A variational principle for fluid sloshing with vorticity, dynamically coupled to vessel motion

Ardakani, Hamid Alemi; Bridges, Thomas J.; Gay–Balmaz, François; Huang, Ying H.; Tronci, Cesare

doi:10.1098/rspa.2018.0642

Cited by 8 publications

(9 citation statements)

References 26 publications

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“…where w is a free variation and [•, •] is a Lie bracket of vector fields. The Euler-Poincaré strategy is used in Cotter and Bokhove [5] and Alemi Ardakani et al [1]. But constrained variations make analysis of the variational principle more difficult.…”

Section: Discussionmentioning

confidence: 99%

The Pressure Boundary Condition and the Pressure as Lagrangian for Water Waves

Bridges

2019

Water Waves

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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. Keywords Oceanography • Lagrangian • Vorticity • Streamfunction • Variational principle where (U , V) is the velocity field evaluated at the free surface, P a (x, t) is the imposed external pressure field, and f (t) is an arbitrary function of time. The function (x, t)

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Section: Discussionmentioning

confidence: 99%

The Pressure Boundary Condition and the Pressure as Lagrangian for Water Waves

Bridges

2019

Water Waves

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“…The detailed derivation is given in various books or papers, e.g., [16,19,52,55], and, thus, it is not repeated here. Inverting this point of view, we may say that, in the present variational formulation, Euler's momentum Equation ( 4) is decomposed into the system of Equations ( 29)- (32). In some sense, Equations ( 28)- (32) provide us with a new formulation of the rotational flow problem.…”

Section: Euler-lagrange Equations Corresponding To Variations Within ...mentioning

confidence: 99%

“…In fact, the only work that the present authors found with some discussion on boundary conditions in this context is the book by Berdichevsky [31] (Section 9.3), where he derives a form of the free-surface dynamic condition, having all the kinematic conditions a priori imposed. In a different direction, which uses Hamilton's principle in conjunction with the constrained variations of the Euler-Poincaré framework, a limited number of works dealing with free-surface boundary conditions have appeared recently (see [32,33], and references therein). Although these works are interesting and illuminating, they are limited to two-dimensional (2D) flows, utilizing a stream function, which drastically simplifies the treatment but is not applicable to 3D flows.…”

Section: History and Background Literaturementioning

confidence: 99%

“…Another direction that is not considered herein (with the exception of the two recent papers [32,33], mentioned above) is 2D free-surface flows with vorticity, although there has been a lot of progress in this direction in the last two decades. This progress has mainly been based on the existence of a scalar stream function, which permits a drastic simplification of the formulation, which is not applicable to 3D problems.…”

Section: Related Research Directions That Are Not Considered In This ...mentioning

confidence: 99%

“…These equations have been obtained (sometimes with a slightly different form) by many authors since 1959, first appearing in [16]. Euler's momentum equation is not directly included in the above list; however, it can be derived by combining Equations ( 29)- (32) and eliminating the fields k, a, A, which do not appear in the standard Eulerian formalism. The detailed derivation is given in various books or papers, e.g., [16,19,52,55], and, thus, it is not repeated here.…”

Section: Euler-lagrange Equations Corresponding To Variations Within ...mentioning

confidence: 99%

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Hamiltonian Variational Formulation of Three-Dimensional, Rotational Free-Surface Flows, with a Moving Seabed, in the Eulerian Description

Mavroeidis

Athanassoulis

2022

Fluids

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Hamiltonian variational principles have provided, since the 1960s, the means of developing very successful wave theories for nonlinear free-surface flows, under the assumption of irrotationality. This success, in conjunction with the recognition that almost all flows in the sea are not irrotational, raises the question of extending Hamilton’s principle to rotational free-surface flows. The Euler equations governing the bulk fluid motion have been derived by means of Hamilton’s principle since the late 1950s. Nevertheless, a complete variational formulation of the rotational water-wave problem, including the derivation of the free-surface boundary conditions, seems to be lacking until now. The purpose of the present work is to construct such a missing variational formulation. The appropriate functional is the usual Hamilton’s action, constrained by the conservation of mass and the conservation of fluid parcels’ identity. The differential equations governing the bulk fluid motion are derived as usually, applying standard methods of the calculus of variations. However, the standard methodology does not provide enough structure to obtain the free-surface boundary conditions. To overcome this difficulty, differential-variational forms of the aforementioned constraints are introduced and applied to the boundary variations of the Eulerian fields. Under this transformation, both kinematic and dynamic free-surface conditions are naturally derived, ensuring the Hamiltonian variational formulation of the complete problem. An interesting feature, appearing in the present variational derivation, is a dual possibility concerning the tangential velocity on the boundary; it may be either the same as in irrotational flow (no condition) or zero, corresponding to the small-viscosity limit. The deeper meaning and the significance of these findings seem to deserve further analysis.

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Kirchhoff’s Insufficiently-Celebrated Equations of Motion

Shashikanth

2021

Dynamically Coupled Rigid Body-Fluid Flow Systems

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A variational principle for fluid sloshing with vorticity, dynamically coupled to vessel motion

Cited by 8 publications

References 26 publications

The Pressure Boundary Condition and the Pressure as Lagrangian for Water Waves

The Pressure Boundary Condition and the Pressure as Lagrangian for Water Waves

Hamiltonian Variational Formulation of Three-Dimensional, Rotational Free-Surface Flows, with a Moving Seabed, in the Eulerian Description

Kirchhoff’s Insufficiently-Celebrated Equations of Motion

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