Abstract
The pressure boundary condition for the full Euler equations with a free surface and general vorticity field is formulated in terms of a generalized Bernoulli equation deduced from the Gavrilyuk–Kalisch–Khorsand conservation law. The use of pressure as a Lagrangian density, as in Luke’s variational principle, is reviewed and extension to a full vortical flow is attempted with limited success. However, a new variational principle for time-dependent water waves in terms of the stream function is found. The variational principle generates vortical boundary conditions but with a harmonic stream function. Other aspects of vorticity in variational principles are also discussed.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ardakani, H. Alemi, Bridges, T.J., Gay-Balmaz, F.,Huang, Y., Tronci, C.: (2018) A variational principle for fluid sloshing with vorticity, dynamically coupled to vessel motion, arXiv:1809.10909
Bateman, H.: Partial Differential Equations. Cambridge University Press, Cambridge (1964)
Casetta, L., Pesce, C.P.: On Seliger and Whitham’s variational principle for hydrodynamic systems from the point of view of “fictitious particles’. Acta Mech. 219, 181–184 (2011)
Clamond, D., Dutykh, D.: Practical use of variational principles for modeling water waves. Phys. D 241, 25–36 (2012)
Cotter, C., Bokhove, O.: Variational water-wave model with accurate dispersion and vertical vorticity. J. Eng. Math. 67, 33–54 (2010)
Fukagawa, H., Fujitani, Y.: Clebsch potentials in the variational principle for a perfect fluid. Prog. Theor. Phys. 124, 517–531 (2010)
Gavrilyuk, S.: Multiphase flow modeling via Hamilton’s Principle. In: dell’Isola, F., Gavrilyuk, S. (eds.) Variational Models and Methods in Solid and Fluid Mechanics. CISM Courses and Lectures, vol. 535. Springer, Vienna (2011)
Gavrilyuk, S., Kalisch, H., Khorsand, Z.: A kinematic conservation law in free surface flow. Nonlinearity 28, 1805–1821 (2015)
Graham, C.R., Henyey, F.S.: Clebsch representation near points where the vorticity vanishes. Phys. Fluids 12, 744 (2000)
Groves, M.D., Wahlén, E.: Existence and conditional energetic stability of solitary gravity–capillary water waves with constant vorticity. Proc. Roy. Soc. Edin. A 145, 791–883 (2015)
Hargreaves, R.: A pressure integral as kinetic potential. Phil. Mag. 16, 436–444 (1908)
Lewis, D., Marsden, J., Montgomery, R., Ratiu, T.: The Hamiltonian structure for dynamic free boundary problems. Phys. D 18, 391–404 (1989)
Luke, J.C.: A variational principle for a fluid with a free surface. J. Fluid Mech. 27, 395–397 (1967)
Seliger, R.E., Whitham, G.B.: Variational principles in continuum mechanics. Proc. Roy. Soc. Lond. A 305, 1–25 (1968)
Wahlén, E.: A Hamiltonian formulation of water waves with constant vorticity. Lett. Math. Phys. 79, 303–315 (2007)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
OpenAccess This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Bridges, T.J. The Pressure Boundary Condition and the Pressure as Lagrangian for Water Waves. Water Waves 1, 131–143 (2019). https://doi.org/10.1007/s42286-019-00001-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s42286-019-00001-0