Numerical Simulation Of Viscoelastic Flow Research Articles

Reynolds and Weissenberg numbers in viscoelastic flows

One of the greatest challenges in performing numerical simulations of viscoelastic flow is how to choose the model parameters in order to be able to achieve quantitative agreement with experimental measurements in benchmark geometries. In the present work, we revisit the original concepts of Reynolds (Re) and Weissenberg (Wi) numbers in order to provide means to adequately choose those parameters. In particular, we highlight the importance of the Oldroyd-B model as a reference for viscoelastic behavior and propose a new methodology for the comparison between general viscoelastic models and the Oldroyd-B model. We identify inconsistencies in the usual approach consensually adopted in the literature. The historical reason for that seems to have its root on the fact that Reynolds and Weissenberg numbers are recognized as expressions and not as concepts. This paradigm allows the choice of parameters of a model as characteristic quantities. We propose here to choose the Oldroyd-B model parameters for the comparison with more complex viscoelastic models so that Re and Wi concepts are maintained. To illustrate this matter, we simulate a flow which is known for capturing an important viscoelastic effect, the transient extrudate swell problem. We tested three viscoelastic models with intermediate levels of complexity, namely PTT, Giesekus, and FENE-P. The traditional and the proposed approaches are compared and the role of the extra parameter, that endows these models with a nonlinear character with respect to the stress, is reinterpreted. It is demonstrated that the nonlinearity introduced by this extra parameter is overestimated in the usual formulation.

Inertia-driven and elastoinertial viscoelastic turbulent channel flow simulated with a hybrid pseudo-spectral/finite-difference numerical scheme

Numerical simulation of viscoelastic flows is challenging because of the hyperbolic nature of viscoelastic constitutive equations. Despite their superior accuracy and efficiency, pseudo-spectral methods require the introduction of artificial diffusion (AD) for numerical stability in hyperbolic problems, which alters the physical nature of the system. This study presents a hybrid numerical procedure that integrates an upwind total variation diminishing (TVD) finite-difference scheme, which is known for its stability in hyperbolic problems, for the polymer stress convection term into an overall pseudo-spectral numerical framework. Numerically stable solutions are obtained for Weissenberg number well beyond O(100) without the need for either global or local AD. Side-by-side comparison with an existing pseudo-spectral code reveals the impact of AD, which is shown to differ drastically between flow regimes. Elastoinertial turbulence (EIT) becomes unphysically suppressed when AD, at any level necessary for stabilizing the pseudo-spectral method, is used. This is attributed to the importance of sharp stress shocks in its self-sustaining cycles. Nevertheless , in regimes dominated by the classical inertial mechanism for turbulence generation, there is still an acceptable range of AD that can be safely used to predict the statistics, dynamics, and structures of drag-reduced turbulence. Detailed numerical resolution analysis of the new hybrid method, especially for capturing the EIT states, is also presented.

A closure model for the drag coefficient of a sphere translating in a viscoelastic fluid

In many large-scale industrial applications dealing with particle-laden viscoelastic fluids, the ensemble-averaged behavior of the mixture is of most interest. The first step to parametrize this behavior is to develop an accurate expression to rapidly evaluate the drag coefficient over a broad range of kinematic parameters. The drag coefficient of a spherical particle translating in a viscoelastic matrix is strongly affected by the viscoelasticity of the fluid. In this study, we aim to parametrize the effects of fluid elasticity, especially the relaxation and retardation times, as well as inertia on the drag coefficient of a sphere translating in a viscoelastic fluid described by the Oldroyd-B model. To this end, we employed three-dimensional direct numerical simulations of viscoelastic flow past a stationary sphere. The accuracy of the numerical formulation is thoroughly tested against a number of benchmark problems consisting of steady flow past a sphere in a bounded circular or square domain filled with either a Newtonian or viscoelastic fluid. Initially, the numerical computations for the drag coefficient over a wide range of geometric and flow parameters are validated by comparison with existing data and drag correction models from the literature. The drag coefficient correction is then evaluated for unconfined flow past a sphere at different Reynolds number, Re, over a wide range of Deborah number, De < 9, and polymer viscosity ratio, 0 < ζ < 1. For small Deborah number (De < 1), the drag coefficient decreases with respect to the Stokes drag coefficient, whereas, at large Deborah number (De > 1), the drag is enhanced due to the large elastic stresses that develop on both the surface and wake of the sphere. These canonical behaviors, observed in the inertia-less flow regime (Re ≤ 1) are amplified as the polymer viscosity ratio approaches unity. At higher Reynolds numbers (Re > 1), the drag coefficient correction arising from viscoelasticity is found to be always bigger than unity, but smaller than the enhancement calculated in the creeping flow limit. In this regime, the formation of an axisymmetric recirculating eddy expands the wake region behind the sphere, and shifts the maximum elastic stresses away from the rear stagnation point towards the flow separation points. Motivated by the self-similarity observed at Re < 1, we propose an approximate model for the viscoelastic drag coefficient based on rational functions, using the lowest possible degree of polynomial functions required to fit the computational data with 95% accuracy, for De ≤ 5 and 0 < ζ < 1. This expression is a key step towards formulating a momentum-exchange model for the flow of dilute suspensions of solid particles in a viscoelastic matrix.

Extensional rheology of a dilute particle-laden viscoelastic solution

Numerical simulations of viscoelastic flow around an isolated particle find that, while the particle contribution to bulk elongational viscosity of a suspension increases at small strains, at larger strains the particle is shielded by stretched polymers, leading to a decrease in relative viscosity.

Macroscopic and mesoscopic simulation of viscoelastic free surface flow in gas‐assisted injection moulding process

AbstractIn this work, an improved simple coupled level‐set and volume of fluid (S‐CLSVOF) method is proposed to trace the moving interfaces in gas assisted injection moulding (GAIM) process based on the viscoelastic FENE‐P model. Firstly, the Kreisselmeier‐Steinhauser (KS) function constructed by means of Boolean operations is adopted as the shape level set (LS) function to precisely represent the complex mould cavities. Then, the benchmark problem of two‐dimensional deformation is used to verify the ability of the S‐CLSVOF method for capturing the moving interfaces. The stress birefringence is calculated and compared with the experiment result. Finally, the proposed method is further applied to the mould filling and gas penetration processes in GAIM. The macroscopic bubble appearance, temperature distribution, and the behaviour of the mesoscopic molecular orientation are shown and analyzed in detail. Due to the complexity of the gas‐liquid interaction, the phenomenon of asymmetrical gas flow is clearly observed in the gas penetration process. The influences of the macroscopic parameters on the macroscopic gas flows and mesoscopic molecular orientation are also discussed, such as insert positions, melt temperatures, and gas delay time. The numerical results illustrate that the coupled method can be applied to the multiscale numerical simulation of viscoelastic flows with complex free surfaces and provide significant guidance for the GAIM process.

Numerical simulation of viscoelastic flows during injection mold filling based on Rolie–Poly model

The feature of the flow front for polystyrene melts in injection molding is investigated in detail in this article. To explore the generation and evolution processes of “V”-shape pattern in the skin areas for polymer melt, a mathematical model based on the co-injection molding is proposed in which the evolution of material line is traced by level set method, and viscoelastic behavior of polystyrenes is described by Rolie–Poly constitutive equation; the molecular orientation, extension and deformation of polystyrenes are also studied in this article. The capabilities of Rolie–Poly constitutive equation is validated by simulating the principal stress difference(PSD) for both an entry-exit slit flow and a cross-slot geometry for polystyrenes melt; then the mold filling process has been analyzed by finite volume method to solve the continuity, momentum, constitutive equation together with level set equation. The numerical results for the evolution processes of “V”-shape pattern shows good agreement with available numerical results and experiments results obtained by injection machine. Furthermore, three main factors that include inlet velocity, molecular weight of polymer and temperature that affecting the formation and form of “V”-shape pattern were discussed, and molecular conformation of polystyrenes melt are also presented.

The fully-implicit log-conformation formulation and its application to three-dimensional flows

The stable and efficient numerical simulation of viscoelastic flows has been a constant struggle due to the High Weissenberg Number Problem. While the stability for macroscopic descriptions could be greatly enhanced by the log-conformation method as proposed by Fattal and Kupferman, the application of the efficient Newton–Raphson algorithm to the full monolithic system of governing equations, consisting of the log-conformation equations and the Navier–Stokes equations, has always posed a problem. In particular, it is the formulation of the constitutive equations by means of the spectral decomposition that hinders the application of further analytical tools. Therefore, up to now, a fully monolithic approach could only be achieved in two dimensions, as, e.g., recently shown by Knechtges et al. (2014).The aim of this paper is to find a generalization of the previously made considerations to three dimensions, such that a monolithic Newton–Raphson solver based on the log-conformation formulation can be implemented also in this case. The underlying idea is analogous to the two-dimensional case, to replace the eigenvalue decomposition in the constitutive equation by an analytically more “well-behaved” term and to rely on the eigenvalue decomposition only for the actual computation. Furthermore, in order to demonstrate the practicality of the proposed method, numerical results of the newly derived formulation are presented in the case of the sedimenting sphere and ellipsoid benchmarks for the Oldroyd-B and Giesekus models. It is found that the expected quadratic convergence of Newton’s method can be achieved.

Numerical simulation of viscoelastic flows using integral constitutive equations: A finite difference approach

This work presents a numerical technique for simulating incompressible, isothermal, viscoelastic flows of fluids governed by the upper-convected Maxwell (UCM) and K–BKZ (Kaye–Bernstein, Kearsley and Zapas) integral models. The numerical technique described herein is an extension of the GENSMAC method to the solution of the momentum and mass conservation equations to include integral constitutive equations. The governing equations are solved by the finite difference method on a staggered grid using a Marker-and-Cell approach. The Finger tensor B t ′ ( t ) is computed in an Eulerian framework using the ideas of the deformation fields method. However, improvements to the deformation fields method are introduced: the Finger tensor B t ′ ( x , t ) is obtained by a second-order accurate method and the stress tensor τ ( x , t ) is computed by a second-order quadrature formula. The numerical method presented in this work is validated by comparing the predictions of velocity and stress fields in two-dimensional fully-developed channel flow of a Maxwell fluid with the corresponding analytic solutions. Furthermore, the flow through a planar 4:1 contraction is investigated and the numerical results were compared with the corresponding experimental data. Finally, the UCM and the K–BKZ models were used to simulate the planar 4:1 contraction flow over a wide range of Reynolds and Weissenberg numbers and the numerical results obtained are in agreement with published data.

Numerical simulations of viscoelastic flows through one slot channel

In this article, the Modified Upper-Convected Maxwell equation (MUCM) is proposed. The viscoelastic polymer solution flow characteristics are described by the numerical method. The stream function contour, velocity contour and stress modulus contour of fluid in slot channel are drawn. The non-Newtonian power law property and viscoelasticity of MUCM fluid influence on the stream function are analyzed. The velocity contour move towards dead oil area with the viscoelasticity increase, flow area increase and the sweep area enlarges, so the sweep efficiency is enhanced.

602 数値シミュレーションによる多層フィルムの溶融押出プロセスの粘弾性流動解析(2)(OS6-1 非ニュートン流体の流動現象,OS6 非ニュートン流体の流動現象,オーガナイズドセッション)

602 数値シミュレーションによる多層フィルムの溶融押出プロセスの粘弾性流動解析(2)(OS6-1 非ニュートン流体の流動現象,OS6 非ニュートン流体の流動現象,オーガナイズドセッション)

602 数値シミュレーションによる多層フィルムの溶融押出プロセスの粘弾性流動解析(1)(OS6-1 非ニュートン流体の流動現象,OS6 非ニュートン流体の流動現象,オーガナイズドセッション)

602 数値シミュレーションによる多層フィルムの溶融押出プロセスの粘弾性流動解析(1)(OS6-1 非ニュートン流体の流動現象,OS6 非ニュートン流体の流動現象,オーガナイズドセッション)

Numerical simulation of viscoelastic flows with Oldroyd-B constitutive equations and novel algebraic stress models

The purpose of the present study is to compare numerical simulations of viscoelastic flows using the differential Oldroyd-B constitutive equations and two newly devised simplified algebraic explicit stress models (AES-models). The flows of a viscoelastic fluid in a 180 ° bent planar channel and in a 4:1 planar contraction are considered to illustrate and support the underlying theory. The flow in the bent channel is used to illustrate the frame-invariant property of the new models in a pure shear flow exhibiting strong streamline curvature. The flow in the 4:1 contraction serves as a benchmark test in a situation where strong elongation occurs. For both geometries, it is found that the predictions of the new AES-models are in good agreement with Oldroyd-B up to Deborah numbers of order 0.5, with a significant reduction in computational effort.

Three-dimensional numerical simulation of viscoelastic flow through an abrupt contraction geometry

AbstractA numerical scheme for simulating three-dimensional viscoelastic flow was developed. The three-dimensional finite volume method (FVM) based on a non-staggered grid and the semi-implicit method for pressure-linked equations (SIMPLE) were adopted to solve the continuity and momentum equations. As we used a non-staggered grid, the momentum interpolation method (MIM) was employed to avoid checkerboard type pressure fields. Viscoelastic properties of the fluid were described by the Phan-Thien and Tanner (PTT) model, which was treated by the compressive interface capturing scheme for arbitrary meshes (CICSAM). The elastic viscous split stress scheme (EVSS) was used to decouple the velocity and the stress fields. Algebraic equations obtained by the above schemes were handled by an iterative solver and a multi-grid method was applied to accelerate convergence. In order to verify the scheme developed in this study, Newtonian flow in a rectangular duct was studied and the resulting velocity fields were compared with the analytical flow fields. Then viscoelastic flow in 4:1 contraction geometry, one of the most frequently used benchmarking problems, was predicted by applying the fully three-dimensional finite volume method.

K-1212 円管押出しに関する粘弾性流動および伝熱の数値シミュレーション(S17-3 数値シミュレーション)(S17 非ニュートン流体の複雑流路内の流れ)

K-1212 円管押出しに関する粘弾性流動および伝熱の数値シミュレーション(S17-3 数値シミュレーション)(S17 非ニュートン流体の複雑流路内の流れ)

1301 有限体積法による粘弾性流体の三次元流動解析(GS-5 数値流体力学と乱流(1))

Three-dimensional numerical simulations of viscoelastic flows in a channel with a cavity are carried out using a finite volume method. The Phan-Thien Tanner model with the elastic viscous split stress (EVSS) formulation is used for the constitutive equation. The governing equations are discretized by a finite volume method for collocated grids. The calculation is carried out at various Weissenberg numbers, We. The results calculated show that at higher We, the main flow reaches deeper section of the cavity and the three-dimensional flow property appears more remarkably.

Numerical Simulation of Viscoelastic Flows through Periodic Undulating Channels.

Numerical Simulation of Viscoelastic Flows through Periodic Undulating Channels.

Three-dimensional numerical simulations of viscoelastic flows – predictability and accuracy

In this paper, fully three-dimensional (3-D) numerical simulations of viscoelastic flows using an implicit finite volume method are discussed with the focus on the predictability and accuracy of the method. The viscoelastic flow problems involving the stress singularity, including plane stick–slip flow, the flow past a junction in a channel, and the 3-D edge flow, are used to test the ability of the method to predict the singularity features with accuracy. The accuracy of the numerical predictions is judged by comparing with the known asymptotic behaviour for Newtonian fluids and some viscoelastic fluids, and the investigations are extended to the viscoelastic cases with unknown singular behaviour. The Phan-Thien–Tanner (PTT) model, and in some cases, the upper-convected Maxwell (UCM) model, are used to describe viscoelastic fluids. The numerical results with mesh refinement show that the accuracy is quite satisfactory, especially for Newtonian flows. For viscoelastic flows, the asymptotic results for the flow around a re-entrant corner for the UCM as well as the PTT fluid are reproduced numerically. In the stick–slip flow, a Newtonian-like asymptotic behaviour is predicted for the UCM fluid. In edge flow, it is verified numerically that the kinematics are Newtonian for viscoelastic fluids described by models with a constant viscosity and a zero second normal stress difference. For viscoelastic fluids described by the models with a shear-thinning viscosity and zero second normal stress difference, the fluid behaves like a power-law fluid, and the difference from its Newtonian kinematics is localized in the region near the singularity, and to capture the asymptotic behaviour, a parameter-dependent mesh has to be used. With the 3-D simulations, it is confirmed that in edge flow, the flow around the edge could not be rectilinear, and some secondary flows on the plane normal to the primary flow direction are expected for viscoelastic fluids described by the models with a shear-dependent second normal stress difference, such as the full PTT model. The strength of the secondary flows will depend on the level of the departure of the second normal stress difference from a fixed constant multiple of viscosity of the fluid.

A finite element method for the axisymmetric three-field Stokes system with a discontinuous pressure

A quadrilateral based velocity-pressure-extrastress tensor mixed finite element method for solving the three-field Stokes system in the axisymmetric case is studied. The method derived from Fortin's Q2 − P1 velocity-pressure element is to be used in connection with the standard Galerkin formulation. This makes it particularly suitable for the numerical simulation of viscoelastic flow. It is proven to be second-order convergent in the natural weighted Sobolev norms, for the system under consideration. The crucial result that the method is uniformly stable is proven for the case of rectangular meshes. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 739–763, 1999

Galerkin/least-square finite-element methods for steady viscoelastic flows

The elastic viscous split stress formulation (EVSS) and the discrete EVSS formulation (DEVSS) are effective in stabilizing numerical simulations of viscoelastic flows and have been widely used. Following the concept of Galerkin least-square perturbations proposed by Hughes et al. [Comput. Meth. Appl. Mech. Eng. 73 (1989) 173–189] and Franca et al. [SIAM J. Numer. Anal. 28(6) (1991) 1680–1697; Comput . Meth. Appl. Mech. Eng. 99 (1992) 209–233; Ibid. 104 (1993) 31–48] we are able to give the DEVSS formulation a new explanation as a perturbation to the Galerkin method based on the strain-rate residual, and furthermore, introduce another stabilized formulation, here named as MIX1, based on the incompressibility residual of the finite element discretizations. The three formulations (EVSS, DEVSS, MIX1), combined with a h– p type finite element algorithm that employs the SUPG technique to solve the viscoelastic constitutive equations are then tested on three benchmark problems: the flow of the upper-convected Maxwell fluid between eccentric cylinders, the flow of the Maxwell fluid around a sphere in a tube and the flow of the Maxwell and Oldroyd-B fluids around a cylinder in a channel. The results are checked with previous published works; good agreement is observed. Our numerical experiments convincingly demonstrate that the MIX1 is an accurate algorithm and convergent in terms of the p-extension, it has the same level of stability and robustness as the DEVSS method and is superior to the EVSS method in some respects. More important is that with MIX1 method one needs not solve for the strain-rate tensor as in EVSS and DEVSS methods, therefore, the CPU time consumption in the MIX1 method especially when using a coupled iteration scheme can be radically reduced. The success of the MIX1 method presents a challenge to the widely accepted concept of making the momentum equation explicitly elliptic.

Linear stability and dynamics of viscoelastic flows using time-dependent numerical simulations

Ultimately, numerical simulation of viscoelastic flows will prove most useful if the calculations can predict the details of steady-state processing conditions as well as the linear stability and non-linear dynamics of these states. We use finite element spatial discretization coupled with a semi-implicit θ-method for time integration to explore the linear and non-linear dynamics of two, two-dimensional viscoelastic flows: plane Couette flow and pressure-driven flow past a linear, periodic array of cylinders in a channel. For the upper convected Maxwell (UCM) fluid, the linear stability analysis for the plane Couette flow can be performed in closed form and the two most dangerous, although always stable, eigenvalues and eigenfunctions are known in closed form. The eigenfunctions are non-orthogonal in the usual inner product and hence, the linear dynamics are expected to exhibit non-normal (non-exponential) behavior at intermediate times. This is demonstrated by numerical integration and by the definition of a suitable growth function based on the eigenvalues and the eigenvectors. Transient growth of the disturbances at intermediate times is predicted by the analysis for the UCM fluid and is demonstrated in linear dynamical simulations for the Oldroyd-B model. Simulations for the fully non-linear equations show the amplification of this transient growth that is caused by non-linear coupling between the non-orthogonal eigenvectors. The finite element analysis of linear stability to two-dimensional disturbances is extended to the two-dimensional flow past a linear, periodic array of cylinders in a channel, where the steady-state motion itself is known only from numerical calculations. For a single cylinder or widely separated cylinders, the flow is stable for the range of Deborah number (De) accessible in the calculations. Moreover, the dependence of the most dangerous eigenvalue on De≡ λV/ R resembles its behavior in simple shear flow, as does the spatial structure of the associated eigenfunction. However, for closely spaced cylinders, an instability is predicted with the critical Deborah number De c scaling linearly with the dimensionless separation distance L between the cylinders, that is, the critical Deborah number De Lc ≡ λV/ L is shown to be an O(1) constant. The unstable eigenfunction appears as a family of two-dimensional vortices close to the channel wall which travel downstream. This instability is possibly caused by the interaction between a shear mode which approaches neutral stability for De ≫ 1 and the periodic modulation caused by the presence of the cylinders. Nonlinear time-dependent simulations show that this secondary flow eventually evolves into a stable limit cycle, indicative of a supercritical Hopf bifurcation from the steady base state.