Variational generalization of the Green–Naghdi and Whitham equations for fluid sloshing in three-dimensional rotating and translating coordinates
Alemi Ardakani, H
Date: 16 April 2021
Journal
European Journal of Mechanics B/Fluids
Publisher
Elsevier / European Mechanics Society (Euromech)
Publisher DOI
Abstract
This paper derives an averaged Lagrangian functional
for dynamic coupling between rigid-body motion and its interior
shallow-water sloshing in three-dimensional rotating and translating coordinates; with a time-dependent rotation vector. A new
set of variational shallow-water equations (SWEs) and generalized Green–Naghdi equations ...
This paper derives an averaged Lagrangian functional
for dynamic coupling between rigid-body motion and its interior
shallow-water sloshing in three-dimensional rotating and translating coordinates; with a time-dependent rotation vector. A new
set of variational shallow-water equations (SWEs) and generalized Green–Naghdi equations for the interior fluid sloshing with
3–D rotation vector and translations, and also the equations of
motion for the linear momentum and angular momentum of the
rigid-body containing shallow water, are derived from the averaged Lagrangian functional, which describes a columnar motion,
by using Hamilton’s principle and the Euler–Poincare variational ´
framework. The generalized Green–Naghdi equations have a
form of potential vorticity (PV) conservation, which can be obtained from the particle-relabeling symmetry, and is a combination of the PV derived by Miles and Salmon (1985) and the PV
derived by Dellar & Salmon (2005) for geophysical fluid dynamics problems, where the rotation vector varies spatially. By applying the assumption of zero-potential-vorticity flow to the averaged Lagrangian functional, a new set of Boussinesq-like evolution equations are derived, which are a generalization of the
Whitham equations for fluid sloshing in three-dimensional rotating and translating coordinates. Moreover, the new variational
principles are appended to Luke’s variational principle to present
a unified variational framework for the hydrodynamic problem
of interactions between gravity-driven potential-flow water waves
and a freely floating rigid-body, dynamically coupled to its interior weakly dispersive nonlinear shallow-water sloshing in three
dimensions.
Mathematics and Statistics
Faculty of Environment, Science and Economy
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