Simulation of the Taylor-Couette flow in a finite geometry by spectral element method

The purpose of the present numerical method is to simulate the flow in between two concentric cylinders of finite axial length. The numerical integration of the transient three-dimensional incompressible Navier-Stokes equations is performed by a Legendre spectral element method in space while the time marching scheme is fully explicit. By taking full advantage of the tensor product bases, it is possible to set up fast diagonalization techniques in order to speed up the solver. Numerical results are compared favorably with experimental data. (C) 2000 IMACS. Published by Elsevier Science B.V. All rights reserved.

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

Geometric, variational discretization of continuum theories

Mathieu Desbrun

This study derives geometric, variational discretization of continuum theories arising in fluid dynamics, magnetohydrodynamics (MHD), and the dynamics of complex fluids. A central role in these discretizations is played by the geometric formulation of fluid dynamics, which views solutions to the governing equations for perfect fluid flow as geodesics on the group of volume-preserving diffeomorphisms of the fluid domain. Inspired by this framework, we construct a finite-dimensional approximation to the diffeomorphism group and its Lie algebra, thereby permitting a variational temporal discretization of geodesics on the spatially discretized diffeomorphism group. The extension to MHD and complex fluid flow is then made through an appeal to the theory of Euler-Poincare systems with advection, which provides a generalization of the variational formulation of ideal fluid flow to fluids with one or more advected parameters. Upon deriving a family of structured integrators for these systems, we test their performance via a numerical implementation of the update schemes on a cartesian grid. Among the hallmarks of these new numerical methods are exact preservation of momenta arising from symmetries, automatic satisfaction of solenoidal constraints on vector fields, good long-term energy behavior, robustness with respect to the spatial and temporal resolution of the discretization, and applicability to irregular meshes. (C) 2011 Elsevier B.V. All rights reserved.

2011

Geometrical Treatise on the Modelling of 3D Particulate Inclusion-Matrix Microstructures with an Application to Historical Stone Masonry Walls

Mahmoud S. M. Shaqfa

As historical stone masonry structures are vulnerable and prone to damage in earthquakes, investigating their structural integrity is important to reduce injuries and casualties while preserving their historical value. Stone masonry is a composite material that is built with stones and binding mortar. Although experimental campaigns are crucial in understanding the structural behaviour of these walls, the wide spectrum of existing stone masonry typologies and the randomness in geometry and material properties render extensive testing campaigns nearly impossible to account for all the uncertainties and variables. Numerical simulations that explicitly represent the microstructure of the wall (i.e., the geometry and arrangement of the stones) can complement experimental studies. However, methods for generating 3D microstructures were lacking.The focus of this dissertation is to study the geometry of the 3D microstructure of stone masonry walls at multiple scales, which opens unprecedented opportunities for studying the micromechanical behaviour of these walls numerically. The first contribution of this thesis was crafting the first 3D virtual microstructure generator that can cover the main typologies of stone masonry that are frequently found in historical buildings. The herein-created microstructure generator borrowed the conventional packing problem analogy to pack the generated stones inside the boundary of the walls following building and physical placement rules to obtain realistic microstructures. The size spectrum, distribution, interlocking and topology of the generated stones were thoroughly investigated. To quickly and efficiently solve the posed packing problem of each stone, we proposed a general-purpose heuristic algorithm that benefits from Pareto's principle which is commonly known as the

80/20

rule. This heuristic was called the Pareto sequential sampling (PSS) algorithm as it depends on sampling most of the solutions, using any design of experiment (DoE) method, from a domain that is known to be a prominent pool of solutions.The second contribution was related to studying the topology and roughness of stones and surfaces. The traditional spherical harmonics expansion was used to study the morphology of closed surfaces of the nonconvex stones to estimate their fractal dimension and uniformly remesh the triangulated surfaces to be used in numerical simulations. As the roughness of natural stones changes on a single stone, we similarly studied the morphology of locally sampled rough patches. We developed the spherical cap harmonics and disk harmonics spectral approaches to provide us with information on the roughness and fractal dimension of any isotropic self-affine rough surface. To speed up the numerical integration of these spectral expansions, we proposed new algorithms based on the well-known Kaczmarz orthogonal projection in a non-memory-intensive manner and using reasonable computational times. The developed algorithms were hinged on the conjugate symmetry of the harmonic bases and the sparsity of the real signals to lower the dimensionality of the problem and accelerate the rate of convergence. These spectral methods are considered general-purpose and have multiple applications in various fields such as medical imaging and computer graphics. With this dissertation, we put forward geometrical and topological concepts that make detailed micromechanical simulations of stone masonry walls practical.

EPFL2022