Finite element methods for linearly perturbed single- and two-phase flows

• Main Author: Hurley, Patrick
• Other Authors: Heil, Matthias; Hazel, Andrew
• Published: University of Manchester 2014
• Subjects: 620
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.618010

This thesis is concerned with the development of a general numerical framework that allows the computation of the evolution of non-axisymmetric perturbations to arbitrary axisymmetric base flows. Both single- and two-phase flows are considered. This framework is developed within oomph-lib, a finite element library for the simulation of multi-physics problems. Following the introduction of the core concepts of the finite element method via the example of a two-phase viscous flow problem in a Cartesian coordinate system, we begin our discussion of the Navier--Stokes equations in cylindrical polar coordinates by deriving the finite element formulation of the equations governing an axisymmetric flow and the conditions to be applied at a free surface. We apply this formulation to two problems involving the relaxation of a free surface in a cylindrical geometry, and validate the numerical results against analytical predictions for the frequency and decay rate obtained from linearised analyses of the respective problems. We then linearise the weak form of the governing equations for a fully three-dimensional flow written in terms of a cylindrical polar coordinate system to obtain the finite element formulation for a linear perturbation to a nonlinear base flow. The solution to the linear problem depends on the base flow and is periodic in the azimuthal direction. We exploit this periodicity by performing a Fourier decomposition in the azimuthal direction, transforming the three-dimensional problem into a series of two-dimensional problems in which the azimuthal mode number appears as a parameter. We implement these equations in oomph-lib and apply this newly-developed methodology to two representative single-phase flow problems. In both cases we demonstrate that our computations match previously published results generated using different numerical methods. Having validated our implementation of the equations governing the perturbation of a (time-dependent) axisymmetric base flow by a linear, non-axisymmetric disturbance, we extend this formulation to include problems containing two immiscible fluids separated by an interface. We derive the equations which need to be applied at the free boundary in such a problem, and augment oomph-lib's existing moving-domain `machinery' to allow the computation of arbitrary perturbations to a `base' free surface position, which is itself an unknown in the problem. We validate this methodology for the case of an axisymmetric disturbance to an axisymmetric base flow, and demonstrate the use of the newly-developed functionality by applying it to a non-axisymmetric relaxing interface problem. We demonstrate that, for all stably stratified configurations of this system, all disturbances to the interface position decay. For an unstably stratified configuration, however, we observe the growth of certain non-axisymmetric modes.