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𝜕𝒖
𝜕𝑡
+ 𝛻
𝒖 2
2
+
1
𝜌
𝛻𝑝 = 0
𝜕𝜌
𝜕𝑡
+ 𝛻 ⋅ 𝜌𝒖 = 0
𝑠 = 𝑐𝑜𝑛𝑠𝑡
𝜕𝜙
𝜕𝑡
+
𝛻𝜙 2
2
+
𝑎2
𝛾 − 1
= 𝐶
𝜕2𝜙
𝜕𝑡2
+ 2𝛻𝜙 ⋅
𝜕𝛻𝜙
𝜕𝑡
= 𝑎2𝛻2𝜙 − 𝛻𝜙 ⋅ 𝛻
𝛻𝜙 2
2
𝐷𝐹
𝐷𝑡
= 0 →
𝜕𝐹
𝜕𝑡
+ 𝒖 ⋅ 𝛻𝐹 = 0
𝒖 → 𝑈∞𝒆𝒙 𝒙 → ∞
𝑧 = 𝑓 𝑥, 𝑦, 𝑡
𝐹 𝑥, 𝑦, 𝑧, 𝑡 = 𝑧 − 𝑓 𝑥, 𝑦, 𝑡 = 0
𝑑𝐹 = 0
(𝒙, 𝑡
(𝒙 + 𝒅𝒙, 𝑡 + 𝑑𝑡
𝐹𝑢 𝑥, 𝑦, 𝑧, 𝑡 = 𝑧 − (𝑧+ 𝑥, 𝑦 + 𝛼 𝑐 − 𝑥 + ℎ 𝑥, 𝑦, 𝑡 )
−
𝜕ℎ
𝜕𝑡
− 𝑢
𝜕𝑧+
𝜕𝑥
− 𝛼 +
𝜕ℎ
𝜕𝑥
− 𝑣
𝜕𝑧+
𝜕𝑦
+
𝜕ℎ
𝜕𝑦
+ 𝑤 = 0
𝐹𝑙 𝑥, 𝑦, 𝑧, 𝑡 = 𝑧 − (𝑧− 𝑥, 𝑦 + 𝛼 𝑐 − 𝑥 + ℎ 𝑥, 𝑦, 𝑡 )
−
𝜕ℎ
𝜕𝑡
− 𝑢
𝜕𝑧−
𝜕𝑥
− 𝛼 +
𝜕ℎ
𝜕𝑥
− 𝑣
𝜕𝑧−
𝜕𝑦
+
𝜕ℎ
𝜕𝑦
+ 𝑤 = 0
𝑧+ 𝑥, 𝑦 + 𝛼(𝑐 − 𝑥)
𝑧− 𝑥, 𝑦 + 𝛼(𝑐 − 𝑥)
ℎ(𝑥, 𝑦, 𝑡)
𝒖 𝑥, 𝑦, 𝑧, 𝑡 = 𝑈∞𝒆𝒙 + 𝒖′
𝑥, 𝑦, 𝑧, 𝑡
𝜙 𝑥, 𝑦, 𝑧, 𝑡 = 𝑈∞ 𝑥 + 𝜙′
𝑥, 𝑦, 𝑧, 𝑡
1 − 𝑀∞
2
𝜕2
𝜙′
𝜕𝑥2
+
𝜕2
𝜙′
𝜕𝑦2
+
𝜕2
𝜙′
𝜕𝑧2
−
2𝑀∞
2
𝑈∞
𝜕2
𝜙′
𝜕𝑡𝜕𝑥
−
𝑀∞
2
𝑈∞
2
𝜕2
𝜙′
𝜕𝑡2
= 0
1
𝑈∞
ቇ
𝜕𝜙′
𝜕𝑧 ±
=
𝜕𝑧±
𝜕𝑥
− 𝛼 +
𝜕ℎ
𝜕𝑥
+
1
𝑈∞
𝜕ℎ
𝜕𝑡
, 𝛻𝜙′ → 0 𝒙 → ∞
𝑐𝑝 = −
2
𝑈∞
2
𝜕𝜙′
𝜕𝑡
+ 𝑈∞
𝜕𝜙′
𝜕𝑥
𝜙′
1 − 𝑀∞
2
𝜕2
𝜙1
𝜕𝑥2
+
𝜕2
𝜙1
𝜕𝑦2
+
𝜕2
𝜙1
𝜕𝑧2
−
2𝑀∞
2
𝑈∞
𝜕2
𝜙
𝜕𝑡𝜕𝑥
−
𝑀∞
2
𝑈∞
2
𝜕2
𝜙
𝜕𝑡2
= 0
1
𝑈∞
ቇ
𝜕𝜙1
𝜕𝑧 ±
=
𝜕𝑧±
𝜕𝑥
− 𝛼 , 𝛻𝜙′
→ 0 𝒙 → ∞
1 − 𝑀∞
2
𝜕2𝜙2
𝜕𝑥2
+
𝜕2𝜙2
𝜕𝑦2
+
𝜕2𝜙2
𝜕𝑧2
−
2𝑀∞
2
𝑈∞
𝜕2𝜙2
𝜕𝑡𝜕𝑥
−
𝑀∞
2
𝑈∞
2
𝜕2𝜙2
𝜕𝑡2
= 0
1
𝑈∞
ቇ
𝜕𝜙2
𝜕𝑧 ±
=
𝜕ℎ
𝜕𝑥
+
1
𝑈∞
𝜕ℎ
𝜕𝑡
, 𝛻𝜙′
→ 0 𝒙 → ∞
0
𝜕2
𝜙
𝜕𝑥2
+
𝜕2
𝜙
𝜕𝑧2
= 0
1
𝑈∞
ቇ
𝜕𝜙′
𝜕𝑧 ±
=
𝜕ℎ
𝜕𝑥
+
1
𝑈∞
𝜕ℎ
𝜕𝑡
, 𝛻𝜙′ → 0 𝒙 → ∞, +Kutta
𝑢 𝑥, 0, 𝑡
𝑤′ 𝑥, 0, 𝑡 = −
1
2𝜋
න
0
∞
2𝑢′
𝜉, 0, 𝑡
𝑥 − 𝜉
𝑑𝜉
𝛾 = 2𝑢′
Γ 𝑥, 𝑡 = ‫׬‬
0
𝑥
2𝑢′
𝜉, 0, 𝑡 𝑑𝜉
𝑢 𝑥, 0, 𝑡
𝑤′ 𝑥, 0, 𝑡 = −
1
2𝜋
න
0
𝑐
2𝑢′ 𝜉, 0, 𝑡
𝑥 − 𝜉
𝑑𝜉 −
1
2𝜋
න
𝑐
∞
2𝑢′ 𝜉, 0, 𝑡
𝑥 − 𝜉
𝑑𝜉
𝑪𝒑
𝜕𝜙′
𝜕𝑡
+ 𝑈∞
𝜕𝜙′
𝜕𝑥
= 0 → 𝜙′
𝑥, 0, 𝑡 = 𝜙′
𝑐, 0, 𝑡 −
𝑥 − 𝑐
𝑈∞
Γ 𝑥, 𝑡 = Γ 𝑐, 𝑡 −
𝑥 − 𝑐
𝑈∞
𝛾 𝑥, 𝑡 = −
1
𝑈∞
𝜕Γ
𝜕𝑡
𝑐, 𝑡 −
𝑥 − 𝑐
𝑈∞
ℎ, 𝑤, 𝑢, Γ, 𝑔 𝑥, 𝑡 = {෠
ℎ, ෝ
𝑤, ො
𝑢, ෠
Γ, ො
𝑔} 𝑥 𝑒−𝑖𝜔𝑡
ෝ
𝒖 𝝃
ෝ
𝑤 𝑥 = −
1
2𝜋
න
0
𝑐
2ො
𝑢 𝜉
𝑥 − 𝜉
𝑑𝜉 + ො
𝑔(𝑥)
ො
𝑔 𝑥 =
𝑖𝜔
2𝜋
෠
Γ 𝑐 න
𝑐
∞
𝑒−𝑖𝜔/𝑈∞ 𝜉−𝑐
𝑥 − 𝜉
𝑑𝜉
ො
𝑢 𝑥 =
1
𝜋
𝑐 − 𝑥
𝑥
න
0
𝑐
(ෝ
𝑤 𝜉 − ො
𝑔 𝜉 )
𝑥 − 𝜉
𝜉
𝑐 − 𝜉
𝑑𝜉
ො
𝑢 𝑥 =
1
𝜋
𝑐 − 𝑥
𝑥
න
0
𝑐
(ෝ
𝑤 𝜉 − ො
𝑔 𝜉 )
𝑥 − 𝜉
𝜉
𝑐 − 𝜉
𝑑𝜉
෠
Γ 𝑐 ෠
Γ 𝑐 = ‫׬‬
0
𝑐
2ො
𝑢 𝑥 𝑑𝑥
෠
Γ 𝑐
෠
Γ 𝑐 =
4 𝑒−𝑖𝑘
‫׬‬
0
𝑐 𝑥
𝑐 − 𝑥
ෝ
𝑤 𝑥 𝑑𝑥
𝜋𝑖𝑘 𝐻1
2
𝑘 + 𝑖𝐻0
2
𝑘
, 𝑘 =
𝜔𝑐
2𝑈∞
𝐻𝑛
(2)
𝐻0
2
𝑘 =
2𝑖
𝜋
න
1
∞
𝑒−𝑖𝑘𝑥
𝑥2 − 1
𝐻1
2
= −
2𝑖
𝜋𝑘
න
1
∞
𝑒−𝑖𝑘𝑥
𝑥2 − 1 3
𝑑𝑥
Ƹ
𝑐𝑝 𝑥 =
2
𝜋
𝑐 − 𝑥
𝑥
න
0
𝑐
𝜉
𝑐 − 𝜉
ෝ
𝑤 𝜉 𝑑𝜉
+
4𝑖𝑘
𝜋𝑐
𝑥 𝑐 − 𝑥 න
0
𝑐
න
0
𝜉
ෝ
𝑤 𝜂 𝑑𝜂
𝑑𝜉
(𝑥 − 𝜉) 𝜉 𝑐 − 𝜉
+
4
𝜋𝑐
1 − 𝐶 𝑘
𝑐 − 𝑥
𝑥
න
0
𝑐
𝜉
𝑐 − 𝜉
ෝ
𝑤 𝜉 𝑑𝜉
𝐶 𝑘 =
𝐻1
2
𝑘
𝐻1
2
𝑘 + 𝑖𝐻0
2
𝑘
𝑘 𝑘 =
𝜔𝑐
2𝑈∞
መ
𝐶𝑙 = −4𝐶 𝑘 න
0
1
𝜉
1 − 𝜉
ෝ
𝑤 𝜉 𝑑𝜉 − 8𝑖𝑘 න
0
1
𝜉 1 − 𝜉 ෝ
𝑤 𝜉 𝑑𝜉
መ
𝐶
𝑚
1
2
= න
0
1
𝜉
1 − 𝜉
− 4 𝜉 1 − 𝜉 ෝ
𝑤 𝜉 𝑑𝜉
+ 𝑖𝑘 න
0
1
න
0
𝜉
ෝ
𝑤 𝜂 𝑑𝜂
1
𝜉 1 − 𝜉
− 8 𝜉 1 − 𝜉 ෝ
𝑤 𝜉 𝑑𝜉
− 2𝐶 𝑘 න
0
1
𝜉
1 − 𝜉
ෝ
𝑤 𝜉 𝑑𝜉
𝐶 𝑘 =
𝐻1
2
𝑘
𝐻1
2
𝑘 + 𝑖𝐻0
2
𝑘
= 𝐹 𝑘 + 𝑖𝐺 𝑘 ≈ 1 −
0.165
1 −
0.0455
𝑘
𝑖
−
0.335
1 −
0.3
𝑘
𝑖
k ≈ 0.4
ෝ
𝑤 𝑥 = 𝑈(−ℎ + 𝛼 𝑏𝑎 − 𝑥 )
𝐿 = 𝜋𝜌𝑏2 ሷ
ℎ + 𝑈 ሶ
𝛼 − 𝑏𝑎 ሷ
𝛼 + 2𝜋𝜌𝑈𝑏𝐶 𝑘 ሶ
ℎ + 𝑈𝛼 + 𝑏
1
2
− 𝑎 ሶ
𝛼
𝑀
= 𝜋𝜌𝑏2 ቊ 𝑏𝑎 ሷ
ℎ − 𝑈𝑏
1
2
− 𝑎 ሶ
𝛼 − 𝑏2
1
8
+ 𝑎2 ሷ
𝛼
𝐿 = 𝜋𝜌𝑏2 ሷ
ℎ + 2𝜋𝜌𝑈𝑏𝐶 𝑘 ሶ
ℎ
1 + 𝑎 𝑏
𝑀𝛼 = 𝜋𝜌𝑏2 ሷ
ℎ𝑏𝑎 + 2𝜋𝜌𝑈𝑏𝐶 𝑘 ሶ
ℎ𝑏 𝑎 +
1
2
ሶ
ℎ
𝜋𝜌𝑏2
𝐿 = 𝜋𝜌𝑏2
𝑈 ሶ
𝛼 − 𝑏𝑎 ሷ
𝛼 + 2𝜋𝜌𝑈𝑏𝐶 𝑘 𝑈𝛼 + 𝑏
1
2
− 𝑎 ሶ
𝛼
1 + 𝑎 𝑏
𝑀
= −𝜋𝜌𝑏2𝑈 ሶ
𝛼𝑏
1
2
− 𝑎 − 𝜋𝜌𝑏4
1
8
+ 𝑎2 ሷ
𝛼
+ 2𝜋𝜌𝑏𝑈𝐶 𝑘 𝑈𝛼 + 𝑏
1
2
− 𝑎 ሶ
𝛼 𝑏 𝑎 +
1
2
𝑤3/4
−𝛽2
𝜕2
𝜙
𝜕𝑥2
+
𝜕2
𝜙
𝜕𝑧2
−
2𝑀∞
2
𝑈∞
𝜕2
𝜙
𝜕𝑡𝜕𝑥
−
𝑀∞
2
𝑈∞
2
𝜕2
𝜙
𝜕𝑡2
= 0
1
𝑈∞
ቇ
𝜕𝜙′
𝜕𝑧 ±
=
𝜕ℎ
𝜕𝑥
+
1
𝑈∞
𝜕ℎ
𝜕𝑡
, 𝛻𝜙′
→ 0 𝒙 → ∞, +Kutta
ℎ 𝑥, 𝑡 = ෠
ℎ 𝑥 𝑒−𝑖𝜔𝑡
෠
𝜙∗
𝑠; 𝑧
෠
𝜙(𝑥, 𝑧) 𝑧 ≥ 0
𝜙 x, z = −
1
𝛽
න
0
𝑥−𝛽𝑧
ෝ
𝑤 𝜉 exp −𝑖
𝜔𝑀∞
2
𝑈∞𝛽2
𝑥 − 𝜉
× 𝐽0
𝜔𝑀∞
2
𝑈∞𝛽2
𝑥 − 𝜉 2 − 𝛽2𝑧2 𝑑𝜉
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03-UnsteadyAero.pdf

  • 1.
  • 2. 𝜕𝒖 𝜕𝑡 + 𝛻 𝒖 2 2 + 1 𝜌 𝛻𝑝 = 0 𝜕𝜌 𝜕𝑡 + 𝛻 ⋅ 𝜌𝒖 = 0 𝑠 = 𝑐𝑜𝑛𝑠𝑡 𝜕𝜙 𝜕𝑡 + 𝛻𝜙 2 2 + 𝑎2 𝛾 − 1 = 𝐶 𝜕2𝜙 𝜕𝑡2 + 2𝛻𝜙 ⋅ 𝜕𝛻𝜙 𝜕𝑡 = 𝑎2𝛻2𝜙 − 𝛻𝜙 ⋅ 𝛻 𝛻𝜙 2 2
  • 3. 𝐷𝐹 𝐷𝑡 = 0 → 𝜕𝐹 𝜕𝑡 + 𝒖 ⋅ 𝛻𝐹 = 0 𝒖 → 𝑈∞𝒆𝒙 𝒙 → ∞ 𝑧 = 𝑓 𝑥, 𝑦, 𝑡 𝐹 𝑥, 𝑦, 𝑧, 𝑡 = 𝑧 − 𝑓 𝑥, 𝑦, 𝑡 = 0 𝑑𝐹 = 0 (𝒙, 𝑡 (𝒙 + 𝒅𝒙, 𝑡 + 𝑑𝑡
  • 4. 𝐹𝑢 𝑥, 𝑦, 𝑧, 𝑡 = 𝑧 − (𝑧+ 𝑥, 𝑦 + 𝛼 𝑐 − 𝑥 + ℎ 𝑥, 𝑦, 𝑡 ) − 𝜕ℎ 𝜕𝑡 − 𝑢 𝜕𝑧+ 𝜕𝑥 − 𝛼 + 𝜕ℎ 𝜕𝑥 − 𝑣 𝜕𝑧+ 𝜕𝑦 + 𝜕ℎ 𝜕𝑦 + 𝑤 = 0 𝐹𝑙 𝑥, 𝑦, 𝑧, 𝑡 = 𝑧 − (𝑧− 𝑥, 𝑦 + 𝛼 𝑐 − 𝑥 + ℎ 𝑥, 𝑦, 𝑡 ) − 𝜕ℎ 𝜕𝑡 − 𝑢 𝜕𝑧− 𝜕𝑥 − 𝛼 + 𝜕ℎ 𝜕𝑥 − 𝑣 𝜕𝑧− 𝜕𝑦 + 𝜕ℎ 𝜕𝑦 + 𝑤 = 0 𝑧+ 𝑥, 𝑦 + 𝛼(𝑐 − 𝑥) 𝑧− 𝑥, 𝑦 + 𝛼(𝑐 − 𝑥) ℎ(𝑥, 𝑦, 𝑡)
  • 5. 𝒖 𝑥, 𝑦, 𝑧, 𝑡 = 𝑈∞𝒆𝒙 + 𝒖′ 𝑥, 𝑦, 𝑧, 𝑡 𝜙 𝑥, 𝑦, 𝑧, 𝑡 = 𝑈∞ 𝑥 + 𝜙′ 𝑥, 𝑦, 𝑧, 𝑡 1 − 𝑀∞ 2 𝜕2 𝜙′ 𝜕𝑥2 + 𝜕2 𝜙′ 𝜕𝑦2 + 𝜕2 𝜙′ 𝜕𝑧2 − 2𝑀∞ 2 𝑈∞ 𝜕2 𝜙′ 𝜕𝑡𝜕𝑥 − 𝑀∞ 2 𝑈∞ 2 𝜕2 𝜙′ 𝜕𝑡2 = 0 1 𝑈∞ ቇ 𝜕𝜙′ 𝜕𝑧 ± = 𝜕𝑧± 𝜕𝑥 − 𝛼 + 𝜕ℎ 𝜕𝑥 + 1 𝑈∞ 𝜕ℎ 𝜕𝑡 , 𝛻𝜙′ → 0 𝒙 → ∞ 𝑐𝑝 = − 2 𝑈∞ 2 𝜕𝜙′ 𝜕𝑡 + 𝑈∞ 𝜕𝜙′ 𝜕𝑥 𝜙′
  • 6. 1 − 𝑀∞ 2 𝜕2 𝜙1 𝜕𝑥2 + 𝜕2 𝜙1 𝜕𝑦2 + 𝜕2 𝜙1 𝜕𝑧2 − 2𝑀∞ 2 𝑈∞ 𝜕2 𝜙 𝜕𝑡𝜕𝑥 − 𝑀∞ 2 𝑈∞ 2 𝜕2 𝜙 𝜕𝑡2 = 0 1 𝑈∞ ቇ 𝜕𝜙1 𝜕𝑧 ± = 𝜕𝑧± 𝜕𝑥 − 𝛼 , 𝛻𝜙′ → 0 𝒙 → ∞ 1 − 𝑀∞ 2 𝜕2𝜙2 𝜕𝑥2 + 𝜕2𝜙2 𝜕𝑦2 + 𝜕2𝜙2 𝜕𝑧2 − 2𝑀∞ 2 𝑈∞ 𝜕2𝜙2 𝜕𝑡𝜕𝑥 − 𝑀∞ 2 𝑈∞ 2 𝜕2𝜙2 𝜕𝑡2 = 0 1 𝑈∞ ቇ 𝜕𝜙2 𝜕𝑧 ± = 𝜕ℎ 𝜕𝑥 + 1 𝑈∞ 𝜕ℎ 𝜕𝑡 , 𝛻𝜙′ → 0 𝒙 → ∞ 0
  • 7. 𝜕2 𝜙 𝜕𝑥2 + 𝜕2 𝜙 𝜕𝑧2 = 0 1 𝑈∞ ቇ 𝜕𝜙′ 𝜕𝑧 ± = 𝜕ℎ 𝜕𝑥 + 1 𝑈∞ 𝜕ℎ 𝜕𝑡 , 𝛻𝜙′ → 0 𝒙 → ∞, +Kutta 𝑢 𝑥, 0, 𝑡 𝑤′ 𝑥, 0, 𝑡 = − 1 2𝜋 න 0 ∞ 2𝑢′ 𝜉, 0, 𝑡 𝑥 − 𝜉 𝑑𝜉 𝛾 = 2𝑢′ Γ 𝑥, 𝑡 = ‫׬‬ 0 𝑥 2𝑢′ 𝜉, 0, 𝑡 𝑑𝜉
  • 8. 𝑢 𝑥, 0, 𝑡 𝑤′ 𝑥, 0, 𝑡 = − 1 2𝜋 න 0 𝑐 2𝑢′ 𝜉, 0, 𝑡 𝑥 − 𝜉 𝑑𝜉 − 1 2𝜋 න 𝑐 ∞ 2𝑢′ 𝜉, 0, 𝑡 𝑥 − 𝜉 𝑑𝜉 𝑪𝒑 𝜕𝜙′ 𝜕𝑡 + 𝑈∞ 𝜕𝜙′ 𝜕𝑥 = 0 → 𝜙′ 𝑥, 0, 𝑡 = 𝜙′ 𝑐, 0, 𝑡 − 𝑥 − 𝑐 𝑈∞ Γ 𝑥, 𝑡 = Γ 𝑐, 𝑡 − 𝑥 − 𝑐 𝑈∞ 𝛾 𝑥, 𝑡 = − 1 𝑈∞ 𝜕Γ 𝜕𝑡 𝑐, 𝑡 − 𝑥 − 𝑐 𝑈∞
  • 9. ℎ, 𝑤, 𝑢, Γ, 𝑔 𝑥, 𝑡 = {෠ ℎ, ෝ 𝑤, ො 𝑢, ෠ Γ, ො 𝑔} 𝑥 𝑒−𝑖𝜔𝑡 ෝ 𝒖 𝝃 ෝ 𝑤 𝑥 = − 1 2𝜋 න 0 𝑐 2ො 𝑢 𝜉 𝑥 − 𝜉 𝑑𝜉 + ො 𝑔(𝑥) ො 𝑔 𝑥 = 𝑖𝜔 2𝜋 ෠ Γ 𝑐 න 𝑐 ∞ 𝑒−𝑖𝜔/𝑈∞ 𝜉−𝑐 𝑥 − 𝜉 𝑑𝜉 ො 𝑢 𝑥 = 1 𝜋 𝑐 − 𝑥 𝑥 න 0 𝑐 (ෝ 𝑤 𝜉 − ො 𝑔 𝜉 ) 𝑥 − 𝜉 𝜉 𝑐 − 𝜉 𝑑𝜉
  • 10. ො 𝑢 𝑥 = 1 𝜋 𝑐 − 𝑥 𝑥 න 0 𝑐 (ෝ 𝑤 𝜉 − ො 𝑔 𝜉 ) 𝑥 − 𝜉 𝜉 𝑐 − 𝜉 𝑑𝜉 ෠ Γ 𝑐 ෠ Γ 𝑐 = ‫׬‬ 0 𝑐 2ො 𝑢 𝑥 𝑑𝑥 ෠ Γ 𝑐 ෠ Γ 𝑐 = 4 𝑒−𝑖𝑘 ‫׬‬ 0 𝑐 𝑥 𝑐 − 𝑥 ෝ 𝑤 𝑥 𝑑𝑥 𝜋𝑖𝑘 𝐻1 2 𝑘 + 𝑖𝐻0 2 𝑘 , 𝑘 = 𝜔𝑐 2𝑈∞ 𝐻𝑛 (2) 𝐻0 2 𝑘 = 2𝑖 𝜋 න 1 ∞ 𝑒−𝑖𝑘𝑥 𝑥2 − 1 𝐻1 2 = − 2𝑖 𝜋𝑘 න 1 ∞ 𝑒−𝑖𝑘𝑥 𝑥2 − 1 3 𝑑𝑥
  • 11. Ƹ 𝑐𝑝 𝑥 = 2 𝜋 𝑐 − 𝑥 𝑥 න 0 𝑐 𝜉 𝑐 − 𝜉 ෝ 𝑤 𝜉 𝑑𝜉 + 4𝑖𝑘 𝜋𝑐 𝑥 𝑐 − 𝑥 න 0 𝑐 න 0 𝜉 ෝ 𝑤 𝜂 𝑑𝜂 𝑑𝜉 (𝑥 − 𝜉) 𝜉 𝑐 − 𝜉 + 4 𝜋𝑐 1 − 𝐶 𝑘 𝑐 − 𝑥 𝑥 න 0 𝑐 𝜉 𝑐 − 𝜉 ෝ 𝑤 𝜉 𝑑𝜉 𝐶 𝑘 = 𝐻1 2 𝑘 𝐻1 2 𝑘 + 𝑖𝐻0 2 𝑘 𝑘 𝑘 = 𝜔𝑐 2𝑈∞
  • 12. መ 𝐶𝑙 = −4𝐶 𝑘 න 0 1 𝜉 1 − 𝜉 ෝ 𝑤 𝜉 𝑑𝜉 − 8𝑖𝑘 න 0 1 𝜉 1 − 𝜉 ෝ 𝑤 𝜉 𝑑𝜉 መ 𝐶 𝑚 1 2 = න 0 1 𝜉 1 − 𝜉 − 4 𝜉 1 − 𝜉 ෝ 𝑤 𝜉 𝑑𝜉 + 𝑖𝑘 න 0 1 න 0 𝜉 ෝ 𝑤 𝜂 𝑑𝜂 1 𝜉 1 − 𝜉 − 8 𝜉 1 − 𝜉 ෝ 𝑤 𝜉 𝑑𝜉 − 2𝐶 𝑘 න 0 1 𝜉 1 − 𝜉 ෝ 𝑤 𝜉 𝑑𝜉
  • 13. 𝐶 𝑘 = 𝐻1 2 𝑘 𝐻1 2 𝑘 + 𝑖𝐻0 2 𝑘 = 𝐹 𝑘 + 𝑖𝐺 𝑘 ≈ 1 − 0.165 1 − 0.0455 𝑘 𝑖 − 0.335 1 − 0.3 𝑘 𝑖 k ≈ 0.4
  • 14. ෝ 𝑤 𝑥 = 𝑈(−ℎ + 𝛼 𝑏𝑎 − 𝑥 )
  • 15. 𝐿 = 𝜋𝜌𝑏2 ሷ ℎ + 𝑈 ሶ 𝛼 − 𝑏𝑎 ሷ 𝛼 + 2𝜋𝜌𝑈𝑏𝐶 𝑘 ሶ ℎ + 𝑈𝛼 + 𝑏 1 2 − 𝑎 ሶ 𝛼 𝑀 = 𝜋𝜌𝑏2 ቊ 𝑏𝑎 ሷ ℎ − 𝑈𝑏 1 2 − 𝑎 ሶ 𝛼 − 𝑏2 1 8 + 𝑎2 ሷ 𝛼
  • 16. 𝐿 = 𝜋𝜌𝑏2 ሷ ℎ + 2𝜋𝜌𝑈𝑏𝐶 𝑘 ሶ ℎ 1 + 𝑎 𝑏 𝑀𝛼 = 𝜋𝜌𝑏2 ሷ ℎ𝑏𝑎 + 2𝜋𝜌𝑈𝑏𝐶 𝑘 ሶ ℎ𝑏 𝑎 + 1 2 ሶ ℎ 𝜋𝜌𝑏2
  • 17. 𝐿 = 𝜋𝜌𝑏2 𝑈 ሶ 𝛼 − 𝑏𝑎 ሷ 𝛼 + 2𝜋𝜌𝑈𝑏𝐶 𝑘 𝑈𝛼 + 𝑏 1 2 − 𝑎 ሶ 𝛼 1 + 𝑎 𝑏 𝑀 = −𝜋𝜌𝑏2𝑈 ሶ 𝛼𝑏 1 2 − 𝑎 − 𝜋𝜌𝑏4 1 8 + 𝑎2 ሷ 𝛼 + 2𝜋𝜌𝑏𝑈𝐶 𝑘 𝑈𝛼 + 𝑏 1 2 − 𝑎 ሶ 𝛼 𝑏 𝑎 + 1 2 𝑤3/4
  • 18. −𝛽2 𝜕2 𝜙 𝜕𝑥2 + 𝜕2 𝜙 𝜕𝑧2 − 2𝑀∞ 2 𝑈∞ 𝜕2 𝜙 𝜕𝑡𝜕𝑥 − 𝑀∞ 2 𝑈∞ 2 𝜕2 𝜙 𝜕𝑡2 = 0 1 𝑈∞ ቇ 𝜕𝜙′ 𝜕𝑧 ± = 𝜕ℎ 𝜕𝑥 + 1 𝑈∞ 𝜕ℎ 𝜕𝑡 , 𝛻𝜙′ → 0 𝒙 → ∞, +Kutta ℎ 𝑥, 𝑡 = ෠ ℎ 𝑥 𝑒−𝑖𝜔𝑡 ෠ 𝜙∗ 𝑠; 𝑧 ෠ 𝜙(𝑥, 𝑧) 𝑧 ≥ 0 𝜙 x, z = − 1 𝛽 න 0 𝑥−𝛽𝑧 ෝ 𝑤 𝜉 exp −𝑖 𝜔𝑀∞ 2 𝑈∞𝛽2 𝑥 − 𝜉 × 𝐽0 𝜔𝑀∞ 2 𝑈∞𝛽2 𝑥 − 𝜉 2 − 𝛽2𝑧2 𝑑𝜉