Discrete Elastic Viscous Split Stress Research Articles

Computational modelling of compressible nonisothermal viscoelastic fluids

The prediction of compressible nonisothermal flows of viscoelastic fluids is important in many industrial processes. Until relatively recently, there was a lack of tractable thermodynamically consistent mathematical models for this class of flows. In this paper a stabilised finite element scheme is presented for the models developed by the authors Mackay and Phillips (2019) which incorporates compressibility and nonisothermal effects. Numerical results are presented for variants of a couple of benchmark problems: the lid driven cavity and the natural convection problems. The temporal discretisation is based on the Taylor–Galerkin method. A compressible version of the discrete elastic viscous split stress (DEVSS) formulation is used to stabilise the numerical scheme. The combined and separate influence of compressible, viscoelastic and thermal effects on the characteristics of these benchmark flows is studied for the first time.

Finite Volume Computational Analysis of the Heat Transfer Characteristic in a Double-Cylinder Counter-Flow Heat Exchanger with Viscoelastic Fluids

This work presents a computational analysis of the heat-exchange characteristics in a double-cylinder (also known as a double-pipe) geometrical arrangement. The heat-exchange is from a hotter viscoelastic fluid flowing in the core (inner) cylinder to a cooler Newtonian fluid flowing in the shell (outer) annulus. For optimal heat-exchange characteristics, the core and shell fluid flow in opposite directions, the so-called counter-flow arrangement.The mathematical modelling of the given problem reduces to a system of nonlinear coupled Partial Differential Equations (PDEs). Specifically, the rheological behaviour of the core fluid is governed by the Giesekus viscoelastic constitutive model. The governing system of coupled nonlinear PDEs is intractable to analytic treatment and hence is solved numerically using Finite Volume Methods (FVM). The FVM numerical methodology is implemented via the open-source software package OpenFOAM. The numerical methods are stabilized, specifically to address numerical instabilities arising from the High Weissenberg Number Problem (HWNP), via a combination of the Discrete Elastic Viscous Stress Splitting (DEVSS) technique and the Log-Conformation Reformulation (LCR) methodology. The DEVSS and LCR stabilization techniques are integrated into the relevant viscoelastic fluid solvers. The novelties of the study center around the simulation and analysis of the optimal heat-exchange characteristics between the heated Giesekus fluid and the coolant Newtonian fluid within a double-pipe counter-flow arrangement. Existing studies in the literature have either focused exclusively on Newtonian fluids and/or on rectangular geometries. The existing OpenFOAM solvers have also largely focused on non-isothermal viscoelastic flows. The relevant OpenFOAM solvers are modified for the present purposes by incorporating the energy equation for viscoelastic fluid flow. The flow characteristics are presented qualitatively (graphically) via the fluid pressure, temperature, velocity, and the polymer-stress components as well as the related normal stress differences. The results illustrate the required decrease in the core fluid temperature in the longitudinal direction due to the cooling effects of the shell fluid, whose temperature predictably increases in the counter-flow direction.

A stabilised finite element framework for viscoelastic multiphase flows using a conservative level-set method

The development of stable numerical schemes for viscoelastic multiphase flows has implications for many areas of engineering applications. The principal original contribution of this paper is the implementation of a conservative level-set method to define implicitly the interface between fluid phases, fully integrated into the mathematical framework of viscoelastic flow. The governing equations are discretized using the finite element method and stabilisation of the constitutive equation is achieved using either the discontinuous Galerkin (DG) or streamline upwinding (SU) method. The discrete elastic viscous stress splitting gradient (DEVSS-G) formulation is also employed in the Navier-Stokes equations to balance the hyperbolic characteristics of the polymeric stress tensor. The numerical scheme is validated with reference to several benchmark problems and excellent quantitative agreement with published data is found for Newtonian and viscoelastic fluids, for both single and multiphase flows. The motion of a gas bubble rising in a viscoelastic fluid is studied in detail. The influence of polymer concentration, surface tension, fluid elasticity and shear-thinning behaviour, on flow features such as the development of filaments and cusps and the generation of negative wakes is explored.

Numerical study on oblique stretching of viscoelastic polymer film

• Numerical format for oblique stretching of viscoelastic film is established. • The application scenarios of the membrane hypothesis are expanded. • Detailed numerical procedure and a complete numerical case are provided. • The influence of the y -direction stretching on thickness uniformity is investigated. Oblique stretching is a special approach for the fabrication of polymer film whose product is mainly used in the optical display field, such as compensation films, retardation films, et al. Nonetheless, the previous research reports on oblique stretching processing are still lacking, and the related numerical research is almost blank. This contribution uses the FEM (finite element method) to simulate the oblique stretching process of viscoelastic polymer film for the first time. The differential viscoelastic PTT (Phan-Thien and Tanner) model, one of the most realistic constitutive models is chosen. Furthermore, the polycarbonate melt was rheologically characterized and used as input material parameters in our simulations. Based on the membrane hypothesis and stabilization algorithms like DEVSS (discrete elastic viscous stress splitting), the specific numerical scheme is derived and the algorithm implementation is summarized. Finally, a complete numerical example of oblique stretching is demonstrated and compared with symmetric stretching. Based on the simulation results, the evolution of thickness and the influence of the y -direction stretching on thickness uniformity are further investigated.

An Experimental Investigation of Viscoelastic Flow in a Contraction Channel.

In order to assess the predictive capability of the S–MDCPP model, which may describe the viscoelastic behavior of the low-density polyethylene melts, a planar contraction flow benchmark problem is calculated in this investigation. A pressure-stabilized iterative fractional step algorithm based on the finite increment calculus (FIC) method is adopted to overcome oscillations of the pressure field due to the incompressibility of fluids. The discrete elastic viscous stress splitting (DEVSS) technique in combination with the streamline upwind Petrov-Galerkin (SUPG) method are employed to calculate the viscoelastic flow. The equal low-order finite elements interpolation approximations for velocity-pressure-stress variables can be applied to calculate the viscoelastic contraction flows for LDPE melts. The predicted velocities agree well with the experimental results of particle imagine velocity (PIV) method, and the pattern of principal stress difference calculated by the S-MDCPP model has good agreement with the results measured by the flow induced birefringence (FIB) device. Numerical and experimental results show that the S-MDCPP model is capable of accurately capturing the rheological behaviors of branched polymers in complex flow.

Numerical simulation of viscoelastic blood flow with hematocrit variation in an arterial segment with two aneurysms

In this study, a viscoelastic model with variable viscosity and relaxation-time is proposed for the simulation of the blood flow in an arterial segment with two aneurysms. The Quemada model is adopted to model both the shear rate-dependent viscosity and hematocrit variation. Available experimental data for the shear rate-dependent relaxation-time of the blood, in a certain range, are fitted and used. The arterial segment with aneurysms is considered as a rigid axisymmetric thin tube with two balloon expansions. The stabilised finite element method with the discrete elastic viscous stress splitting (DEVSS) method is used to solve the governing equations to overcome the numerical instabilities. Numerical results including velocity profiles, shear rate distribution and viscosity contours are obtained under different values of red blood cells (RBCs) concentrations. The results indicate that the hematocrit variation has a significant effect on the flow regimes and hemodynamics factors such as the wall shear stress (WSS). Hence, the shear thinning property should not be ignored for blood flow simulations.

Stabilized variational formulation of an oldroyd-B fluid flow equations on a Graphic Processing Unit (GPU) architecture

The governing equations of the flow of an oldroyd-B fluid are discretized using the finite element method. To overcome the convective nature of the momentum equation, the Galerkin/Least-Squares Finite Element Method (GLS/FEM) is used while the Discrete Elastic–Viscous Stress-Splitting (DEVSS) method is used to overcome the instability due to the absence of diffusion in the constitutive equations. The discretized equations are implemented on a hybrid system between the Graphics Processing Unit (GPU) architecture using Compute-Unified-Device-Architecture (CUDA) and a multi-core CPU. The implementation is applied successfully to simulate the blood flow in abdominal aortic aneurysm. To accelerate application performance on the GPU several optimized approaches are adopted. The most significant approach is the coloring technique that is used to assemble the global matrix. Numerical experiments show that the hybrid CPU-GPU implementation has a 26 time speedup over the multi-core CPU implementations.

Numerical Study of Shear Banding in Flows of Fluids Governed by the Rolie-Poly Two-Fluid Model via Stabilized Finite Volume Methods

The flow of viscoelastic fluids may, under certain conditions, exhibit shear-banding characteristics that result from their susceptibility to unusual flow instabilities. In this work, we explore both the existing shear banding mechanisms in the literature, namely; constitutive instabilities and flow-induced inhomogeneities. Shear banding due to constitutive instabilities is modelled via either the Johnson–Segalman or the Giesekus constitutive models. Shear banding due to flow-induced inhomogeneities is modelled via the Rolie–Poly constitutive model. The Rolie–Poly constitutive equation is especially chosen because it expresses, precisely, the shear rheometry of polymer solutions for a large number of strain rates. For the Rolie–Poly approach, we use the two-fluid model wherein the stress dynamics are coupled with concentration equations. We follow a computational analysis approach via an efficient and versatile numerical algorithm. The numerical algorithm is based on the Finite Volume Method (FVM) and it is implemented in the open-source software package, OpenFOAM. The efficiency of our numerical algorithms is enhanced via two possible stabilization techniques, namely; the Log-Conformation Reformulation (LCR) and the Discrete Elastic Viscous Stress Splitting (DEVSS) methodologies. We demonstrate that our stabilized numerical algorithms accurately simulate these complex (shear banded) flows of complex (viscoelastic) fluids. Verification of the shear-banding results via both the Giesekus and Johnson-Segalman models show good agreement with existing literature using the DEVSS technique. A comparison of the Rolie–Poly two-fluid model results with existing literature for the concentration and velocity profiles is also in good agreement.

Benchmark solutions of the stabilized computations of flows of fluids governed by the Rolie-Poly constitutive model

Recent studies demonstrate that flow induced non-uniformities of concentration can trigger shear banding in the flow of certain viscoelastic fluids. These studies show that the driving mechanisms for such shear banding are related to the coupling of the polymer stresses to an inhomogeneous concentration profile. The Rolie-Poly (RP) viscoelastic constitutive model has been used in such studies since it has been comprehensively subjected to extensive experimental validation with regards to shear banding and has the demonstrated ability to accurately express the rheology of polymer solutions for a wide range of strain rates. The primary aim of this work is to develop an efficient computational methodology that could be used to accurately simulate the flow of complex fluids governed by the Rolie-Poly constitutive equation. The development of such a computational platform is crucially important for the purposes of our follow up studies on the computational analysis of shear banding phenomena by coupling polymer stress with inhomogeneous concentration profile. Our numerical algorithms will be based on the finite volume method (FVM) and will be implemented on the open source software package OpenFOAM®. In this paper, we will present both validation results as well as new benchmark results from our FVM based OpenFOAM® numerical solver for flow of fluids governed by the Rolie-Poly constitutive model. We use two well-known benchmark problems, the lid-driven cavity flow and the 4:1 planar contraction flow problems. In order to stabilize the numerical algorithm at high Weissenberg numbers, we employ either of two stabilization techniques; the Discrete Elastic Viscous Stress Splitting (DEVSS) technique as well as the Log-Conformation Reformulation (LCR) methodology. Validation of our results is done by comparing our (stabilized) numerical results, against data from existing literature. The numerical results obtained for the contraction flow using the LCR stabilization approach are in good agreement with the existing literature for a wider range of Weissenberg numbers. The DEVSS method shows a good agreement only for lower Weissenberg numbers. For the lid-driven cavity flow, good agreement with the existing literature is observed for low Weissenberg numbers using either of the two stabilization techniques.

The cell-based smoothed finite element method for viscoelastic fluid flows using fractional-step schemes

This paper generalizes the cell-based smoothed finite element method (CS-FEM) to viscoelastic fluid flows under the fractional-step umbrella. The characteristic-based split (CBS) scheme is employed to decouple the Navier–Stokes equations and the Oldroyd-B constitutive model. Moreover, the discrete elastic-viscous-split-stress (DEVSS) procedure and stabilized pressure gradient projection (SPGP) are introduced into a second-order CBS framework. The projections of velocity and pressure gradients therein combine to form the DEVSS-G/CBS(B)-SPGP algorithm that further stabilizes the discretized model. Both proposed schemes are suitable for use in CS-FEM because they accept equal low-order interpolations for primitive variables. The spatial discretization is based upon bilinear four-node quadrilateral elements where all gradient-related items are smoothed without any methodological difficulties. In particular, the cell-based smoothing concept enables accurate evaluation of viscoelastic fluid force. The developed methods are validated against previously published data for several benchmark problems. A satisfactory agreement exists between the present and earlier results.

Finite element simulation of three-dimensional viscoelastic flow at high Weissenberg number based on the log-conformation formulation

Viscoelasticity is an important characteristic of many complex fluids such as polymer solutions and melts. Understanding the viscoelastic behavior of such complex fluids presents mathematical, modeling and computational challenges, particularly in the case of fluids affected by elastic turbulence at high Weissenberg number. A numerical methodology based on the penalty finite element method with a decoupled algorithm is presented in the study to simulate three-dimensional flow of viscoelastic fluids. The discrete elastic viscous split stress (DEVSS) formulation in cooperating with log-conformation formulation transformation is employed to improve computational stability at high Weissenberg number. The momentum equation is calculated after introducing an ellipticity factor and the constitutive equation is calculated based on the logarithm of the conformation tensor. The finite element-finite difference formulations of governing equations are derived. The planar contraction as a representative benchmark problem is used to test the robustness of the numerical method to predict real flow patterns of viscoelastic fluids at different Weissenberg numbers. The simulation results predicted with differential constitutive models based on the logarithm of the conformation tensor agree well with Quinzani’s experimental results. Both the stability and the accuracy are improved compared with traditional calculation method. The numerical methodology proposed in the study can well predict complex flow patterns of viscoelastic fluids at high Weissenberg number.

Nonlinear stability and dynamics of nonisothermal film casting

A well-known Phan-Thien and Tanner differential viscoelastic constitutive equation has been employed to analyze the nonlinear stability and dynamics of nonisothermal film casting. A finite element method combined with stabilization techniques including discrete elastic viscous stress splitting and the streamline upwind Petrov–Galerkin method was first performed for the numerical calculation of transient film casting. Therefore, the stability analysis of film casting was carried out based on a wider parameter space of processing and the rheological properties of the polymer melt, during which the critical draw ratio for the occurrence of draw resonance (Drc) was taken as an indicator for flow stability. Unlike the results predicted by the upper-convected Maxwell model, the stabilizing region upon an upper critical draw ratio is nonexistent, while more than one peak of Drc was observed with varying aspect ratios, which was dominated by two deformation types named planar and transitional deformations, respec...

A comparative study of viscoelastic planar contraction flow for polymer melts using molecular constitutive models

The benchmark problem of planar contraction flow is often used to evaluate the stability of numerical algorithms and the predictability of viscoelastic constitutive models in predicting the rheological behavior of polymer fluids. In this paper, for the simulation of planar contraction flow, the single/simplified modified double convected Pom-Pom (S-MDCPP) model recently proposed is adopted, and in the frame of iterative stabilized fractional step algorithm, the discrete elastic viscous stress splitting (DEVSS) technique in combination with the streamline upwind Petrov-Galerkin (SUPG) method is taken to serve as a stable numerical solution scheme for the viscoelastic flow, where the equal low-order interpolations are applied to the independent solution variables by virtue of stabilized form of mass conservation equation constructed based-on the finite incremental calculus method. The comparison of rheological behaviors by S-MDCPP model with those by experiment results and DCPP model is taken on the aspects of the material functions of shear viscosities and uniaxial elongation viscosities, stresses fields through the way of flow induced birefringence in the planar contraction flow, with the aim of assessing the constitutive models. In addition, the effects of the constitutive parameters of the S-MDCPP model and Weissenberg number on the rheological behaviors of polymer melts are discussed, with the flow field responses such as velocity, pressure, stress components, and backbone stretch included. Through these comparative studies, the capabilities of the S-MDCPP constitutive model are justified for the applications in branched polymer melts characterizations and numerical modeling.

Finite element simulation of three-dimensional viscoelastic planar contraction flow with multi-mode FENE-P constitutive model

Viscoelasticity is a characteristic of many complex fluids like polymer melts, petroleum, blood, etc. The investigation of viscoelastic flow mechanism has practical significance in both scientific and engineering field. Owing to strongly nonlinear, numerical method becomes a practical way to solve viscoelastic flow problem. In the study, the mathematical model of three-dimensional flow of viscoelastic fluids is established. The planar contraction flow as a benchmark problem for the numerical investigation of viscoelastic flow is solved by using the penalty finite element method with a decoupled algorithm. The multi-mode finitely extensible nonlinear elastic dumbbell with a Peterlin closure approximation (FENE-P) constitutive model is used to describe the viscoelastic rheological properties. The discrete elastic viscous split stress formulation in cooperating with the inconsistent streamline upwind scheme is employed to improve the computation stability. The numerical methods proposed in the study can be well used to predict complex flow patterns of viscoelastic fluids.

Continuous modeling and simulation of flow-swell-crystallization behaviors of viscoelastic polymer melts in the hollow profile extrusion process

The continuous viscoelastic flow, swell and crystallization behaviors of polymer melts experienced within and through out of the die/mould can significantly influence the variation of internal structure, and hence determine the performance and dimension of final products in practical processing. However, traditionally theoretical and experimental methods are difficult to be afforded to this problem. In the study, the continuous viscoelastic flow, swell and crystallization behaviors of polymer melts within and through out of practically used die/mould are investigated by means of finite element-finite difference simulation. The mathematical model of three-dimensional viscoelastic flow of polymer melts is established with Phan-Thien and Tanner constitutive model. The streamface–streamline method is introduced to adjust the swelling free surface of the extrudate. The influence of the thermal and flow state on the crystallization phenomenon are discriminated using a modified Schneider’s approach. A decoupled solving algorithm for the governing equations is introduced and the corresponding penalty finite element-finite difference model is derived. The discrete elastic viscous split stress algorithm incorporating the streamline upwind scheme is adopted to improve calculation stability. The essential viscoelastic flow characteristics, extrudate swell phenomenon and flow induced crystallization of polymer melts in the hollow profile extrusion process are investigated based on the proposed mathematical model and numerical methods. Both the distribution of flow variables within the extrusion die and the variation of crystallization kinetic process within the shaped mould are successfully predicted by means of finite element-finite difference simulation of flow-swell-crystallization continuous behaviors of viscoelastic polymer melts.

Numerical investigation of the crystallization and orientation behavior in polymer processing with a two-phase model

The crystallization and orientation behavior of a polymeric material can significantly influence the performance of products in practical processing. In this study, the variations in morphology that occur during solidification in polymer processing are mathematically modeled using a two-phase model. The amorphous phase is approximated as a finite extensible nonlinear elastic dumbbell with a Peterlin closure approximation (FENE-P) fluid, and the semi-crystalline phase is modeled as rigid rods oriented within the flow field. The crystallization and orientation behavior are numerically investigated using the penalty finite element–finite difference method with a decoupled algorithm. The evolution of the crystallization process is described by Schneider's equation, which differentiates between the effects of thermal and flow states. The hybrid closure approximation is adopted for the calculation of the three-dimensional orientation tensor. The discrete elastic viscous split stress (DEVSS) algorithm, which incorporates the streamline upwind scheme, is introduced to improve calculation stability. The variations in morphology during polymer processing are successfully predicted using the proposed mathematical model and numerical method. The influence of processing conditions on the crystallization and orientation behavior is further discussed.

A new discrete EVSS method for the viscoelastic flows

In this paper, we propose a new discrete elastic–viscous-split-stress (EVSS) method for the viscoelastic flows obeying an Oldroyd-B type constitutive equation. Supposing that the velocity and pressure are approximated by the Taylor–Hood elements, and applying the orthogonal projection technique, we introduce two Gauss integrations as a stabilizing term in the discrete EVSS method. Compared with the popular one, the new method does not need to introduce any extra variable and reduces the degrees of freedom of the discrete system a lot, but gets the same stabilized result. Additionally, some numerical experiments are shown to verify its efficiency.

Numerical investigation of the thermally and flow induced crystallization behavior of semi-crystalline polymers by using finite element–finite difference method

The thermally and flow induced crystallization behavior of semi-crystalline polymer in processing can significantly influence the quality of final products. The investigation of its mechanism has both scientific and industrial interest. A mathematical model in three dimensions for thermally and flow induced crystallization of polymer melts obeying differential Phan-Thien and Tanner (PTT) constitutive model has been developed and solved by using the finite element–finite difference method. A penalty method is introduced to solve the nonlinear governing equations with a decoupled algorithm. The corresponding finite element–finite difference model is derived by using the discrete elastic viscous split stress algorithm incorporating the streamline upwind scheme. A modified Schneider's approach is employed to discriminate the relative roles of the thermal state and the flow state on the crystallization phenomenon. The thermally and flow induced crystallization characteristics of polypropylene is investigated based on the proposed mathematical model and numerical scheme. The half crystallization time of polypropylene in a cooled couette flow configuration obtained by simulation are compared with Koscher's experimental results, which show that they agree well with each other. Two reasons to cause crystallization of polypropylene in pipe extrusion process including the thermal state and the flow state are investigated. Both the crystalline distribution and crystalline size of polypropylene are obtained by using the finite element–finite difference simulation of three-dimensional thermally and flow induced crystallization. The effects of processing conditions including the volume flow rate and the temperature boundary on the crystallization kinetics process are further discussed.

Modeling and simulation of three-dimensional planar contraction flow of viscoelastic fluids with PTT, Giesekus and FENE-P constitutive models

The planar contraction flow is a representative benchmark problem that is often used as a stringent test for the robustness of the numerical method to predict real flow patterns of viscoelastic fluids. A numerical methodology based on the penalty finite element method with a decoupled algorithm is presented in the study to simulate three-dimensional flow of viscoelastic fluids through planar contraction. The viscoelastic rheological responses are described by using three kind differential constitutive models including the Phan-Thien–Tanner (PTT) model, the Giesekus model and the finite extensible nonlinear elastic dumbbell with a peterlin closure approximation (FENE-P) model. The discrete elastic viscous split stress (DEVSS) formulation in cooperating with the inconsistent streamline upwind (SU) scheme is employed to improve the computational stability. The simulation results of flow velocity and stress on different sections of the flow field are compared with Quinzani’s experimental results that detected by laser-doppler velocimetry and flow-induced birefringence technologies. It is found that the simulation results predicted with three kind differential constitutive models agree well with the experimental results. The numerical methodology proposed in the study can be used successfully to predict complex flow patterns of viscoelastic fluids.

Numerical simulation of three-dimensional viscoelastic planar contraction flow using the software OpenFOAM

In this study, we have used a highly accurate and novel approach to solve numerically for true three-dimensional (3D) viscoelastic flows into sudden contractions. Motivation for this development has been for the advancement of 3D viscoelastic flows in complex geometries, where this new methodology is available as a general solver, written for the open source code OpenFOAM ( Weller et al., 1998). The proposed approach is able to include the multiple relaxation times of differential constitutive equations, and has been performed using the Finite Volume Method (FVM), based on a Discrete Elastic Viscous Split Stress (DEVSS) technique ( Fortin et al., 1997). In this work, both the Giesekus and Phan-Thien–Tanner (PTT) shear-thinning models were implemented to reproduce flow through a planar 4:1 contraction, where numerical convergence was achieved for a Weissenberg number ( We) of 2.9. Direct comparison with experimental data and literature involving 2D and 3D numerical simulations shows this method to be both stable and effective.