High order methods for the approximation of the incompressible Navier-Stokes equations in a moving domain

In this paper we address the numerical approximation of the incompressible Navier-Stokes equations in a moving domain by the spectral element method and high order time integrators. We present the Arbitrary Lagrangian Eulerian (ALE) formulation of the incompressible Navier-Stokes equations and propose a numerical method based on the following kernels: a Lagrange basis associated with Fekete points in the spectral element method context, BDF time integrators, an ALE map of high degree, and an algebraic linear solver. In particular, the high degree ALE map is appropriate to deal with a computational domain whose boundary is described with curved elements. Finally, we apply the proposed strategy to a test case. (C) 2011 Elsevier B.V. All rights reserved.

Spectral element approximation of the incompressible Navier-Stokes equations in a moving domain and applications

Gonçalo Pena

In this thesis we address the numerical approximation of the incompressible Navier-Stokes equations evolving in a moving domain with the spectral element method and high order time integrators. First, we present the spectral element method and the basic tools to perform spectral discretizations of the Galerkin or Galerkin with Numerical Integration (G-NI) type. We cover a large range of possibilities regarding the reference elements, basis functions, interpolation points and quadrature points. In this approach, the integration and differentiation of the polynomial functions is done numerically through the help of suitable point sets. Regarding the differentiation, we present a detailed numerical study of which points should be used to attain better stability (among the choices we present). Second, we introduce the incompressible steady/unsteady Stokes and Navier-Stokes equations and their spectral approximation. In the unsteady case, we introduce a combination of Backward Differentiation Formulas and an extrapolation formula of the same order for the time integration. Once the equations are discretized, a linear system must be solved to obtain the approximate solution. In this context, we consider the solution of the whole system of equations combined with a block type preconditioner. The preconditioner is shown to be optimal in terms of number of iterations used by the GMRES method in the steady case, but not in the unsteady one. Another alternative presented is to use algebraic factorization methods of the Yosida type and decouple the calculation of velocity and pressure. A benchmark is also presented to access the numerical convergence properties of this type of methods in our context. Third, we extend the algorithms developed in the fixed domain case to the Arbitrary Lagrangian Eulerian framework. The issue of defining a high order ALE map is addressed. This allows to construct a computational domain that is described with curved elements. A benchmark using a direct method to solve the linear system or the Yosida-q methods is presented to show the convergence orders of the method proposed. Finally, we apply the developed method with an implicit fully coupled and semi-implicit approach, to solve a fluid-structure interaction problem for a simple 2D hemodynamics example.

EPFL2009

Simulation of shear-driven flows

Roland Bouffanais

The research work reported in the present dissertation is aimed at the analysis of complex physical phenomena involving instabilities and nonlinearities occurring in fluids through state-of-the-art numerical modeling. Solutions of intricate fluid physics problems are devised in two particularly arduous situations: fluid domains with moving boundaries and the high-Reynolds-number regime dominated by nonlinear convective effects. Shear-driven flows of incompressible Newtonian fluids enclosed in cavities of varying geometries are thoroughly investigated in the two following frameworks: transition with a free surface and confined turbulence. The physical system we consider is made of an incompressible Newtonian fluid filling a bounded, or partially bounded cavity. A series of shear-driven flows are easily generated by setting in motion some part of the container boundary. These driven-cavity flows are not only technologically important, they are of great scientific interest because they display almost all physical fluid phenomena that can possibly occur in incompressible flows, and this in the simplest geometrical settings. Thus corner eddies, secondary flows, longitudinal vortices, complex three-dimensional patterns, chaotic particle motions, nonuniqueness, transition, and turbulence all occur naturally and can be studied in the same geometry. This facilitates the comparison of results from experiments, analysis, and computation over the whole range of Reynolds numbers. The flows under consideration are part of a larger class of confined flows driven by linear or angular momentum gradients. This dissertation reports a detailed study of a novel numerical method developed for the simulation of an unsteady free-surface flow in three-space-dimensions. This method relies on a moving-grid technique to solve the Navier-Stokes equations expressed in the arbitrary Lagrangian-Eulerian (ALE) kinematics and discretized by the spectral element method. A comprehensive analysis of the continuous and discretized formulations of the general problem in the ALE frame, with nonlinear, non-homogeneous and unsteady boundary conditions is presented. In this dissertation, we first consider in the internal turbulent flow of a fluid enclosed in a bounded cubical cavity driven by the constant translation of its lid. The solution of this flow relied on large-eddy simulations, which served to improve our physical understanding of this complex flow dynamics. Subsequently, a novel subgrid model based on approximate deconvolution methods coupled with a dynamic mixed scale model was devised. The large-eddy simulation of the lid-driven cubical cavity flow based on this novel subgrid model has shown improvements over traditional subgrid-viscosity type of models. Finally a new interpretation of approximate deconvolution models when used with implicit filtering as a way to approximate the projective grid filter was given. This led to the introduction of the grid filter models. Through the use of a newly-developed method of numerical simulation, in this dissertation we solve unsteady flows with a flat and moving free-surface in the transitional regime. These flows are the incompressible flow of a viscous fluid enclosed in a cylindrical container with an open top surface and driven by the steady rotation of the bottom wall. New flow states are investigated based on the fully three-dimensional solution of the Navier-Stokes equations for these free-surface cylindrical swirling flows, without resorting to any symmetry properties unlike all other results available in the literature. To our knowledge, this study delivers the most general available results for this free-surface problem due to its original mathematical treatment. This second part of the dissertation is a basic research task directed at increasing our understanding of the influence of the presence of a free surface on the intricate transitional flow dynamics of shear-driven flows.

EPFL2007

Numerical simulation of thermal problems coupled with magnetohydrodynamic effects in aluminium cell

Jacques Rappaz

A system of partial differential equations describing the thermal behavior of aluminium cell coupled with magnetohydrodynamic effects is numerically solved. The thermal model is considered its a two-phases Stefan problem which consists of a non-linear convection-diffusion heat equation with Joule effect as a source. The magnetohydrodynamic fields are governed by Navier-Stokes and by static Maxwell equations. A pseudo-evolutionary scheme (Chernoff) is used to obtain the stationary solution giving the temperature and the frozen layer profile for the simulation of the ledges in the cell. A numerical approximation using a finite element method is formulated to obtain the fluid velocity, electrical potential, magnetic induction and temperature. An iterative algorithm and 3-D numerical results are presented. (C) 2008 Elsevier Inc. All rights reserved.