Vincent Legat

The Lagrangian particle method for macroscopic and micro-macro viscoelastic flow computations

We propose a new numerical technique, referred to as the Lagrangian Particle Method (LPM), for computing time-dependent viscoelastic flows using either a differential constitutive equation (macroscopic approach) or a kinetic theory model (micro-macro approach). In LPM, the Eulerian finite element solution of the conservation equations is decoupled from the Lagrangian computation of the extra-stress at a number of discrete particles convected by the flow. In the macroscopic approach, the extra-stress carried by the particles is obtained by integrating the constitutive equation along the particle trajectories. In the micro-macro approach, the extra-stress is computed by solving along the particle paths the stochastic differential equation associated with the kinetic theory model. Results are given for the start-up flow between slightly eccentric rotating cylinders, using the FENE and FENE-P dumbbell models for dilute polymer solutions

A new modeling framework for sea-ice mechanics based on elasto-brittle rheology

Abstract We present a new modeling framework for sea-ice mechanics based on elasto-brittle (EB) behavior. the EB framework considers sea ice as a continuous elastic plate encountering progressive damage, simulating the opening of cracks and leads. As a result of long-range elastic interactions, the stress relaxation following a damage event can induce an avalanche of damage. Damage propagates in narrow linear features, resulting in a very heterogeneous strain field. Idealized simulations of the Arctic sea-ice cover are analyzed in terms of ice strain rates and contrasted to observations and simulations performed with the classical viscous–plastic (VP) rheology. the statistical and scaling properties of ice strain rates are used as the evaluation metric. We show that EB simulations give a good representation of the shear faulting mechanism that accommodates most sea-ice deformation. the distributions of strain rates and the scaling laws of ice deformation are well captured by the EB framework, which is not the case for VP simulations. These results suggest that the properties of ice deformation emerge from elasto-brittle ice-mechanical behavior and motivate the implementation of the EB framework in a global sea-ice model.

The backward-tracking Lagrangian particle method for transient viscoelastic flows

A new Lagrangian particle method for solving transient viscoelastic flow for both macroscopic and microscopic stress equations is proposed. In this method, referred to as the backward-tracking Lagrangian particle method (BLPM), we specify the particle locations and calculate the trajectories leading to these locations. This backward tracking process is stopped after a specified time (possibly only a single time step), and the initial configuration for the Lagrangian integration of the stress is obtained by interpolating a stored Eulerian field at that time. In order to demonstrate the accuracy, efficiency and stability of the method, we consider two benchmark problems in the context of the FENE dumbbell kinetic theory of dilute polymer solutions and its FENE-P approximate constitutive equation: the high eccentricity journal bearing flow and the 4:1 contraction flow. With the help of these examples, we show in which manner accurate and stable results can be obtained, for transients of both polymer stress and stream function, with a minimum number of particles and a minimum particle path length

The Consistent Streamline Upwind Petrov-galerkin Method for Viscoelastic Flow Revisited

In an earlier paper, Marchal and Crochet introduced two mixed finite-element methods for calculating viscoelastic flow. The first one, based on a consistent streamline-upwind/Petrov-Galerkin integration of the constitutive equations, was disregarded because it produced wiggles in the numerical calculation of the stick-slip flow of a Maxwell fluid. The second method, based on a non-consistent streamline-upwind integration, was found to be stable at high values of the Weissenberg number; however, it introduces artificial diffusivity of the order of the element size, h, which decreases with mesh refinement. In the present paper, we re-examine the first method. We show that it is both stable and accurate for solving flows of a Maxwell fluid in smooth geometries. The test problems are the flow around a sphere in a tube and the flow through a corrugated tube. The results coincide with those of other accurate methods for solving the same problems. Finally, we show that the results obtained with the second method converge towards the most accurate results when the element size decreases. In particular, we show that the velocity field is little affected by numerical diffusion in the stress constitutive equations.

New closure approximations for the kinetic theory of finitely extensible dumbbells

We address the closure problem for the most elementary non-linear kinetic model of a dilute polymeric solution, known as the Warner finitely extensible non-linear elastic (FENE) dumbbell model. In view of the closure problem, the FENE theory cannot be translated into an equivalent macroscopic constitutive equation for the polymer contribution to the stress tensor. We present a general framework for developing closure approximations, based on the concept of canonical distribution subspace first introduced by Verleye and Dupret (in: Developments in Non-Newtonian Flows, AMD-Vol. 175, ASME, New York, 1993, 139-163) in the context of fiber suspension modeling. The classical consistent pre-averaging approximation due to Peterlin (that yields the FENE-P constitutive equation) is obtained from the canonical approach as the simplest first-order closure model involving only the second moment of the configuration distribution function. A second-order closure model (referred to as FENE-P-2) is derived, which involves the second and fourth moments of the distribution function. We show that the FENE-P-2 model behaves like the FENE-P equation with a reduced extensibility parameter. In this respect, it is a close relative of the FENE-P* equation proposed by van Heel et al. (J. Non-Newton. Fluid Mech., 1998, in press). Inspired by stochastic simulation results for the FENE theory, we propose a more sophisticated second-order closure model (referred to as FENE-L). The rheological response of the FENE-P, FENE-P-2 and FENE-L closure models are compared to that of the FENE theory in various time-dependent, one-dimensional elongational flows. Overall, the FENE-L model is found to provide the best agreement with the FENE results. In particular, it is capable of reproducing the hysteretic behaviour of the FENE model, also observed in recent experiments involving polystyrene-based Boger fluids (Doyle et al., J. Non-Newton. Fluid Mech., submitted), in stress versus birefringence curves during startup of flow and subsequent relaxation

Finite element simulation of a filament stretching extensional rheometer

The present paper is devoted to the numerical study of an extensional rheometer where a fluid sample is stretched between two plates. We use the FENE-CR model in order to simulate a Boger fluid stretched in the device. The analysis shows that the flow is not purely extensional, and that experimental data have to be analysed with caution, in particular if the extensional viscosity is not very high. In general, numerical simulations allow us to better understand experimental data

On the hysteretic behaviour of dilute polymer solutions in relaxation following extensional flow

The hysteretic behaviour of dilute polymer solutions in relaxation following extensional now is studied in the framework of three distinct theoretical models. For ideal kinematics of uniaxial elongation, we show that the kinetic theory of FENE dumbbells and its FENE-L approximation present a hysteresis when plotting polymer stress versus average molecular extension. Similar behaviour is obtained for ideal extensional kinematics using a FENE-P constitutive equation with a spectrum of finite extensibility parameters. Finally, a numerical simulation of the filament-stretching device shows that spatial inhomogeneities of the stress and average conformation fields also lead to hysteretic behaviour with a single-mode FENE-CR constitutive equation. In all three cases, hysteretic behaviour results from the combined effect of dispersity and non-linearity. We also address the validity of the stress-optic law for FENE dumbbells in relaxation following start-up of uniaxial extension. The simulation results show that the stress-optic coefficient remains constant at low strains only. Plots of stress-optic coefficient versus birefringence show hysteresis as well. This rules out a modified stress-optic law for FENE dumbbells wherein the stress-optic coefficient would be a function of the second moment of the configuration distribution function alone. Finally, it is shown in the Appendix that a proper selection of the spectrum of finite extensibilities can be made so that the multi-mode FENE-P model gives essentially the same stress response as the kinetic theory of FENE dumbbells in transient uniaxial extension

Practical Evaluation of 4 Mixed Finite-element Methods for Viscoelastic Flow

We apply four stress-velocity-pressure algorithms to calculate four benchmark problems, i.e., the flow of a Maxwell fluid around a sphere, through a wavy tube, through an abrupt contraction, and in circular extrusion. For every flow, we use only one mesh, i.e., the same number of velocity nodes and the same boundary conditions for all algorithms. The meshes are neither too coarse nor too refined in order to provide us with a practical evaluation of the methods, i.e., a simple mixed method MIX0, the 4 X 4 element, and two types of interpolation for elastic-viscous split stress (EVSS). We also investigate three methods of integration of the constitutive equations: Galerkin, SUPG, and SU. The performance of 4 X 4 and the high-order EVSS are about the same. It is shown that the performance of MIX0 can be remarkably stable and accurate with smooth problems or leads to very poor results in more difficult cases. The low-order EVSS method is accurate, stable, and cheap in computer time. It should be a good candidate for three-dimensional developments.

Advection schemes for unstructured grid ocean modelling

We study advection schemes for unstructured grid ocean models. Four linear advection schemes are investigated by solving a scalar transport equation. Schemes under consideration include continuous, nonconforming and discontinuous finite elements and finite volumes. A comprehensive derivation of the numerical schemes is presented and conservation and dispersion properties are discussed. An assessment is made by performing the test problem introduced by Hecht et al. [J. Geophys. Res. 100 (1995) 20763] in which a passive scalar field is advected through an analytical Stommel gyre. It is found that continuous finite elements and finite volumes have some difficulties to represent accurately solutions with steep gradients. As a result they are prone to generate unphysical oscillations. On the other hand, discontinuous and non-conforming finite element schemes perform better. This is due to their higher flexibility that makes them better suited to highly sheared flows

The adaptive Lagrangian particle method for macroscopic and micro–macro computations of time-dependent viscoelastic flows

We propose a new numerical technique, referred to as the Adaptive Lagrangian Particle Method (ALPM), for computing time-dependent viscoelastic flows using either a differential constitutive equation (macroscopic approach) or a kinetic theory model (micro-macro approach). In ALPM, the Eulerian finite element solution of the conservation equations is decoupled from the Lagrangian computation of the extra-stress at a number of discrete particles convected by the how. In the macroscopic approach, the extra-stress carried by the particles is obtained by integrating the constitutive equation along the particle trajectories. In the micro-macro approach, the extra-stress is computed by solving along the particle paths the stochastic differential equation associated with the kinetic theory model. At each time step, ALPM automatically enforces that all elements of the mesh have a number of Lagrangian particles ranging within a user-specified interval. Results are given for the start-up flow between highly eccentric rotating cylinders, using the FENE and FENE-P dumbbell models for dilute polymer solutions