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Structure preserving numerical methods for fluid-structure interactions

Project Description

The University of Exeter EPSRC DTP (Engineering and Physical Sciences Research Council Doctoral Training Partnership) is offering up to 4 fully funded doctoral studentships for 2019/20 entry. Students will be given sector-leading training and development with outstanding facilities and resources. Studentships will be awarded to outstanding applicants, the distribution will be overseen by the University’s EPSRC Strategy Group in partnership with the Doctoral College.


Dr Hamid Alemi Ardakani, Department of Mathematics, College of Engineering, Mathematics and Physical Sciences
Professor John Thuburn, Department of Mathematics, College of Engineering, Mathematics and Physical Sciences

Project description:

This PhD studentship research project concerns the mathematical modelling of dynamic coupling between a suspended container with variable cross-section, with rigid or elastic pivoting rods, undergoing pendulum-like oscillations and its interior shallow fluid sloshing in three dimensions. The simplest model for the coupled fluid-body dynamics in three dimensions is the motion of a spherical pendulum with interior fluid motion. The coupled fluid sloshing and rigid body dynamics is of great practical interest in the terrestrial transport of liquids, sloshing of fluid in ships and on-board spacecrafts, motion of water waves in ocean wave energy converters and fuel in aviation and astrodynamics. Although there are many publications on theoretical and numerical analysis of fluid sloshing dynamics (e.g. Ibrahim, A., 2005. Liquid Sloshing Dynamics: Theory and Applications. Cambridge University Press, and Faltinsen, O.M., Timokha, A.N., 2009.
Sloshing. Cambridge University Press, and references therein) there are a few numbers of research works on structure preserving numerical methods for the coupled rigid body dynamics and its interior inviscid and incompressible fluid motion in two or three dimensions. In two horizontal space dimensions the exact material conservation of potential vorticity is crucial as it is an important dynamical variable in shallow water hydrodynamics. For dynamic coupling between rigid body motion and its interior shallow water sloshing in two horizontal
space dimensions to improve the simulation of nonlinear aspects of the fluid sloshing over a flat bottom or inside a vessel with variable cross-section, a potential enstrophy and energy conserving scheme is desirable. The aim is to start with the simplest problem in the coupled fluidbody dynamics in two horizontal space dimensions and in Eulerian coordinates, and develop energy and potential vorticity preserving integrators. The two key themes of the research project are:

1. Write the forced two-dimensional shallow water equations and the nonlinear integrodifferential equation for the rigid body motion in Nambu’s bracket form (see Nambu, Y., 1973. Generalized Hamiltonian dynamics. Phys. Rev. D 7, 2405-2412.), and develop energy and potential enstrophy conserving schemes by extending the theoretical and numerical discretisations of Salmon (2005) to the coupled fluid-body interactions (see Salmon, R., 2005. A general method for conserving quantities related to potential vorticity in numerical methods. Nonlinearity 18, R1-R16.)

2. Formulation of the coupled fluid-body equations in Hamiltonian and Poisson bracket form and extension of the energy and potential enstrophy preserving schemes of Salmon (2004) to this problem (see Salmon, R., 2004. Poisson-bracket approach to the construction of energyand potential-enstrophy-conserving algorithms for the shallow-water equations. J. Atmos. Sci. 61, 2016-2036.)

The goal is to compare the Nambu’s bracket solver with the Hamiltonian/Poisson bracket integrator to investigate which solver is more useful for further theoretical and numerical developments of the coupled interactions between rigid body motion and its interior fluid
sloshing in full 3D rotating and translating coordinates.

Funding Notes

For successful eligible applicants the studentship comprises:

- An index-linked stipend for up to 3.5 years full time (currently £14,777 per annum for 2018/19), pro-rata for part-time students.
- Payment of University tuition fees (UK/EU)
- Research Training Support Grant (RTSG) of £5,000 over 3.5 years, or pro-rata for part-time students

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