Numerical study of transitions in Taylor-Couette flow

This thesis is divided into two parts. In the first one the creation of a complete numerical tool, from the mesh generation to the data treatment, is presented. However time consuming this part has been, this is not the most important one. The second and more important part is concerned with the description and physical interpretation of the results obtained with the numerical tool previously developed. The numerical tool, which exists in its quasi-final version since mid-1997, fully satisfies the following requirements. It is specifically designed to study the flow in the annular space between coaxial, differentially-rotating cylinders of finite length. The spectral element method is used for the space discretization. The study of transition requires the high accuracy warranted by this type of method. The numerical scheme has also to be efficient. This requirement is satisfied, due to the fully-explicit time scheme adopted, where both diffusive and non-linear terms of the Navier-Stokes equations are treated explicitly, and the direct inversion of the pseudo-Laplacian matrix applied to the pressure. This inversion is performed in the most efficient way with a fast diagonalization technique. In the Reynolds numbers range we are interested in, the time-step limitation due to the linear viscous term is only slightly more stringent than the one due to the non-linear term. The last requirement that has been fulfilled has been to design as simple a code as possible. The entire code is constructed from a number of well-known algorithms, fitted together to enhance efficiency. However, the way we regularize the boundary conditions is new. It represents the physics more precisely than Tavener et al. [65]. There is a second original feature in our code, which is linked to the time scheme. We derived our time discretization from the scheme of Gavrilakis et al. [31]. Their scheme cannot be applied to cylindrical coordinates as it is. We thus had to modify it. The second part of the thesis consists of the numerical study of the first transitions of the Taylor-Couette flow in a finite-length geometry. The aspect ratio between the length of the cylinders and the gap between them has been chosen equal to twelve; this is small enough for the effects of the upper and lower boundaries of the flow to be significant. It is believed that these end-effects play a non-negligible role in the transition of the flow. On the other hand, the aspect ratio is large enough to make qualitative comparisons with the infinite-length case, which has been studied extensively both theoretically and numerically. The transition process depends on a relatively large number of parameters. Our investigation focuses on the case where the cylinders rotate in opposite directions. The study of counter-rotating Taylor-Couette flow for a large but finite aspect ratio is the main originality of this thesis. Furthermore, the physical mechanism of the appearance of the second bifurcation in classical Taylor-Couette flow, where only the inner cylinder is rotating, is described in the light of the non-linear interaction between velocity and vorticity.

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 approximation of the unsteady Navier-Stokes equations with an application to the flow past a square cylinder

Francesco Casalegno

The goal of this project is to numerically solve the Navier-Stokes equations by using different numerical methods with particular emphasis on solving the problem of the flow past a square cylinder. In particular, we use the finite element method based on P2/P1 elements for the velocity and pressure fields for the spatial approximation, while the backward Euler method (with semi-implicit treatment of the nonlinear term) for the time discretization. We firstly test the numerical schemes on a benchmark problem with known exact solution. Then, we discuss in detail the advantages, in terms of computational costs, in using the algebraic Chorin-Temam method with additional implementation improvements. We finally investigate the problem of the two-dimensional flow past a square cylinder, focusing our attention on the range 0.1-300 for the Reynolds number (Re). We describe the two different regimes associated to the steady and the unsteady flows and we remark as the latter is in fact due to a Hopf bifurcation of the system. We also discuss the relation between the Strouhal and Reynolds numbers concluding that the Strouhal number attains its maximum value in the range 169-170 for the Reynolds number. In particular, a cubic model is proposed, showing very good matching with observed data and a better fitting than other models available in literature.

2013

A posteriori error estimation for partial differential equations with random input data

Diane Sylvie Guignard

This thesis is devoted to the derivation of error estimates for partial differential equations with random input data, with a focus on a posteriori error estimates which are the basis for adaptive strategies. Such procedures aim at obtaining an approximation of the solution with a given precision while minimizing the computational costs. If several sources of error come into play, it is then necessary to balance them to avoid unnecessary work. We are first interested in problems that contain small uncertainties approximated by finite elements. The use of perturbation techniques is appropriate in this setting since only few terms in the power series expansion of the exact random solution with respect to a parameter characterizing the amount of randomness in the problem are required to obtain an accurate approximation. The goal is then to perform an error analysis for the finite element approximation of the expansion up to a certain order. First, an elliptic model problem with random diffusion coefficient with affine dependence on a vector of independent random variables is studied. We give both a priori and a posteriori error estimates for the first term in the expansion for various norms of the error. The results are then extended to higher order approximations and to other sources of uncertainty, such as boundary conditions or forcing term. Next, the analysis of nonlinear problems in random domains is proposed, considering the one-dimensional viscous Burgers' equation and the more involved incompressible steady-state Navier-Stokes equations. The domain mapping method is used to transform the equations in random domains into equations in a fixed reference domain with random coefficients. We give conditions on the mapping and the input data under which we can prove the well-posedness of the problems and give a posteriori error estimates for the finite element approximation of the first term in the expansion. Finally, we consider the heat equation with random Robin boundary conditions. For this parabolic problem, the time discretization brings an additional source of error that is accounted for in the error analysis. The second part of this work consists in the analysis of a random elliptic diffusion problem that is approximated in the physical space by the finite element method and in the stochastic space by the stochastic collocation method on a sparse grid. Considering a random diffusion coefficient with affine dependence on a vector of independent random variables, we derive a residual-based a posteriori error estimate that controls the two sources of error. The stochastic error estimator is then used to drive an adaptive sparse grid algorithm which aims at alleviating the so-called curse of dimensionality inherent to tensor grids. Several numerical examples are given to illustrate the performance of the adaptive procedure.