Planar Contraction Flow Research Articles

High order asymptotic solution of planar contraction flow with a slip boundary condition

The creeping flow of an incompressible viscous fluid in a planar contraction channel (Jeffery-Hamel flow) is investigated using slip boundary conditions. A high order asymptotic solution is presented to describe the velocity field accurately. Results indicate that the effect of slip is strong in the region close to the sink, while it is almost negligible in the far field, and in some cases, multi-extrema and reverse values appear in the velocity profile near the sink.

Extensional response of the pom-pom model through planar contraction flows for branched polymer melts

One objective of this study is to assess the influence of the number of dangling arms at each end of the pom-pom molecule on flow characteristics in a 4:1 planar rounded-corner contraction, where varying the number of arms affects the level of entanglement in the system. For these complex flows, we evaluate the major influence of extensional viscosity and Trouton ratio when comparing kinetic-based as opposed to phenomenological network-based models. The stability of the proposed extended pom-pom model (XPP) is investigated for a range of Weissenberg numbers utilising a hybrid finite-element/volume scheme. For this scheme, we solve the momentum-continuity equations by a fractional-staged Taylor–Galerkin/pressure-correction procedure and invoke a cell-vertex fluctuation distribution finite volume scheme for the constitutive law. Distinction may be drawn between fluid response in the extension-hardening regime and those displaying a degree of softening. In particular, trends in salient-corner vortex-intensity response can be associated with the extensional properties of the extended version of the pom-pom model. Critical levels of solution elasticity attainable depend on precise extensional viscosity characteristics. The influence of various model parameters upon field response is described through rates of deformation, stress and stretch. This would include consideration for parameters governing anisotropy ( α ), and the ratio of stretch to orientation relaxation times ( ɛ ).

High-resolution finite element simulation of 4:1 planar contraction flow of viscoelastic fluid

In this work, we present high-resolution solutions for viscoelastic 4:1 planar contraction flow problems using a transient finite element method based on the fractional step method (FSM) and stabilization techniques (DEVSS-G/DG) with linear equal-order interpolation function. The Oldroyd-B model was used as the constitutive equation. A parallel multi-frontal algorithm was implemented to enhance computational speed and all solutions were obtained on a parallel machine. The vortex intensity and the re-attachment length of corner vortex show good mesh-convergent behavior and are compared with previous results from the literature. In particular, the present results are in good agreement with the predictions of the high-resolution finite volume method of Alves et al. [15]. This may be the first case that quantitative agreement is obtained between studies using different numerical methods for the benchmark problem of 4:1 planar contraction flow. As there has been little quantitative agreement in the previous investigations and only few simulation results with highly refined meshes exit, this study may well be regarded as accurate and meaningful in the sense that reasonable convergence is achieved for prediction of 4:1 planar contraction flow using transient finite element methods.

Three-dimensional numerical simulation of viscoelastic contraction flows using the Pom–Pom differential constitutive model

A three-dimensional finite element method is applied for analyzing the viscoelastic flow of branched LDPE through 3D planar contraction geometry. For describing the viscoelastic properties of the flowing material, a recently proposed eXtended Pom–Pom model is used. The main purpose of our study is to evaluate the ability of this model to capture some specific 3D effects in the case of converging entry flow of relatively high strain-hardening polyethylene melts. A special attention is paid to the vortex region. Like in our previous study [I. Sirakov, A. Ainser, J. Guillet, Three-dimensional numerical simulation of viscoelastic planar contraction flow, in: Proceedings of the Fourth International ESAFORM Conference, Liege, 23–25 April 2001] (where the Phan-Thien Tanner (PTT) model was used), the calculations predict an open vortex in which the material is involved in a complicated spiral motion, directed from the vertical plane of symmetry to the side wall of the contraction. Quantitative comparisons with experiments are done on the vortex size in the width direction (contraction z-axis).

Time-Weissenberg Number Superposition in 4:1 Planar Contraction Flow of a Viscoelastic Fluid

Vortex enhancement mechanism in 4:1 planar contraction flow of a viscoelastic fluid has been investigated, where the Oldroyd-B model was considered as a constitutive equation. Finite element method was used based on the fractional step method for time marching and DEVSS/DG as a stabilizing scheme. The prediction of vortex dynamics agreed well with the experimental results. At first, a lip vortex was generated at the reentrant corner at low Weissenberg number (We). As We increases, the size of the salient corner vortex becomes reduced, while the lip vortex becomes more pronounced and finally takes over the corner vortex. The trajectory of a lip vortex at high We was found to follow the steady state positions of the lip vortex at lower We, which means that the fluid has a memory and there exists a simple relationship between We and time. Accordingly, a simple universal behavior was found to dominate the vortex enhancement mechanism in the 4:1 planar contraction flow of the Oldroyd-B fluid.

Numerical simulations of the planar contraction flow for a polyethylene melt using the XPP model

The Discrete Elastic Viscous Stress Splitting Technique in combination with the Discontinuous Galerkin (DEVSS/DG) method is used to simulate a low density polyethylene melt flowing in a transient contraction flow problem. Numerical results using the original and a modified form of the eXtended Pom–Pom (XPP) model are compared to numerical results obtained with the exponential form of the Phan–Thien Tanner (PTT-a) and the Giesekus model and to experimental data of velocities and stresses. These models are known to be well capable of predicting all characteristic features encountered experimentally. Curiously, the eXtended Pom–Pom mode, formulated with the same non-affine or irreversible stretch dynamics as the original Pom–Pom model, encounters convergence problems using the DEVSS/DG method. A slight modification of the stretch dynamics such that it becomes consistent with other viscoelastic models and in agreement with a modification of the stretch dynamics based on non-equilibrium thermodynamics by Van Meerveld [J. Non-Newtonian Fluid Mech. 108 (2002) 291] gives a more numerical stable behaviour and steady state could be reached. From this it is clear that physical and numerical issues still play a mixed role in numerical viscoelastic flow problems.

Planar contraction flow with a slip boundary condition

Inertialess planar contraction flow with a Navier (linear) slip boundary condition has been studied for Newtonian and inelastic non-Newtonian fluids. For Newtonian fluids there is a region of curved streamlines as the flow adjusts from an upstream no-slip region with radial streamlines to a downstream slip-flow region that also has radial streamlines. Power-law fluids exhibit qualitatively different transverse flows for n>0.5 and n<0.5, with radial flow for n=0.5; it is the downstream flow that approaches no-slip for n<0.5, while the upstream approaches slip flow. Carreau–Yasuda fluids show a transition from Newtonian behavior upstream to power-law behavior downstream. The most interesting case is a Carreau–Yasuda fluid with n<0.5, where the azimuthal velocity changes direction, causing significant curvature in the streamlines. The streamline curvature may be relevant to the onset of entry flow instabilities at high stress levels in polymer processing.

Benchmark solutions for the flow of Oldroyd-B and PTT fluids in planar contractions

The paper presents very accurate numerical results for the vortex size, the vortex intensity and the Couette correction, in planar contraction flows of Oldroyd-B and PTT fluids ( ε=0.25) with both the linear and the exponential stress function, and with a solvent viscosity ratio equal to 1/9. The accuracy of these results is quantified, being generally below 1% (0.3% for most results), and the finest mesh employed had over 1 million degrees of freedom. Such degree of mesh fineness is shown to be required for accurate results with the Oldroyd-B fluid, especially at high Deborah numbers, but the shear-thinning PTT fluid in general does not require the finest meshes. In terms of level of elasticity, steady results for the PTT fluid could be obtained for values of the Deborah number in excess of 100 (linear PTT) and 10,000 (exponential PTT).

Highly elastic solutions for Oldroyd-B and Phan-Thien/Tanner fluids with a finite volume/element method: planar contraction flows

A hybrid finite volume/element method is analysed through the computation of creeping flows of viscoelastic fluids in plane 4:1 sharp and rounded-corner contraction geometries. Simulations are presented for three models: a constant viscosity Oldroyd-B fluid, and Phan-Thien/Tanner (PTT) shear thinning fluids of exponential and linear approximation form. A Taylor–Galerkin/pressure-correction scheme is implemented as the base time-stepping framework. The momentum equations are solved by a finite element method, whilst the constitutive equations are solved by a finite volume approach. Mesh convergence is analysed via refinement around the contraction to capture boundary layers and flow structure. Pressure drop is shown to increase with flow rate for a fixed fluid. For the Oldroyd-B model, singular behaviour is reported in the main stress component as one approaches the corner in the rounded, as with the sharp geometry. Velocity components display an asymptotic trend with a positive slope. Higher values of Weissenberg numbers ( We) are reached with these finite volume schemes compared to their finite element counterparts, attributing this to superior accuracy properties.

Finite volume multigrid method of the planar contraction flow of a viscoelastic fluid

This paper reports on a numerical algorithm for the steady flow of viscoelastic fluid. The conservative and constitutive equations are solved using the finite volume method (FVM) with a hybrid scheme for the velocities and first-order upwind approximation for the viscoelastic stress. A non-uniform staggered grid system is used. The iterative SIMPLE algorithm is employed to relax the coupled momentum and continuity equations. The non-linear algebraic equations over the flow domain are solved iteratively by the symmetrical coupled Gauss–Seidel (SCGS) method. In both, the full approximation storage (FAS) multigrid algorithm is used. An Oldroyd-B fluid model was selected for the calculation. Results are reported for planar 4:1 abrupt contraction at various Weissenberg numbers. The solutions are found to be stable and smooth. The solutions show that at high Weissenberg number the domain must be long enough. The convergence of the method has been verified with grid refinement. All the calculations have been performed on a PC equipped with a Pentium III processor at 550 MHz. Copyright © 2001 John Wiley & Sons, Ltd.

Finite volume multigrid method of the planar contraction flow of a viscoelastic fluid

AbstractThis paper reports on a numerical algorithm for the steady flow of viscoelastic fluid. The conservative and constitutive equations are solved using the finite volume method (FVM) with a hybrid scheme for the velocities and first‐order upwind approximation for the viscoelastic stress. A non‐uniform staggered grid system is used. The iterative SIMPLE algorithm is employed to relax the coupled momentum and continuity equations. The non‐linear algebraic equations over the flow domain are solved iteratively by the symmetrical coupled Gauss–Seidel (SCGS) method. In both, the full approximation storage (FAS) multigrid algorithm is used. An Oldroyd‐B fluid model was selected for the calculation. Results are reported for planar 4:1 abrupt contraction at various Weissenberg numbers. The solutions are found to be stable and smooth. The solutions show that at high Weissenberg number the domain must be long enough. The convergence of the method has been verified with grid refinement. All the calculations have been performed on a PC equipped with a Pentium III processor at 550 MHz. Copyright © 2001 John Wiley &amp; Sons, Ltd.

A cell-vertex finite volume/element method on triangles for abrupt contraction viscoelastic flows

A stable and accurate implementation is presented of a cell-vertex hybrid finite volume/element scheme based upon triangular meshes. This scheme is novel to this domain and is applied to the numerical solution of Oldroyd model fluids in abrupt planar contraction flows. All important has been the use of non-conservative flux representation, consistency in treatment of transients, fluxes and sources, and a recovery technique for velocity gradients. Linear stress representation, with non-recovered stress gradients, has proved crucial to widening stability thresholds. Solution smoothness at the higher levels of elasticity results, so that converged solutions are attainable. We have also highlighted the importance of incorporating a reduced-integration local discontinuity capturing technique for the re-entrant corner solution. A diminishing lip vortex with mesh refinement is reported. With increasing elasticity, lip vortex growth and a diminishing salient corner vortex is noted. In addition, a trailing-edge vortex is found to accompany the onset of a lip vortex. Agreement with analytical theory is observed through the asymptotic behaviour of velocities and stresses near the re-entrant corner, a region where the loss of evolution of the discrete system is investigated.

An adaptation of the separable KBKZ equation for comparable response in planar and axisymmetric flow

A specialisation of the ‘separable’ form of the KBKZ equation [M.H. Wagner, J. Non-Newtonian Fluid Mech. 4 (1978) 39–55] is presented that allows strain-hardening response to both planar and uniaxial elongation, while simultaneously giving shear-thinning behaviour. The specialisation expresses the strain-damping function in terms of a single invariant measure of strain. It is demonstrated that comparable planar and uniaxial elongational viscosities are predicted for IUPAC LDPE, with an over-prediction of the first normal stress difference in shear flow. It is shown that it is possible to fit transient data, for an LDPE melt that exhibits strain-hardening, for both planar and uniaxial elongational data simultaneously. It is shown that the new damping model gives vortex growth for simulations of both planar and axisymmetric contraction flows of LDPE, in contrast to a popular existing model that has been demonstrated to fail to predict vortex growth in planar contraction flow [P. Olley, R. Spares, P.D. Coates, J. Non-Newtonian Fluid Mech. 86 (1999) 337–357]. The finite-element based method used to find the flow solution is described including a novel method for implementing streamline tracking. Details are given of the iterative method used to compute flow fields, according to the stress field, and for the associated method to implement under-relaxation. Two new vortex measures are defined in order to quantify simulated vortices that can have an ‘hourglass’ shape (a shape that has been reported experimentally for LDPE contraction flows). To allow a broad assessment, comparisons are given as geometry type, contraction ratio, damping model, and the number of available strain-damping parameters are varied. Results for planar and axisymmetric contractions are compared, and contraction ratios of 8:1 and 4:1 are studied for planar flow. Results obtained using the new damping model are compared with well-established results obtained using an existing damping model for 4:1 axisymmetric contraction flow. The quantitative effects on flow simulations from using ‘single- β’ and ’multiple- β’ damping functions are assessed.

Upwind local differential quadrature method for solving incompressible viscous flow

The differential quadrature method (DQM) is able to obtain quite accurate numerical solutions of differential equations with few grid points and less computational effort. The successful applications of DQM have been reported to solve various problems of fluid mechanics. But, owing to the limitation of its quadrature rules, DQM is convenient only for regular regions and lacks upwind mechanism to characterize the convection of the fluid flow. In the present paper, a local differential quadrature method (LDQM) with upwind mechanism is proposed to solve incompressible viscous fluid flow problems in irregular regions. As its application, planar contraction flows in irregular regions is numerically solved and excellent numerical results are obtained for coarse meshes.

Numerical breakdown at high Weissenberg number in non-Newtonian contraction flows

Planar contraction flows of non-Newtonian fluids with integral constitutive models are studied to investigate the problem of numerical breakdown at high Weissenberg or Debrorah numbers. Spurious shear stress extrema are found on the wall downstream of the re-entrant corner for both sharp and rounded corners. Moreover, a non-monotonic relation between shear stress and strain rate is found when the Deborah number limit is approached, which correlates with these shear extrema. This strongly suggests that non-monotonicity between shear stress and strain rate may be responsible for the Deborah number limit problem in contraction flow simulations. This non-monotonicity is caused by the inaccuracy of the quadrature, using constitutive equations that do not have shear stress maxima when exactly evaluated. This conclusion agrees with recent analytical findings by others that inaccuracy of the integration along the streamlines – either by numerical integration or asymptotic approximation – makes the problem ill-conditioned, with spurious growth occurring on the wall downstream of the re-entrant corner.

Comparison of a quasi-Newtonian fluid with a viscoelastic fluid in planar contraction flow

The flow of a 5.0 wt.% solution of polyisobutylene in tetradecane through a planar 4 : 1 contraction exhibiting a shear thinning viscosity is simulated using the flow-type sensitive quasi-Newtonian fluid model. The shear viscosity is fitted by the Giesekus model, which, with the chosen parameters, leads to an extension thickening elongational viscosity. The stress and velocity fields of the numerical simulations are compared with the experimental results of Quinzani et al. [J. Non-Newtonian Fluid Mech. 52 (1994) 1–36] and the numerical results of the viscoelastic simulation using the Giesekus model of Azaiez et al. [J. Non-Newtonian Fluid Mech. 62 (1996) 253–277]. It can be shown that the quasi-Newtonian fluid qualitatively predicts the essential features of the flow in the vicinity of the contraction.

Numerical investigations of Lagrangian unsteady extensional flows of viscoelastic fluids in 3-D rectangular ducts with sudden contractions

A numerical study focusing on the macroscopic flow patterns and the Lagrangian unsteady extensional flow behaviour of a viscoelastic fluid in a rectangular duct with a sudden contraction is carried out using a three-dimensional (3-D) finite volume method (FVM). The flow geometry consists of an upstream duct with a square cross-section and a smaller downstream duct with a square or a rectangular cross-section, which corresponds to the so-called „square/square” (Sq/Sq) contraction and the planar contraction, respectively. The Phan-Thien Tanner (PTT) model, including the upper convected Maxwell (UCM) model as a special case, is adopted for the viscoelastic fluid description. Different vortex features observed experimentally in the two different types of contraction flows for the same fluid are predicted. It is found numerically that the macroscopic flow features are well correlated to the fluid‘s Lagrangian unsteady extensional behaviour in the Sq/Sq (or similarly in the axi-symmetric) contraction flow. In the planar contraction flow, however, the predicted Lagrangian unsteady extensional behaviour is different from its steady-state case, and no direct correlation can be made. The large transient extensional stress and the bounded transient extensional viscosity in the planar contraction flow may be responsible for the differences in the vortex patterns in the two contraction flows of viscoelastic fluid with a constant viscosity.

Three dimensional numerical simulations of viscoelastic flows through planar contractions

We present in this paper a fully three dimensional (3D) convergent numerical study of planar viscoelastic contraction flows. A 3D finite volume method (FVM) with the primary variable elastic viscous split stress (EVSS) formulation is employed, and a very efficient 3D block solver coupled with block correction is developed to speed up the convergence rate. Full 3D simulations of viscoelastic flows in 4:1 planar abrupt contractions are carried out using experimental conditions. Upstream vortex patterns comparable with the existing flow visualisation observations are captured using the upper convected Maxwell model for a Boger fluid and the Phan-Thien–Tanner model for a shear thinning fluid. Comprehensive comparisons between numerical simulation results and data measured in the dynamic fields in a 4:1 planar abrupt contraction are made, and the results indicate that the experimental measurements can be quantitatively reproduced if the fluid is well characterised by an appropriate viscoelastic model. It is confirmed numerically that the shear thinning of the fluid reduces the intensity of the singularity of viscoelastic flow near the re-entrant corner. With the Oldroyd-B model, by extensive computations on successively refined meshes with the minimum dimensionless size being 0.16–0.014 on the contraction plane in 2D configuration, it is revealed that, although the asymptotic flow behavior near the re-entrant corner and the build-up of the overall pressure and extensional stresses as well as the kinematic behavior along the centreline are insensitive to mesh refinement, completely different vortex activities may be predicted if the mesh is not sufficiently fine. It is verified numerically that, depending on the flow inertia and rheological properties of fluids, both the lip vortex mechanism and the corner vortex mechanism may be responsible for the vortex activities of viscoelastic fluids in 4:1 planar contraction flow, and the elasticity number E and Mach number M of the flow can be used to determine the vortex mechanism approximately. It is clear that the development process of the vortex activities could be underestimated with 2D simplification, and overpredicted with the creeping flow assumption, particularly when R e>0.5. Therefore, in planar contraction flow analyses, numerical artifacts may be produced with a coarse mesh, and 2D flow simulation is only a good approximation to the fully 3D flow if the upstream aspect ratio W/ H in the experiment is at least 10.

Entry and exit flows of casson fluids

AbstractEntry and exit flows through abrupt contractions are studied numerically for Casson fluids exhibiting a yield stress. The Casson constitutive equation recommended for describing blood flow is used with an appropriate modification proposed by Papanastasiou, which applies everywhere in the flow field in both yielded and practically unyielded regions. The emphasis is on determining the extent and shape of unyielded/yielded regions along with the swelling ratio of the free stream for planar and axisymmetric contraction flows for the whole range of Casson numbers. The results for pressure are used to determine the excess pressure losses that give rise to entrance, exit, and the total end correction.

Effect of ingredients on the rheological properties of extruded corn meal

A slit rheometer, mounted on a square cross section barrel extension, was attached to the exit of a single screw extruder to facilitate the simultaneous measurement of steady shear viscosity (ηsh), apparent planar extensional viscosity (ηpa), and the first normal stress differences (N1). The hole pressure method was used to determine N1 and the entrance pressure drop method with Cogswell’s analysis for abrupt planar contraction flow was used to determine ηpa. The effect of slot dimensions on N1 was investigated using low density polyethylene. Increasing the depth and width of the slot by 50% produced no significant change in the hole pressure. N1 predicted using the Higashitani–Pritchard–Baird (HPB) equation compared favorably with extrapolated N1 values obtained on a cone and plate rheometer. Addition of salt and sugar to corn grits affected the rheological properties of the melt, as observed by the changes in ηsh, N1, and ηpa. Of the three material functions, N1 and ηpa were more sensitive to the addition of ingredients. Corn grits exhibited thinning behavior in shear and planar extensional flows.