Simulation of Time-Dependent Viscoelastic Fluid Flows by Spectral Elements

The research work reported in this dissertation is aimed to develop efficient and stable numerical schemes in order to obtain accurate numerical solution for viscoelastic fluid flows within the spectral element context. The present research consists in the transformation of a large class of differential constitutive models into an equation where the main variable is the logarithm of the conformation tensor or a quantity related to it in a simple way. Particular cases cover the Oldroyd-B fluid and the FENE-P model. Applying matrix logarithm formulation in the framework of the spectral element method is a new type of approach that according to our knowledge no one has implemented before. The reformulation of the classical constitutive equation using a new variable namely the logarithmic formulation, enforces the eigenvalues of the conformation tensor to remain positive for all steps of the simulation. However, satisfying the symmetric positive definiteness of the conformation tensor during the simulation is the necessary condition for stability; but definitely, it is not the sufficient condition to reach meaningful results. The main effort of this research is devoted to introduce a new algorithm in order to overcome the drawback of direct reformulating the classical constitutive equation to the logarithmic one. To evaluate the capability of the extended matrix logarithm formulation, comprehensive studies have been done based on the linear stability analysis to show the influence of this method on the resulting eigenvalue spectra and explain its success to tackle high Weissenberg numbers. With this new method one can treat high Weissenberg number flows at values of practical interest. One of the worst obstacles for numerical simulation of viscoelastic fluids is the presence of spurious modes during the simulation. At high Weissenberg number, many schemes suffer from instabilities and numerical convergence may not be attainable. This is often attributed to the presence of solution singularities due to the geometry, the dominant non-linear terms in the constitutive equations, or the change of type of the underlying mixed-form differential system. Refining the mesh proved to be not very helpful. In this study, to understand more deeply the mechanism of instability generation a comprehensive study about the growth of spurious modes with time evolution, mesh refinement, boundary conditions and Weissenberg number or any other affected parameters has been performed. Then to get rid of these spurious modes the filter based stabilization of spectral element methods proposed by Boyd was applied with success.

Simulation of flows of viscoelastic fluids at high Weissenberg number using a filter-based stabilization of the spectral element method

Michel Deville, Nicolas Fietier, Azadeh Jafari

The challenge for computational rheologists is to develop efficient and stable numerical schemes in order to obtain accurate numerical solutions for the governing equations at values of practical interest of the Weissenberg number. One of the associated problems for numerical simulation of viscoelastic fluids is that the accuracy of the results when approaching critical values at which numerical instabilities occur is very low and refining the mesh proved to be not very helpful. In order to investigate the numerical instability generation a comprehensive study about the growth of spurious modes with time evolution, mesh refinement, boundary conditions and Weissenberg number or any other affected parameters is performed on the planar Poiseuille channel flow. To get rid of these spurious modes the filter based stabilization of spectral element methods proposed by Boyd (1998) [1] is applied. This filter technique is very useful to eliminate spurious modes for one element decomposition, while in the case of multi-element configuration, the performance of this technique is not ideal. Since the performance of filter-based stabilization of spectral element acts very well for one element decomposition, a possible remedy to solve the associated problem of multi-element decomposition is mesh-transfer technique which means: first mapping the multi-element configuration to one element configuration, applying filter-based stabilization technique to this new topology and hereafter transferring the filtered variables to the original configuration. This way of implementing filtering is very useful for the Oldroyd-B fluids when a moderate number of grid points is used. (C) 2011 Elsevier Ltd. All rights reserved.

2012

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

Time-dependent algorithms for the simulation of viscoelastic flows with spectral element methods: applications and stability

Michel Deville, Nicolas Fietier

This paper presents the development of spectral element methods to simulate unsteady flows of viscoelastic fluids using a closed-form differential constitutive equation. The generation and decay Poiseuille planar flows are considered as benchmark problems to test the abilities of our computational method to deal with truly time-dependent flows. Satisfactory results converging toward steady-state regimes have been obtained for the flow through a four-to-one planar abrupt contraction with unsteady algorithms. Time-dependent simulations of viscoelastic flows are prone to numerical instabilities even for simple geometrical configurations. Possible methods to improve the numerical stability of the computational algorithms are discussed in view of the results carried out with numerical simulations for the flows through a straight channel and the four-to-one contraction.

2003

Simulation of shear-driven flows

Roland Bouffanais

EPFL2007