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March  2020, 12(1): 25-52. doi: 10.3934/jgm.2020003

Poisson brackets for the dynamically coupled system of a free boundary and a neutrally buoyant rigid body in a body-fixed frame

Banavara N. Shashikanth

Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, NM 88003-8001, USA

Received  January 2019 Published  January 2020

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.

Keywords: Wave-body coupled dynamics, Momentum laws, Poisson brackets, Lie-Poisson bracket, Zakharov bracket.
Mathematics Subject Classification: Primary: 37K05, 74F10, 76B07; Secondary: 53D17.
Citation: Banavara N. Shashikanth. Poisson brackets for the dynamically coupled system of a free boundary and a neutrally buoyant rigid body in a body-fixed frame. Journal of Geometric Mechanics, 2020, 12 (1) : 25-52. doi: 10.3934/jgm.2020003
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 Google Scholar

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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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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L. D. Faddeev and L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons, Classics in Mathematics, Springer, 2007.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[11]

F. John, On the motion of floating bodies. I., Comm. Pure App. Math., 2 (1949), 13-57.  doi: 10.1002/cpa.3160020102.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[16]

H. Lamb, phHydrodynamics, 6$^{th}$ edition, Dover, New York, 1932.  Google Scholar

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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.  Google Scholar

[18]

T. Lundgren and P. Koumoutsakos, On the generation of vorticity at a free surface, J. Fluid Mech., 382 (1999), 351-366.   Google Scholar

[19]

T. Miloh, Hamilton's principle, Lagrange's method, and ship motion theory, J. Ship Research, 28 (1984), 229-237.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

L. M. Milne-Thomson, Theoretical Hydrodynamics, 5$^{th}$ edition, Dover, New York, 1996. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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 Google Scholar

[26] P. G. Saffman, Vortex Dynamics,, Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, 1992.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

E. van Daalen, Numerical and Theoretical Studies of Water Waves and Floating Bodies, PhD Thesis, University of Twente, 1993.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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 Google Scholar

[2] G. K. Batchelor, An Introduction to Fluid Dynamics,, Cambridge University Press, Cambridge, 1999.   Google Scholar
[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

L. D. Faddeev and L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons, Classics in Mathematics, Springer, 2007.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[11]

F. John, On the motion of floating bodies. I., Comm. Pure App. Math., 2 (1949), 13-57.  doi: 10.1002/cpa.3160020102.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[16]

H. Lamb, phHydrodynamics, 6$^{th}$ edition, Dover, New York, 1932.  Google Scholar

[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.  Google Scholar

[18]

T. Lundgren and P. Koumoutsakos, On the generation of vorticity at a free surface, J. Fluid Mech., 382 (1999), 351-366.   Google Scholar

[19]

T. Miloh, Hamilton's principle, Lagrange's method, and ship motion theory, J. Ship Research, 28 (1984), 229-237.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

L. M. Milne-Thomson, Theoretical Hydrodynamics, 5$^{th}$ edition, Dover, New York, 1996. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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 Google Scholar

[26] P. G. Saffman, Vortex Dynamics,, Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, 1992.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

E. van Daalen, Numerical and Theoretical Studies of Water Waves and Floating Bodies, PhD Thesis, University of Twente, 1993.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  Schematic perspective of a rigid body $ B $ beneath the free surface $ \Sigma_f $ of water. The bottom flat surface $ \mathcal{S} $ is shown by the dashed rectangle. Both $ \Sigma_f $ and $ \mathcal{S} $ extend to infinity in the $ x $ and $ y $ (horizontal) directions. In the text, the origin of the spatial frame $ xyz $ is located at the center of the disc $ C_R \subset \mathcal{S} $
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Figure 2.  A vertical slice of the setup in Figure 1, shown along with the body-fixed frame
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