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November  2014, 19(9): 2767-2783. doi: 10.3934/dcdsb.2014.19.2767

Straightforward approximation of the translating and pulsating free surface Green function

Zhi-Min Chen 1,

1. 

Ship Science, University of Southampton, Southampton SO17 1BJ, United Kingdom

Received  October 2013 Revised  March 2014 Published  September 2014

The translating and pulsating free surface Green function represents the velocity potential of a three-dimensional free surface source advancing in waves. This function involves singular wave integral, which is troublesome in numerical computation. In the present study, a regular wave integral approach is developed for the discretisation of the singular wave integral in a whole space harmonic function expansion, which permits the free surface wave produced by the fluid motion to be decomposed by plane regular propagation waves. This approximation gives rise to a simple and straightforward evaluation of the Green function. The algorithm is validated from comparisons between present numerical results and existing numerical data.
Keywords: hydrodynamics., numerical approximation, surface gravity wave, Free surface Green function.
Mathematics Subject Classification: 31B10, 35Q35, 76B07, 76B2.
Citation: Zhi-Min Chen. Straightforward approximation of the translating and pulsating free surface Green function. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2767-2783. doi: 10.3934/dcdsb.2014.19.2767
References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards Mathematics Series 55, Washington, USA, 1964. doi: 10.1119/1.1972842.  Google Scholar

[2]

N. F. Bondarenko, M. Z. Gak and F. V. Dolzhansky, Laboratory and theoretical models of a plane periodic flow, Izv. Atmos. Oceanic Phys., 15 (1979), 711-716. Google Scholar

[3]

M. Bessho, On the fundamental singularity in a theory of motions in a seaway, Memories of the Defense Academy Japan, 17 (1977), 95-105.  Google Scholar

[4]

J. G. Charney and J. G. DeVore, Multiple flow equilibria in the atmosphere and blocking, J. Atmos. Sci., 36 (1979), 1205-1216. doi: 10.1175/1520-0469(1979)036<1205:MFEITA>2.0.CO;2.  Google Scholar

[5]

Z. -M. Chen and W. G. Price, Secondary fluid flows driven electromagnetically in a two-dimensional extended duct, Proc. R. Soc. Lond. Ser. A, 461 (2005), 1659-1683. doi: 10.1098/rspa.2005.1454.  Google Scholar

[6]

Z. -M. Chen, A vortex based panel method for potential flow simulation around a hydrofoil, J. Fluids Struct., 28 (2012), 378-391. doi: 10.1016/j.jfluidstructs.2011.10.003.  Google Scholar

[7]

Z. -M. Chen, Harmonic function expansion for translating Green functions and dissipative free-surface waves, Wave Motion, 50 (2013), 282-294. doi: 10.1016/j.wavemoti.2012.09.005.  Google Scholar

[8]

Z. -M. Chen, Regular wave integral approach to the prediction of hydrodynamic performance of submerged spheroid, Wave Motion, 51 (2014), 193-205. doi: 10.1016/j.wavemoti.2013.06.005.  Google Scholar

[9]

G. Dagan and T. Miloh, Free-surface flow past oscillating singularities at resonant frequency, J. Fluid Mech., 120 (1982), 139-54. doi: 10.1017/S0022112082002705.  Google Scholar

[10]

L. K. Forbes, An algorithm for 3-dimensional free-surface problems in hydrodynamics, J. Comput. Phys., 82 (1989), 330-347. doi: 10.1016/0021-9991(89)90052-1.  Google Scholar

[11]

J. Grue and E. Palm, Wave radiation and wave diffraction from a submerged body in a uniform current, J. Fluid Mech., 151 (1985), 257-78. doi: 10.1017/S0022112085000957.  Google Scholar

[12]

M. D. Haskind, On wave motion of a heavy fluid, Prikl. Mat. Mekh., 18 (1954), 15-26.  Google Scholar

[13]

T. H. Havelock, Wave resistance, Proc. R. Soc. Lond. Ser. A, 118 (1928), 24-33. doi: 10.1098/rspa.1928.0033.  Google Scholar

[14]

T. H. Havelock, The theory of wave resistance, Proc. R. Soc. Lond. Ser. A, 138 (1932), 339-348. doi: 10.1098/rspa.1932.0188.  Google Scholar

[15]

A. J. Hess and A. M. O. Smith, Calculation of non-lifting potential flow about arbitrary three-dimensional bodies, J. Ship Res., 8 (1964), 22-24. Google Scholar

[16]

A. J. Hess and A. M. O. Smith, Calculation of potential flow about arbitrary bodies, Prog. Aeronautical Sci., 8 (1966), 1-138. doi: 10.1016/0376-0421(67)90003-6.  Google Scholar

[17]

J. L. Hess and D. C. Wilcox, Progress in the Solution of the Problem of a Three-Dimensional Body Oscillating in the Presence of a Free Surface, Final technical report, McDonnell Douglas Company Rep. DAC 67647, 1969. Google Scholar

[18]

R. B. Inglis and W. G. Price, Calculation of the velocity potential of a translating, pulsating source, Transactions of the Royal Institution of Naval Architects, 123 (1980), 163-175. Google Scholar

[19]

H. Iwashita and M. Ohkusu, The Green function method for ship motions at forward speed, Ship Tech. Res., 39 (1992), 3-21. Google Scholar

[20]

Y. Liu and D. K. P. Yue, On the solution near the critical frequency for an oscillating and translating body in or near a free surface, J. Fluid Mech., 254 (1993), 251-66. doi: 10.1017/S0022112093002113.  Google Scholar

[21]

A. Mo and E. Palm, On radiated and scattered waves from a submerged elliptic cylinder in a uniform current, J. Ship Res., 31 (1987), 23-33. Google Scholar

[22]

J. N. Newman, Algorithms for the free-surface Green function, J. Engng. Math., 19 (1985), 57-67. doi: 10.1007/BF00055041.  Google Scholar

[23]

J. N. Newman, Evaluation of the wave-resistance Green function: Part 1 - The double integral, J. Ship Res., 31 (1987), 79-90. Google Scholar

[24]

J. N. Newman, Evaluation of the wave-resistance Green function: Part 2 - the single integral on the centerplane, J. Ship Res., 31 (1987), 145-150. Google Scholar

[25]

F. Noblesse, Alternative integral representations for the Green function of the theory of ship wave resistance, J. Engng. Math., 15 (1981), 241-265. doi: 10.1007/BF00042923.  Google Scholar

[26]

F. Noblesse, The Green function in the theory of radiation and diffraction of regular water waves by a body, J. Engng. Math., 16 (1982), 137-169. doi: 10.1007/BF00042551.  Google Scholar

[27]

J. V. Wehausen and E. V. Laitone, Surface waves, in Fluid Dynamics III, (Eds. S. Flugge, C. Truesdell), in Handbuch der Physik 9, Springer, Berlin, 1960, 446-778.  Google Scholar

[28]

Y. Zhang and S. Zhu, Resonant interaction between a uniform current and an oscillating object, Appl. Ocean Res., 27 (1995), 259-264. doi: 10.1016/0141-1187(95)00018-6.  Google Scholar

show all references

References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards Mathematics Series 55, Washington, USA, 1964. doi: 10.1119/1.1972842.  Google Scholar

[2]

N. F. Bondarenko, M. Z. Gak and F. V. Dolzhansky, Laboratory and theoretical models of a plane periodic flow, Izv. Atmos. Oceanic Phys., 15 (1979), 711-716. Google Scholar

[3]

M. Bessho, On the fundamental singularity in a theory of motions in a seaway, Memories of the Defense Academy Japan, 17 (1977), 95-105.  Google Scholar

[4]

J. G. Charney and J. G. DeVore, Multiple flow equilibria in the atmosphere and blocking, J. Atmos. Sci., 36 (1979), 1205-1216. doi: 10.1175/1520-0469(1979)036<1205:MFEITA>2.0.CO;2.  Google Scholar

[5]

Z. -M. Chen and W. G. Price, Secondary fluid flows driven electromagnetically in a two-dimensional extended duct, Proc. R. Soc. Lond. Ser. A, 461 (2005), 1659-1683. doi: 10.1098/rspa.2005.1454.  Google Scholar

[6]

Z. -M. Chen, A vortex based panel method for potential flow simulation around a hydrofoil, J. Fluids Struct., 28 (2012), 378-391. doi: 10.1016/j.jfluidstructs.2011.10.003.  Google Scholar

[7]

Z. -M. Chen, Harmonic function expansion for translating Green functions and dissipative free-surface waves, Wave Motion, 50 (2013), 282-294. doi: 10.1016/j.wavemoti.2012.09.005.  Google Scholar

[8]

Z. -M. Chen, Regular wave integral approach to the prediction of hydrodynamic performance of submerged spheroid, Wave Motion, 51 (2014), 193-205. doi: 10.1016/j.wavemoti.2013.06.005.  Google Scholar

[9]

G. Dagan and T. Miloh, Free-surface flow past oscillating singularities at resonant frequency, J. Fluid Mech., 120 (1982), 139-54. doi: 10.1017/S0022112082002705.  Google Scholar

[10]

L. K. Forbes, An algorithm for 3-dimensional free-surface problems in hydrodynamics, J. Comput. Phys., 82 (1989), 330-347. doi: 10.1016/0021-9991(89)90052-1.  Google Scholar

[11]

J. Grue and E. Palm, Wave radiation and wave diffraction from a submerged body in a uniform current, J. Fluid Mech., 151 (1985), 257-78. doi: 10.1017/S0022112085000957.  Google Scholar

[12]

M. D. Haskind, On wave motion of a heavy fluid, Prikl. Mat. Mekh., 18 (1954), 15-26.  Google Scholar

[13]

T. H. Havelock, Wave resistance, Proc. R. Soc. Lond. Ser. A, 118 (1928), 24-33. doi: 10.1098/rspa.1928.0033.  Google Scholar

[14]

T. H. Havelock, The theory of wave resistance, Proc. R. Soc. Lond. Ser. A, 138 (1932), 339-348. doi: 10.1098/rspa.1932.0188.  Google Scholar

[15]

A. J. Hess and A. M. O. Smith, Calculation of non-lifting potential flow about arbitrary three-dimensional bodies, J. Ship Res., 8 (1964), 22-24. Google Scholar

[16]

A. J. Hess and A. M. O. Smith, Calculation of potential flow about arbitrary bodies, Prog. Aeronautical Sci., 8 (1966), 1-138. doi: 10.1016/0376-0421(67)90003-6.  Google Scholar

[17]

J. L. Hess and D. C. Wilcox, Progress in the Solution of the Problem of a Three-Dimensional Body Oscillating in the Presence of a Free Surface, Final technical report, McDonnell Douglas Company Rep. DAC 67647, 1969. Google Scholar

[18]

R. B. Inglis and W. G. Price, Calculation of the velocity potential of a translating, pulsating source, Transactions of the Royal Institution of Naval Architects, 123 (1980), 163-175. Google Scholar

[19]

H. Iwashita and M. Ohkusu, The Green function method for ship motions at forward speed, Ship Tech. Res., 39 (1992), 3-21. Google Scholar

[20]

Y. Liu and D. K. P. Yue, On the solution near the critical frequency for an oscillating and translating body in or near a free surface, J. Fluid Mech., 254 (1993), 251-66. doi: 10.1017/S0022112093002113.  Google Scholar

[21]

A. Mo and E. Palm, On radiated and scattered waves from a submerged elliptic cylinder in a uniform current, J. Ship Res., 31 (1987), 23-33. Google Scholar

[22]

J. N. Newman, Algorithms for the free-surface Green function, J. Engng. Math., 19 (1985), 57-67. doi: 10.1007/BF00055041.  Google Scholar

[23]

J. N. Newman, Evaluation of the wave-resistance Green function: Part 1 - The double integral, J. Ship Res., 31 (1987), 79-90. Google Scholar

[24]

J. N. Newman, Evaluation of the wave-resistance Green function: Part 2 - the single integral on the centerplane, J. Ship Res., 31 (1987), 145-150. Google Scholar

[25]

F. Noblesse, Alternative integral representations for the Green function of the theory of ship wave resistance, J. Engng. Math., 15 (1981), 241-265. doi: 10.1007/BF00042923.  Google Scholar

[26]

F. Noblesse, The Green function in the theory of radiation and diffraction of regular water waves by a body, J. Engng. Math., 16 (1982), 137-169. doi: 10.1007/BF00042551.  Google Scholar

[27]

J. V. Wehausen and E. V. Laitone, Surface waves, in Fluid Dynamics III, (Eds. S. Flugge, C. Truesdell), in Handbuch der Physik 9, Springer, Berlin, 1960, 446-778.  Google Scholar

[28]

Y. Zhang and S. Zhu, Resonant interaction between a uniform current and an oscillating object, Appl. Ocean Res., 27 (1995), 259-264. doi: 10.1016/0141-1187(95)00018-6.  Google Scholar

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