A free surface frequency domain green function with viscous dissipation and partial reflections from side walls

Abstract

The free-surface Green function method is widely used in solving the radiation or diffraction problems caused by a ship or ocean structure oscillating on the waves. In the context of inviscid potential flow, hydrodynamic problems such as multi-body interaction and tank side wall effect cannot be properly dealt with based on the traditional free-surface frequency domain Green function method, in which the water viscosity is omitted and the energy dissipation effect is absent. In this paper, an open-sea Green function with viscous dissipation was presented within the theory of visco-potential flow. Then the tank Green function with a partial reflection from the side walls in wave tanks was formulated as a formal sum of open-sea Green functions representing the infinite images between two parallel side walls of the source in the tank. The new far-field characteristics of the tank Green function is vitally important for improving the validity of side-wall effects evaluation, which can be used in supervising the tank model tests.

Keywords

Green function / viscous dissipation / side wall effect / partial reflection

References

[1] Abramowitz M., Stegun I.A. Handbook of mathematical functions, 1964, New York: Dover Publications

[2] Chen X. On the side wall effects upon bodies of arbitrary geometry in wave tanks. Applied Ocean Research, 1994, 16: 337-345

[3] Chen Xiaobo (2004). Hydrodynamics in offshore and naval applications-Part I. Keynote Lecure at the 6th International Conference on Hydrodynamics, Perth, Australia.

[4] Chen Xiaobo, Dias F (2010). Visco-potential flow and time-harmonic ship waves. 25th IwwwFB, Harbin, China.

[5] Chen Xiaobo, Lu Dongqiang (2009). Time-harmonic ship waves with the effect of surface tension and fluid viscosity. 24th IwwwFB, Zelenogork, Russia.

[6] Dias F., Dyachenko A.F., Zakharov V.E. Theory of weakly damped free-surface flows: a new formulation based on potential flow solutions. Physics letters, A, 2008, 372: 1297-1302

[7] Dutykh D., Dias F. Viscous potential free-surface in a fluid layer of finite depth. C. R. Acad. Sci. Paris, Ser. I, 2007, 345: 113-118

[8] Evans D.V. The use of porous screens as wave dampers in narrow wave tanks. Journal of Engineering Mathematics, 1990, 24: 203-212

[9] John F. On the motion of floating bodies II:simple harmonic motions. Communications on Pure and Applied Mathematics, 1950, 3: 45-101

[10] Kashiwagi M. Radiation and diffraction forces acting on an offshore-structure model in a towing tank. International Journal of Offshore and Polar Engineering, 1991, 1(2): 101-107

[11] Linton C.M. On the free-surface Green’s function for channel problems. Applied Ocean Research, 1993, 15: 263-267

[12] Linton C.M., Evans D.V. The radiation and scattering of surface waves by a vertical circular cylinder in a channel. Phil. Trans. R. Soc. Lond. A., 1992, 338: 325-357

[13] McIver P. The wave field scattered by a vertical cylinder in narrow wave tank. Applied Ocean Research, 1993, 15: 25-37

[14] McIver P., Benneett G.S. Scattering of water waves by axisymmetric bodies in a channel. Journal of Engineering Mathematics, 1993, 27: 1-29

[15] Newman J.N. Double precision evaluation of the oscillatory source potential. Journal of Ship Research, 1984, 28(3): 151-154

[16] Qin Hongde, Shen Jing (2010). Green function with dissipation and side wall effect in wave tanks. 25th IwwwFB, Harbin, China.

[17] Shen Jing, Qin Hongde (2011). Tank Green function with partial reflections from side walls. 26th IwwwFB, Athens (Greece).

[18] Taylor R.E., Hung S.M. Mean drift forces on an articulated column oscillating in a wave tank. Applied Ocean Research, 1985, 7(2): 66-78

[19] Vazquez J.H., Williams A.N. Hydrodynamic loads on a three-dimensional body in a narrow tank. Journal of Offshore Mechanics and Arctic Engineering, 1994, 116: 117-121

[20] Xia Jinzhu (2002). Some insight into the Green function of the channel problem. 17th IwwwFB, Cambridge, UK.

[21] Yeung R.W., Sphaier S.H. Wave-interference effects on a truncated cylinder in a channel. Journal of Engineering Mathematics, 1989, 23(2): 95-117