Jean-Robert Clermont

Non-isothermal viscoelastic flow computations in an axisymmetric contraction at high Weissenberg numbers by a finite volume method

Abstract This paper presents a numerical study of non-isothermal viscoelastic flows in a 4:1 axisymmetric abrupt contraction. The model retained for the simulations is an upper convected Maxwell (UCM) constitutive equation whose temperature dependence is described by a WLF equation. General thermodynamical framework leads to introduce the complex energy conversion mechanism from elastic to thermal energy occurring in viscoelastic fluid flow. The mass, momentum, constitutive and energy equations are discretized using a finite volume method on a staggered grid with upwind scheme for the convective-type terms. A decoupled algorithm stabilized by a pseudo-transient stress term and an elastic viscous stress splitting (EVSS) formulation are employed. Convergence is reached by a fixed-point iteration. The Stokes problem is solved by a fast augmented Lagrangian method and convective-type equations by a bi-conjugate gradient stabilized iterative procedure. Elastic effects are investigated in both isothermal and non-isothermal flow situations. Thermodynamical behaviour related to pure energy elasticity and pure entropy elasticity is considered. Various temperature boundary conditions corresponding to an external cooling of the flow are prescribed at the wall in order to investigate the temperature dependence in flows with large temperature gradients. Without encountering any upper limit for convergence, the present method provides solutions up to Weissenberg number We=10.00 and is able to take into account great temperature changes. General difficulties involved in thermal control processing are underlined.

Numerical simulation of extrudate swell for Oldroyd-B fluids using the stream-tube analysis and a streamline approximation

Abstract In this paper, the stream-tube method is applied to the numerical simulation of axisymmetric extrudate swell of Oldroyd-B fluids. The analysis permits computation of the flow by considering stream tubes separately, from the wall to the inside, in a mapped computational domain where the streamlines are parallel and straight. The present study follows a previous paper on the swelling problem for a fluid obeying a memory-integral equation. A simpler numerical model is proposed here, in which the singularity in the vicinity of the junction point between the wall and the free surface, considered in the previous work, is not examined explicitly. Several values of the parameter β of the Oldroyd-B model, from β = 0.01 to β = 1 (Maxwell model) are considered. The algorithm converges for a broad range of Weissenberg numbers. Results are consistent with the numerical data in the literature. CPU time and storage area are reduced.

Calculation of main flows of a memory-integral fluid in an axisymmetric contraction at high Weissenberg numbers

Abstract A memory-integral equation is considered for numerical flow simulation in a four-to-one circular contraction. The stream-tube analysis, which has been used previously for viscoelastic flow in the swelling problem, is now considered for a duct flow where recirculations are encountered. The method enables computation of the main flow field in a mapped domain of the physical domain where the transformed streamlines are parallel and straight. The primary unknowns of the problem are the transformation and the pressure, and a simple scheme may be developed for evaluating the kinematic quantities involved by a memory-integral equation. The Goddard-Miller constitutive model, already investigated for simulation of the swelling flow of a polystyrene material, is employed. Time evolution and related kinematic quantities are easily taken into account for the computation of stresses. A mixed formulation is adopted and the relevant non-linear equations are solved numerically by the Levenberg-Marquardt algorithm. A satisfactory convergence is obtained up to a Weissenberg number of 30. The results, though related to the main flow in the circular abrupt contraction, clearly show the growing effects of the singularity at the section of contraction and the importance of the recirculating zone near the salient corner.

Experimental and numerical study of entry flow of low-density polyethylene melts

The present work deals with experimental and numerical features of entry flows of two polyethylene melts, namely a linear low-density polyethylene (LLDPE) and a low-density polyethylene (LDPE) in an axisymmetric converging geometry. The study also involves rheological characterization of the polymers and determination of flow parameters, at 160°C. For both fluids, the data are fed into a viscoelastic integral Wagner constitutive equation. The numerical flow simulations are performed by using a stream-tube mapping analysis. Consideration of a sub-domain of the total flow domain, the “peripheral stream tube”, close to the wall of the converging duct permits to relate the results of the numerical simulation to experimental flow characteristics as total and entrance pressure drops. The agreement is good for the total pressure losses, but, concerning LDPE, a lack of consistency remains for the entrance pressure drop.

Calculation of kinematic histories in two- and three-dimensional flows using streamline coordinate functions

In this paper, the theoretical elements of the stream-tube method are considered for the evaluation of strain-rate and strain histories required for the descriptions of memory-integral equations. One interest of the stream-tube analysis lies in the fact that the computation is performed in a transformed domain of the physical domain where the mapped streamlines are parallel and straight. Unknown mapping functions are used for analytic expressions of tensor components, for two- and three-dimensional flow situations. Calculations in three-dimensional flows for corotational models indicate that the relevant analytic expressions deduced from corotating frame determination are too complicated to be realistic for computation, but are exploitable for two-dimensional flow simulations. Strain histories are presented for two- and three-dimensional flows and may be applied to evaluation of stresses in both cases, when using codeformational constitutive equations.

Experimental and numerical study of the swelling of a viscoelastic liquid using the stream tube method and a kinematic singularity approximation

Abstract The present paper deals with experimental and numerical aspects of the swelling problem. The rheological characterization of an industrial polystyrene and determination of flow parameters (jet surface, velocities, etc.) are invoved. The data are fed into a viscoelastic integral constitutive equation. The swell problem is analysed using the stream tube method and a specific kinematic approximation of the singularity at the junction point between the wall and the free surface. A method is proposed for determining the swelling ratio by considering approximate functions for the streamlines. The numerical results are consistent with those of the literature, for low Weissenberg numbers. Moreover, the present analysis enables us, using approximate expansions for the kinematic functions in the singularity domain, to reach higher Weissenberg numbers than 20. In this case, the numerical predictions of the model are still consistent with the experimental data, particularly for those of the jet surface and velocity profiles.

A numerical approach for computing flows by local transformations and domain decomposition using an optimization algorithm

This paper presents theoretical information for numerical simulation of two- and three-dimensional flows, based on the concepts of streamlines and local transformation functions associated with domain decomposition. In this approach, in addition to the pressure, the primary unknowns are the local mapping functions between the physical sub-domains and transformed domains where the mapped streamlines are parallel straight lines. This makes it easy to handle time-dependent constitutive equations for complex fluids. To solve the governing equations that also involve compatibility equations between sub-domains, an optimization algorithm is set up in order to compute the unknowns related to the streamlines iteratively. Applications are given for two-dimensional flows between eccentric cylinders, using different constitutive equations.

Numerical simulation of the die swell problem of a Newtonian fluid by using the concept of stream function and a local analysis of the singularity at the corner

Abstract The die-swell problem is reconsidered by using the concept of stream tubes for the incompressible axisymmetric case. Using some assumptions and analytical equations, it is possible to study the influence of the singularity at the corner where the free surface forms, in addition to the upstream and downstream boundary conditions. A minimization technique is used for the determination of the coefficients related to the analytical equations of the streamlines. This enables us to compute the flow field in the jet for a Newtonian fluid, the swelling ratio of which is found to be 12%.

Computations of non‐isothermal viscous and viscoelastic flows in abrupt contractions using a finite volume method

A finite volume method is applied to numerical simulations of steady isothermal and non‐isothermal flows of fluids obeying different constitutive equations: Newtonian, purely viscous with shear‐thinning properties (Carreau law) and viscoelastic Upper Convected Maxwell differential model whose temperature dependence is described by a William‐Landel‐Ferry equation. The flow situations concern various abrupt axisymmetric contractions from 2:1 to 16:1. Such flow geometries are involved in polymer processing operations. The governing equations are discretized on a staggered grid with an upwind scheme for the convective‐type terms and are solved by a decoupled algorithm, stabilized by a pseudo‐transient stress term and an elastic viscous stress splitting technique. The numerical results highlight the influence of temperature on the flow situations, and also the complex behaviour of the materials under non‐isothermal conditions.

Finite element and stream-tube formulations for flow computations: two-dimensional applications

This paper presents a velocity/pressure/function-of-transformation formulation to simulate two- and three-dimensional flows by a finite element Galerkin technique associated to stream-tube analysis. The latter method involves an unknown transformation. Applications are given for plane and axisymmetric flows with slip or no-slip conditions at the wall and with free surfaces. In the mapped domain, two-dimensional streamline Hermite elements, well-adapted to various boundary conditions, are considered. The numerical results for a Newtonian fluid are found to be consistent with those of the literature and highlight singularity effects that arise at the transition zone of change of boundary conditions, in stick-slip and swell flow problems.