船尾迹的波动理论
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摘要
当船在水面上移动时，在船的后面会有一个V形尾迹，这个尾迹的角度是38°56′，称为经典Kelvin尾迹，Kelvin尾迹波系由扩散波和横向波组成，在两条Kelvin臂上这两个波系相交，这里波浪起伏最为明显，SAR(合成孔径雷达)的图像能显示Kelvin尾迹，对此许多学者作了研究。
本文着重研究船行波几个问题，从理论的角度上进行分析，解释和发现了有关船行波的一些现象，本文创新之处和主要结果有：
一、消除了粘性流体中船行波临界边的奇性。
本论文获得了粘性流体中的船行波模态，而且可以清晰的看到粘性对船行波的影响，当粘性渐渐增加时，发现扩散波的影响渐渐消失，变成以横向波为主。还得到了船行波的速度场，速度在跨过临界边时连续变化，在临界边的外侧，色散关系虽然没有实数解，因而没有波动，但是仍存在扰动。
二、对窄船行波的形成进行了初步的探索和研究。
我们参考了有限翼展水翼的升力理论，建立了马蹄涡与自由面相互作用的模型，发现马蹄涡跟自由面作用的时候，在尾迹上中间是一个相对平坦的区域，过了一定的角度范围，变成相对强烈震荡得区域，我们很准确的得到了这个角度的很简单的代数关系式，怀疑这个平坦区域到震荡区域的交接处可能对SAR得有着重要的影响。另外，考虑到水下物体(如水下潜艇)的情形，有可能存在圆涡，我们也初步研究了圆涡与自由面的相互作用。
三、研究在有限水深理想流体中的船行波
有限深船行波得到解析式很难，因为有双曲函数的关系，我们推广了Radko的方法，研究了Froude数对船行波态的影响，搞清楚了一些规律。
As is well known, when a ship moves on the water, there are steady V-shaped waves behind it. The wake angle is 38°56′and is usually called the Kelvin ship wave. The Kelvin wave system is composed of diverging waves and transverse waves. At the two Kelvin arms diverging waves and transverse waves intersect. There the undulation is especially obvious. Synthetic Aperture Radar (SAR) images of the sea surface can show the Kelvin wakes with many studies on it. Many scholars have also studied the waves including regarding the water as an inviscid fluid and as viscous fluid.
Some problems about ship waves are investigated and analyzed theoretically in this dissertation. some new phenomena about ship waves have been explained and obtained. The originalities and the main results in this dissertation are:
1.The singularity at the critical lines of ship waves on a viscous fluid is avoided.
The ship waves mode on a viscous fluid have been obtained. The effect of viscosity on ship waves can been seen clearly. When the viscosity become bigger, the effect of diverging waves disappears slowly and transverse waves dominates. The velocity fields are also obtained. The velocity changes continuously on the critical lines. At the outside of the critical lines, the dispersive relation has no real solution, but the disturbances still exist.
2. The formation of narrow ship waves has been explored and studied
The paper makes reference to the lifting theory of the submerged hydrofoil of finite span and establishes a model of the interaction between a submerged horseshoe vortex and a free surface far from the ship. smooth region. In the region with an included angle of just a few degrees the free surface is smooth. Then the fast oscillation begins when the angle becomes larger. We get the exact angle expression. It is trusted that this region has important effect on the SAR image of narrow V-images. In addition, the interaction between a submerged circle vortex and a free surface has also been considered. The circle vortex could be exist if the submerged body such as a submarine moves under the water.
3. Study the ship waves on an ideal fluid of finite depth
It's difficulty to obtain the expression of wave elevation because of the hyperbolic function. We extended Radko's method and studied the effects of Froude number on ship wave modes and made some phenomena clear.
本文着重研究船行波几个问题，从理论的角度上进行分析，解释和发现了有关船行波的一些现象，本文创新之处和主要结果有：
一、消除了粘性流体中船行波临界边的奇性。
本论文获得了粘性流体中的船行波模态，而且可以清晰的看到粘性对船行波的影响，当粘性渐渐增加时，发现扩散波的影响渐渐消失，变成以横向波为主。还得到了船行波的速度场，速度在跨过临界边时连续变化，在临界边的外侧，色散关系虽然没有实数解，因而没有波动，但是仍存在扰动。
二、对窄船行波的形成进行了初步的探索和研究。
我们参考了有限翼展水翼的升力理论，建立了马蹄涡与自由面相互作用的模型，发现马蹄涡跟自由面作用的时候，在尾迹上中间是一个相对平坦的区域，过了一定的角度范围，变成相对强烈震荡得区域，我们很准确的得到了这个角度的很简单的代数关系式，怀疑这个平坦区域到震荡区域的交接处可能对SAR得有着重要的影响。另外，考虑到水下物体(如水下潜艇)的情形，有可能存在圆涡，我们也初步研究了圆涡与自由面的相互作用。
三、研究在有限水深理想流体中的船行波
有限深船行波得到解析式很难，因为有双曲函数的关系，我们推广了Radko的方法，研究了Froude数对船行波态的影响，搞清楚了一些规律。
As is well known, when a ship moves on the water, there are steady V-shaped waves behind it. The wake angle is 38°56′and is usually called the Kelvin ship wave. The Kelvin wave system is composed of diverging waves and transverse waves. At the two Kelvin arms diverging waves and transverse waves intersect. There the undulation is especially obvious. Synthetic Aperture Radar (SAR) images of the sea surface can show the Kelvin wakes with many studies on it. Many scholars have also studied the waves including regarding the water as an inviscid fluid and as viscous fluid.
Some problems about ship waves are investigated and analyzed theoretically in this dissertation. some new phenomena about ship waves have been explained and obtained. The originalities and the main results in this dissertation are:
1.The singularity at the critical lines of ship waves on a viscous fluid is avoided.
The ship waves mode on a viscous fluid have been obtained. The effect of viscosity on ship waves can been seen clearly. When the viscosity become bigger, the effect of diverging waves disappears slowly and transverse waves dominates. The velocity fields are also obtained. The velocity changes continuously on the critical lines. At the outside of the critical lines, the dispersive relation has no real solution, but the disturbances still exist.
2. The formation of narrow ship waves has been explored and studied
The paper makes reference to the lifting theory of the submerged hydrofoil of finite span and establishes a model of the interaction between a submerged horseshoe vortex and a free surface far from the ship. smooth region. In the region with an included angle of just a few degrees the free surface is smooth. Then the fast oscillation begins when the angle becomes larger. We get the exact angle expression. It is trusted that this region has important effect on the SAR image of narrow V-images. In addition, the interaction between a submerged circle vortex and a free surface has also been considered. The circle vortex could be exist if the submerged body such as a submarine moves under the water.
3. Study the ship waves on an ideal fluid of finite depth
It's difficulty to obtain the expression of wave elevation because of the hyperbolic function. We extended Radko's method and studied the effects of Froude number on ship wave modes and made some phenomena clear.
引文
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2. Allen, R.E The effect of interference and viscous effects in the Kelvin ship-wave problem [J]. Journal of Fluid Mechanics 1968, Vol.34: 417-421.
3. Ammicht.E. Free surface and surface tension effects on submerged bodies [J]. Jouranl of Engineering Mathematics 1979, Vol.13 No.4: 327-338.
4. Anderson, B.J. On ship waves in shallow water[C]. In Proceedings of the 1976 International Seminar on Wave Resistance 1976, 341-347. New York: Office of Naval Research
5. Bentwich, M., Miloh, T. The unsteady matched Stokes-Oseen solution for the flow past a sphere [J]. Journal of Fluid Mechanics 1978, Vol.38 No.2: 17-32.
6. Bessho, M. On a consistent linearized theory of the wave-making resistance of ships [J]. Journal of Ship Research 1994, Vol.38 No.2: 83-96
7. Bikitimirov, Y.K. On solving the problem of ship motion in shallow water using the second-order approximation[C]. Proceedings of the International Serminar on Wave Resistance 1976, 1-6.
8. Blake, J.R., Chwang. A.T. Fundamental singularities of viscous flow. Part Ⅰ. The image system in the vicinity of a stationary no-slip boundary [J]. Journal of Engineering Mathematics 1974, Vol.8: 23-29.
9. Brard. R. Viscosity, wake and ship waves [J]. Journal of Ship Research 1970, Vol.14: 207-240.
10. Brard. R. The representaion of a given ship form by singularity distributions when the boundary condition on the free surface is linearized [J]. Journal of Ship Research 1972, Vol.16: 79-92.
11. Brown. E.D., Buchsbaum, S.B., Hall, R.E., Penhume, J.P., Schmitte, K.E, Watson, K.M., Wyatt, D.C. Observations of a nonlinear solitary wave packet in the Kelvin wake of a ship [J]. Journal of Fluid Mechanics 1989, Vol.204: 263-293.
12. Carlaw, H.J., Jaegar, J.C. Conduction of Heat in Solids[M]. London: Oxford University Press, 1959.
13. Chan, A.T., Chwang, A.T. Interaction of laminar far wake with a free [J]. Physics of Fluids 1996, Vol.8 No.2: 421-429.
14. Chan, A.T., Chwang, A.T. Ship waves on a viscous fluid of finite depth [J]. Physics of Fluids 1997, Vol.9 No.4: 940-944.
15. Chester, C., Friedman, B. & Ursell, F. Anestension of the method of steepest descent[C]. Proceedings of the Cambridge Philosophical Society 1957, Vol.53: 599-610.
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