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Poisson brackets for the dynamically coupled system of a free boundary and a neutrally buoyant rigid body in a body-fixed frame
Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, NM 88003-8001, USA |
The fully coupled dynamic interaction problem of the free surface of an incompressible fluid and a rigid body beneath it, in an inviscid, irrotational framework and in the absence of surface tension, is considered. Evolution equations of the global momenta of the body+fluid system are derived. It is then shown that, under fairly general assumptions, these evolution equations combined with the evolution equation of the free-surface, referred to a body-fixed frame, is a Hamiltonian system. The Poisson brackets of the system are the sum of the canonical Zakharov bracket and the non-canonical Lie-Poisson bracket. Variations are performed consistent with the mixed Dirichlet-Neumann problem governing the system.
References:
[1] |
G. A. Athanassoulis, Analytical dynamics of wave body interaction, in Mathematical techniques for water waves (ed. B. N. Mandal), Advances in Fluid Mechanics, 8, Computational Mechanics Publications, Southampton, Boston (1997), 79–154. Available from: urlhttps://www.researchgate.net/publication/266301554_Analytical_dynamics_of_wave-body_interaction |
[2] |
G. K. Batchelor, An Introduction to Fluid Dynamics,, Cambridge University Press, Cambridge, 1999.
![]() ![]() |
[3] |
M. Brøns, M. C. Thompson, T. Leweke and K. Hourigan,
Vorticity generation and conservation for two-dimensional interfaces and boundaries, J. Fluid Mech., 758 (2014), 63-93.
|
[4] |
R. Camassa and D. D. Holm,
An integrable shallow water equation with peaked solitons, Physical Review Letters, 71 (1993), 1661-1664.
doi: 10.1103/PhysRevLett.71.1661. |
[5] |
A. Compelli, R. Ivanov and M. Todorov, Hamiltonian models for the propogation of irrotational surface gravity waves over a variable bottom, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 376 (2018), 20170091, 15 pp.
doi: 10.1098/rsta.2017.0091. |
[6] |
L. D. Faddeev and L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons, Classics in Mathematics, Springer, 2007. |
[7] |
C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura,
Method for solving the Korteweg-de Vries equation, Phys, Rev. Lett., 19 (1967), 1095-1097.
|
[8] |
F. Gay-Balmaz, J. E. Marsden and T. S. Ratiu,
Reduced variational formulations in free boundary continuum mechanics, J. Nonlinear Sci., 22 (2012), 463-497.
doi: 10.1007/s00332-012-9143-4. |
[9] |
R. Ivanov,
Two-component integrable systems modelling shallow water waves: The constant vorticity case, Wave Motion, 46 (2009), 389-396.
doi: 10.1016/j.wavemoti.2009.06.012. |
[10] |
R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge Texts in Applied Mathematics, Cambridge University Press, 1997.
doi: 10.1017/CBO9780511624056.![]() ![]() ![]() |
[11] |
F. John,
On the motion of floating bodies. I., Comm. Pure App. Math., 2 (1949), 13-57.
doi: 10.1002/cpa.3160020102. |
[12] |
F. John,
On the motion of floating bodies. Ⅱ. Simple harmonic motions, Comm. Pure App. Math., 3 (1950), 45-101.
doi: 10.1002/cpa.3160030106. |
[13] |
G. Kirchhoff,
Ueber die Bewegung eines Rotationskörpers in einer Flüssigkeit, Journal Für Die Reine Und Angewandte Mathematik (Crelle's Journal), 71 (1869), 237-262.
doi: 10.1515/crll.1870.71.237. |
[14] |
E. A. Kuznetsov and V. P. Ruban,
Cherenkov interaction of vortices with a free surface, J. Exp. Theor. Phys., 88 (1999), 492-505.
doi: 10.1134/1.558820. |
[15] |
S. D. Kelly and H. Xiong, Self-propulsion of a free hydrofoil with localized discrete vortex shedding: Analytical modeling and simulation, Theoretical and Computational Fluid Dynamics, 24, (2010), 45–50. |
[16] |
H. Lamb, phHydrodynamics, 6$^{th}$ edition, Dover, New York, 1932. |
[17] |
D. Lewis, J. Marsden, R. Montgomery and T. Ratiu,
The Hamiltonian structure for dynamic free boundary problems, Physica D, 18 (1986), 391-404.
doi: 10.1016/0167-2789(86)90207-1. |
[18] |
T. Lundgren and P. Koumoutsakos,
On the generation of vorticity at a free surface, J. Fluid Mech., 382 (1999), 351-366.
|
[19] |
T. Miloh,
Hamilton's principle, Lagrange's method, and ship motion theory, J. Ship Research, 28 (1984), 229-237.
|
[20] |
T. Miloh,
A note on impulsive sphere motion beneath a free-surface, J. Engg. Math., 41 (2001), 1-11.
doi: 10.1023/A:1011895510151. |
[21] |
J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, Texts in Applied Mathematics 17, Second Edition, Springer-Verlag, 1999.
doi: 10.1007/978-0-387-21792-5. |
[22] |
L. M. Milne-Thomson, Theoretical Hydrodynamics, 5$^{th}$ edition, Dover, New York, 1996. |
[23] |
E. P. Rood, Vorticity interactions with a free surface, in Fluid Vortices (ed. S. I. Green), Fluid Mechanics and Its Applications, 30, Springer, Dordrecht, (1995), 687–730. |
[24] |
A. Rouhi and J. Wright,
Hamiltonian formulation for the motion of vortices in the presence of a free surface for ideal flow, Phys. Rev E, 48 (1993), 1850-1865.
doi: 10.1103/PhysRevE.48.1850. |
[25] |
N. Salvesen, E. O. Tuck and O. Faltinsen, Ship motions and sea loads, Transactions of the Society of Naval Architects and Marine Engineers, 6 (1970), 1–30. Available from: www.marinecontrol.org/References/papers/SalvesenTuckFaltinsen1970.PDF |
[26] |
P. G. Saffman, Vortex Dynamics,, Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, 1992.
![]() ![]() |
[27] |
B. N. Shashikanth,
Kirchhoff's equations of motion via a constrained Zakharov system, J. Geom. Mech., 8 (2016), 461-485.
doi: 10.3934/jgm.2016016. |
[28] |
B. N. Shashikanth, J. E. Marsden, J. W. Burdick and S. D. Kelly,
The Hamiltonian structure of a 2-D rigid cylinder interacting dynamically with $N$ point vortices, Phys. Fluids, 14 (2002), 1214-1227.
doi: 10.1063/1.1445183. |
[29] |
B. N. Shashikanth,
Poisson brackets for the dynamically interacting system of a 2D rigid boundary and $N$ point vortices: The case of arbitrary smooth cylinder shapes, Reg. Chaotic Dyn., 10 (2005), 1-14.
doi: 10.1070/RD2005v010n01ABEH000295. |
[30] |
B. N. Shashikanth, A. Sheshmani, S. D. Kelly and J. E. Marsden,
Hamiltonian structure for a neutrally buoyant rigid body interacting with $N$ vortex rings of arbitrary shape: The case of arbitrary smooth body shape, Theoretical and Computational Fluid Dynamics, 22 (2008), 37-64.
doi: 10.1007/s00162-007-0065-y. |
[31] |
P. Tallapragada and S. D. Kelly,
Propulsion of a spherical body shedding coaxial vortex rings in an ideal fluid, Regular and Chaotic Dynamics, 18 (2013), 21-32.
doi: 10.1134/S1560354713010024. |
[32] |
F. Ursell,
On the heaving motion of a circular cylinder on the surface of a fluid, Quart. J. Mech. App. Math., Ⅱ (1949), 218-231.
doi: 10.1093/qjmam/2.2.218. |
[33] |
F. Ursell,
Short surface waves due to an oscillating immersed body, Proc. R. Soc. Lond. A, 220 (1953), 90-103.
doi: 10.1098/rspa.1953.0174. |
[34] |
E. van Daalen, Numerical and Theoretical Studies of Water Waves and Floating Bodies, PhD Thesis, University of Twente, 1993. |
[35] |
E. F. G. van Daalen, E. van Groesen and P. J. Zandbergen, A Hamiltonian formulation for nonlinear wave-body interactions, Proceedings of the Eighth International Workshop on Water Waves and Floating Bodies, St John's, Newfoundland, Canada, 1993,159–163. Available from: http://www.iwwwfb.org/Abstracts/iwwwfb08/iwwwfb08-41.pdf. |
[36] |
J. Vankerschaver, E. Kanso and J. Marsden:,
The geometry and dynamics of interacting rigid bodies and point vortices, J. Geom. Mech., 1 (2009), 223-266.
doi: 10.3934/jgm.2009.1.223. |
[37] |
J. V. Wehausen,
The motion of floating bodies, Ann. Rev. Fluid Mech., 3 (1971), 237-268.
doi: 10.1146/annurev.fl.03.010171.001321. |
[38] |
G. X. Wu and R. Eatock Taylor,
The coupled finite element and boundary element analysis of nonlinear interactions between waves and bodies, Ocean Engg., 30 (2003), 387-400.
doi: 10.1016/S0029-8018(02)00037-9. |
[39] |
V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Appl. Mech. Tech. Phys., 9 (1968), 190–194. Originally published in Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi, 9 (1968), 86–94.
doi: 10.1007/BF00913182. |
show all references
References:
[1] |
G. A. Athanassoulis, Analytical dynamics of wave body interaction, in Mathematical techniques for water waves (ed. B. N. Mandal), Advances in Fluid Mechanics, 8, Computational Mechanics Publications, Southampton, Boston (1997), 79–154. Available from: urlhttps://www.researchgate.net/publication/266301554_Analytical_dynamics_of_wave-body_interaction |
[2] |
G. K. Batchelor, An Introduction to Fluid Dynamics,, Cambridge University Press, Cambridge, 1999.
![]() ![]() |
[3] |
M. Brøns, M. C. Thompson, T. Leweke and K. Hourigan,
Vorticity generation and conservation for two-dimensional interfaces and boundaries, J. Fluid Mech., 758 (2014), 63-93.
|
[4] |
R. Camassa and D. D. Holm,
An integrable shallow water equation with peaked solitons, Physical Review Letters, 71 (1993), 1661-1664.
doi: 10.1103/PhysRevLett.71.1661. |
[5] |
A. Compelli, R. Ivanov and M. Todorov, Hamiltonian models for the propogation of irrotational surface gravity waves over a variable bottom, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 376 (2018), 20170091, 15 pp.
doi: 10.1098/rsta.2017.0091. |
[6] |
L. D. Faddeev and L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons, Classics in Mathematics, Springer, 2007. |
[7] |
C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura,
Method for solving the Korteweg-de Vries equation, Phys, Rev. Lett., 19 (1967), 1095-1097.
|
[8] |
F. Gay-Balmaz, J. E. Marsden and T. S. Ratiu,
Reduced variational formulations in free boundary continuum mechanics, J. Nonlinear Sci., 22 (2012), 463-497.
doi: 10.1007/s00332-012-9143-4. |
[9] |
R. Ivanov,
Two-component integrable systems modelling shallow water waves: The constant vorticity case, Wave Motion, 46 (2009), 389-396.
doi: 10.1016/j.wavemoti.2009.06.012. |
[10] |
R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves, Cambridge Texts in Applied Mathematics, Cambridge University Press, 1997.
doi: 10.1017/CBO9780511624056.![]() ![]() ![]() |
[11] |
F. John,
On the motion of floating bodies. I., Comm. Pure App. Math., 2 (1949), 13-57.
doi: 10.1002/cpa.3160020102. |
[12] |
F. John,
On the motion of floating bodies. Ⅱ. Simple harmonic motions, Comm. Pure App. Math., 3 (1950), 45-101.
doi: 10.1002/cpa.3160030106. |
[13] |
G. Kirchhoff,
Ueber die Bewegung eines Rotationskörpers in einer Flüssigkeit, Journal Für Die Reine Und Angewandte Mathematik (Crelle's Journal), 71 (1869), 237-262.
doi: 10.1515/crll.1870.71.237. |
[14] |
E. A. Kuznetsov and V. P. Ruban,
Cherenkov interaction of vortices with a free surface, J. Exp. Theor. Phys., 88 (1999), 492-505.
doi: 10.1134/1.558820. |
[15] |
S. D. Kelly and H. Xiong, Self-propulsion of a free hydrofoil with localized discrete vortex shedding: Analytical modeling and simulation, Theoretical and Computational Fluid Dynamics, 24, (2010), 45–50. |
[16] |
H. Lamb, phHydrodynamics, 6$^{th}$ edition, Dover, New York, 1932. |
[17] |
D. Lewis, J. Marsden, R. Montgomery and T. Ratiu,
The Hamiltonian structure for dynamic free boundary problems, Physica D, 18 (1986), 391-404.
doi: 10.1016/0167-2789(86)90207-1. |
[18] |
T. Lundgren and P. Koumoutsakos,
On the generation of vorticity at a free surface, J. Fluid Mech., 382 (1999), 351-366.
|
[19] |
T. Miloh,
Hamilton's principle, Lagrange's method, and ship motion theory, J. Ship Research, 28 (1984), 229-237.
|
[20] |
T. Miloh,
A note on impulsive sphere motion beneath a free-surface, J. Engg. Math., 41 (2001), 1-11.
doi: 10.1023/A:1011895510151. |
[21] |
J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, Texts in Applied Mathematics 17, Second Edition, Springer-Verlag, 1999.
doi: 10.1007/978-0-387-21792-5. |
[22] |
L. M. Milne-Thomson, Theoretical Hydrodynamics, 5$^{th}$ edition, Dover, New York, 1996. |
[23] |
E. P. Rood, Vorticity interactions with a free surface, in Fluid Vortices (ed. S. I. Green), Fluid Mechanics and Its Applications, 30, Springer, Dordrecht, (1995), 687–730. |
[24] |
A. Rouhi and J. Wright,
Hamiltonian formulation for the motion of vortices in the presence of a free surface for ideal flow, Phys. Rev E, 48 (1993), 1850-1865.
doi: 10.1103/PhysRevE.48.1850. |
[25] |
N. Salvesen, E. O. Tuck and O. Faltinsen, Ship motions and sea loads, Transactions of the Society of Naval Architects and Marine Engineers, 6 (1970), 1–30. Available from: www.marinecontrol.org/References/papers/SalvesenTuckFaltinsen1970.PDF |
[26] |
P. G. Saffman, Vortex Dynamics,, Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, 1992.
![]() ![]() |
[27] |
B. N. Shashikanth,
Kirchhoff's equations of motion via a constrained Zakharov system, J. Geom. Mech., 8 (2016), 461-485.
doi: 10.3934/jgm.2016016. |
[28] |
B. N. Shashikanth, J. E. Marsden, J. W. Burdick and S. D. Kelly,
The Hamiltonian structure of a 2-D rigid cylinder interacting dynamically with $N$ point vortices, Phys. Fluids, 14 (2002), 1214-1227.
doi: 10.1063/1.1445183. |
[29] |
B. N. Shashikanth,
Poisson brackets for the dynamically interacting system of a 2D rigid boundary and $N$ point vortices: The case of arbitrary smooth cylinder shapes, Reg. Chaotic Dyn., 10 (2005), 1-14.
doi: 10.1070/RD2005v010n01ABEH000295. |
[30] |
B. N. Shashikanth, A. Sheshmani, S. D. Kelly and J. E. Marsden,
Hamiltonian structure for a neutrally buoyant rigid body interacting with $N$ vortex rings of arbitrary shape: The case of arbitrary smooth body shape, Theoretical and Computational Fluid Dynamics, 22 (2008), 37-64.
doi: 10.1007/s00162-007-0065-y. |
[31] |
P. Tallapragada and S. D. Kelly,
Propulsion of a spherical body shedding coaxial vortex rings in an ideal fluid, Regular and Chaotic Dynamics, 18 (2013), 21-32.
doi: 10.1134/S1560354713010024. |
[32] |
F. Ursell,
On the heaving motion of a circular cylinder on the surface of a fluid, Quart. J. Mech. App. Math., Ⅱ (1949), 218-231.
doi: 10.1093/qjmam/2.2.218. |
[33] |
F. Ursell,
Short surface waves due to an oscillating immersed body, Proc. R. Soc. Lond. A, 220 (1953), 90-103.
doi: 10.1098/rspa.1953.0174. |
[34] |
E. van Daalen, Numerical and Theoretical Studies of Water Waves and Floating Bodies, PhD Thesis, University of Twente, 1993. |
[35] |
E. F. G. van Daalen, E. van Groesen and P. J. Zandbergen, A Hamiltonian formulation for nonlinear wave-body interactions, Proceedings of the Eighth International Workshop on Water Waves and Floating Bodies, St John's, Newfoundland, Canada, 1993,159–163. Available from: http://www.iwwwfb.org/Abstracts/iwwwfb08/iwwwfb08-41.pdf. |
[36] |
J. Vankerschaver, E. Kanso and J. Marsden:,
The geometry and dynamics of interacting rigid bodies and point vortices, J. Geom. Mech., 1 (2009), 223-266.
doi: 10.3934/jgm.2009.1.223. |
[37] |
J. V. Wehausen,
The motion of floating bodies, Ann. Rev. Fluid Mech., 3 (1971), 237-268.
doi: 10.1146/annurev.fl.03.010171.001321. |
[38] |
G. X. Wu and R. Eatock Taylor,
The coupled finite element and boundary element analysis of nonlinear interactions between waves and bodies, Ocean Engg., 30 (2003), 387-400.
doi: 10.1016/S0029-8018(02)00037-9. |
[39] |
V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Appl. Mech. Tech. Phys., 9 (1968), 190–194. Originally published in Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi, 9 (1968), 86–94.
doi: 10.1007/BF00913182. |


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