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102
where the wake function is approximated by a cosine distribution, K=0.41 and
C=2.05. Substituting (40) into (39) and integrating yields two equations of the
form
F1
duβ
1 dδ
du
dU
+ F2
+ F3 τ = F4
+Φ
dη
δ dη
dη
dη
(41)
A third equation of this form can be obtained by substituting ξ = δ in (40) and
differentiating the equation with respect to the station, which corresponds to the
case α = ∞. The coefficients Fn and the function Φ are given as follows:
for α = 0 (momentum equation)
⎡3
u Uu ⎤
1
u2
F1 = ⎢ u2β − Uuβ + 2 τ2 + 1.58949uβ τ − τ ⎥
⎣8
2
K
K
K ⎦
3
1
u
F2 = ⎡⎢ uβ − U + 1.58949 τ ⎤⎥
⎣4
2
K⎦
1⎡ u
⎤
F3 = ⎢ 4 τ + 1.58949uβ − U ⎥
⎦
K⎣ K
u ⎤
⎡
F4 = ⎢uβ + 2 τ ⎥
⎣
K⎦
Φ=−
(42)
uτ2
δ
for α = 1 (momentum of momentum equation)
⎡ 3 u 2 1 uτU uτ uβ ⎛ 3 2
1
5
1 2 ⎤
⎜ + 2 − 0.16701⎞⎟ + uβ2 ⎛⎜ − 2 ⎞⎟ − Uuβ ⎛⎜ − 2 ⎞⎟ ⎥
F1 = ⎢ τ2 −
+
⎝ 16 π ⎠
⎝ 2 π ⎠⎦
⎠
⎣4 K
2 K
K ⎝4 π
⎡ u ⎛ 3 2 0.16701⎞
⎤
⎟ + uβ ⎛⎜ 1 − 22 ⎞⎟ − U ⎛⎜ 1 − 12 ⎞⎟ ⎥
F2 = ⎢ τ ⎜ + 2 −
⎝2 π ⎠
⎝ 4 π ⎠⎦
2 ⎠
⎣ K ⎝8 π
F3 =
1 ⎡ uτ U
⎛ 5 2 3 ∗0.16701⎞⎟ ⎤
− + uβ ⎜ + 2 −
⎢
⎠ ⎥⎦
⎝8 π
K⎣K 4
2
⎡3 u
⎛ 3 3 ⎞⎤
F4 = ⎢ τ + uβ ⎜ − 2 ⎟ ⎥
⎝ 4 π ⎠⎦
⎣4 K
Φ=−
1
δ
δ 2 ∫0
τ
1 U2
dξ = − Cτ
ρ
2
δ
for α = ∞ (differentiated skin friction law)
(43)