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

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.

Simulation of Time-Dependent Viscoelastic Fluid Flows by Spectral Elements

Azadeh Jafari

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.

EPFL2011

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

Computational fluid dynamics for naval engineering problems

Nicola Parolini

The subject of this thesis is the numerical simulation of viscous free-surface flows in naval engineering applications. State-of-the-art numerical methods based on the solution of the Navier-Stokes equations are used to predict the flow around different classes of boats. We investigate the role of the Computational Fluid Dynamics in the design of racing boats, such as America's Cup yachts and Olympic class rowing hull. The mathematical models describing the different aspects of the physical problem, as well as the numerical methods adopted for their solution, are introduced and critically discussed. The different phases of the overall numerical simulation procedure, from grid generation through the solution of the flow equations to the post-processing of the results, are described. We present the numerical simulations that have been performed to investigate the role of different design parameters in the conception of America's Cup yachts and we describe how the results obtained from the simulations are integrated into the overall design process. The free-surface flow around an Olympic rowing boat is also considered. We propose a simplified approach to take into account the effect of the boat dynamics in the prediction of the hydrodynamic forces acting on the boat. Based on the results of the simulations, we propose a new design concept and we investigate its potential benefits on the boat performances. One of the aspects that is found to be not completely satisfactory, within the standard numerical methods adopted, is the modelling of complex free-surface flows. The second part of this thesis is devoted to a more theoretical and methodological investigation of this aspect. In particular, we present and analyse a new numerical method based on the level set approach for the solution of two-fluid flows. The numerical scheme based on a finite element discretization is introduced and different critical aspects of its implementation are discussed. In particular, we present and analyse a new technique for the stabilization of the advection equation associated to the level set problem. Moreover, we propose a new reinitialization procedure for the level set function which plays a crucial role in the accuracy of the algorithm. The convergence properties of this procedure are analysed and comparisons with more standard approaches are presented. Finally, the proposed method has been used to solve a variety of test cases concerning time dependent two-fluid viscous flows. The results of the simulation are presented and discussed.