ = 0.0
F3 =
(44)
In order to close the set of three equations represented by (41), the term Φ in (43)
must be evaluated. The shear stress coefficient is assumed to satisfy the following
relation:
dCτ λ ˆ
= C − Cτ
dη δ τ
(
)
(45)
∫ τ dξ
(46)
where
Cτ =
1
2
1
2 ρU δ
δ
0
The following parameters were determined empirically:
λ = 0.15
1.0 ⎞
⎛
⎟
Cˆτ = 0.025⎜ 1.0 −
⎝
H⎠
2
(47)
The turbulent boundary layer solution is repeatedly stepped downstream from the
starting point, where the transition analysis supplied the initial conditions. H
from (37) is used to determine Cˆτ and Cτ is taken to be equal to Cˆτ at the starting
point. Basing an initial guess at the value of uτ on the value of Cf at the starting
point, initial values of uβ, uτ, and δ are determined iteratively, using the following
definitions:
u ⎤
⎡ uτ
+ β ⎥
⎣ KU 2U ⎦
δ* = δ⎢
⎡ ⎛ uτ ⎞ 2 3 ⎛ uβ ⎞ 2
uτ uβ ⎤
θ = δ − δ ⎢ 2⎝
+
+
1.58949
U 2 ⎥⎦
8⎝U⎠
⎣ KU ⎠
*
(48)