Numerical Simulation Of Viscoelastic Flow Research Articles

Numerical Simulation of Viscoelastic Flow in Three-Dimensional Rectangular Abrupt Contraction Channels

Numerical simulation of viscoelastic flow in two three-dimensional (3-D) rectangular abrupt contraction channels was carried out using the Giesekus model. The HSMAC (Highly Simplified Marker and Cell) scheme was applied to solve the discretized basic equations. 3-D flow patterns and the distribution of vortex size were investigated to elucidate 3-D flow property. Several 3-D flow phenomena such as a spiral flow were predicted. The effect of wall was strong when the aspect ratio of the channel (width/height) was small, and the 3-D flow behavior was notable near the channel wall: the vortex size drastically changed near the wall.

A stable finite element method for the axisymmetric three-field Stokes system

A rectangular based velocity-pressure-extra-stress tensor mixed finite element method for solving the three-field Stokes system in the axisymmetric case is studied. The method is to be used in connection with the standard Galerkin formulation, which 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.

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.

Numerical Simulation of Viscoelastic Flow due to a Rotating Disc Enclosed in a Cylindrical Casing with Large Axial Clearance.

Numerical Simulation of Viscoelastic Flow due to a Rotating Disc Enclosed in a Cylindrical Casing with Large Axial Clearance.

Numerical Calculation of Viscoelastic Flows through Eccentric Abrupt Contraction.

Numerical simulations of viscoelastic flows through an eccentric four-to-one abrupt contraction are carried out using the Giesekus model. The SMAC (Simplified-Marker-and-Cell) method is used to analyze the three-dimensional flows. The velocity profiles along the path line passing through the center of the exit exhibit an overshoot near the entry section, and at high Weissenberg numbers an undershoot follows the overshoot. The magnitude of the stress along the same path line has a peak near the entry section, and its slow relaxation process indicates that a large downstream length is necessary for fully developed stress conditions to exist. The peak is lower than that for the flow through the concentric four-to-one abrupt contraction ; the decrease in the peak amplitude is understood to be due to the distortion of the path line in the eccentric geometry. A corner vortex, the height of which is a maximum at the widest corner, grows as the Weissenberg number increases. Furthermore, the tangential flow toward the widest section inside the vortex is determined.

Effect of constitutive models on the numerical simulation of viscoelastic flow at an entry region

AbstractThe poor solutions of numerical simulations at a contraction has often been attributed to the numerical scheme used. However, the solution of a numerical simulation is also highly dependent on the constitutive equation. A study of various constitutive equations on the simulation of the 4:1 contraction flow is made in this paper, and their effects on the results analyzed. The constitutive models considered include the Upper Convected Maxwell model, the Oldroyd‐B model, the White‐Metzner model, the Phan‐Thien‐Tanner model and the Giesekus‐Leonov model. It was found that although the Phan‐Thien‐Tanner model gave the best results, the solution at higher shear rates were still not satisfactory.

An adaptive viscoelastic stress splitting scheme and its applications: AVSS/SI and AVSS/SUPG

We report an adaptive viscoelastic stress splitting (AVSS) scheme, which was found to be extremely robust in the numerical simulation of viscoelastic flow involving steep stress boundary layers. The scheme is different from the elastic viscous split stress (EVSS) in that the local Newtonian component is allowed to depend adaptively on the magnitude of the local elastic stress. Two decoupled versions of the scheme were implemented for the Upper Convected Maxwell (UCM) model: the streamline integration (AVSS/SI), and the streamline upwind Petrov-Galerkin (AVSS/SUPG) methods of integrating the stress. The implementations were benchmarked against the known analytic Poiseuille solution, and no upper limiting Weissenberg number was found (the computation was stopped at Weissenberg number of O(10 4)). The flow past a sphere in a tube was solved next, giving convergent results up to a Weissenberg number of 3.2 with the AVSS/SI and 1.55 with the AVSS/SUPG (both were decoupled schemes; without the adaptive scheme, the limiting Weissenberg number for the decoupled streamline integration method was about 0.3). These results show that the decoupled AVSS is more stable than the corresponding EVSS, and the SI is more robust than SUPG in solving the constitutive equation of hyperbolic type. In addition, we found a very long stress wake behind the sphere, and a weak vortex in the rear stagnation region at a Weissenberg number above W i of about 1.6, which does not seem to increase in size or strength with increasing W i .

Numerical simulation of viscoelastic flow using flux difference splitting at moderate Reynolds numbers

An efficient numerical method for solving the Navier-Stokes equations is extended to allow solution of the equations of the Maxwell and Oldroyd B models of viscoelasticity at moderate Reynolds numbers where inertial terms cannot be ignored. The continuity equation is treated by the artificial compressibility method and the resulting set of equations is split into a hyperbolic set and one that contributes to the mixed character of the system. A flux difference splitting scheme is used to approximate the hyperbolic set and an implicit method is used to time march to steady state. The method is used to solve the channel entry problem. A grid refinement study is done in the entry region of the channel and the solution is matched to a downstream region solution. The latter is compared with the prediction of linear perturbation analysis. Very good agreement is found for the rate of development. Flow over a flat plate is studied and leading edge solutions are qualitatively compared to linear solutions, while the downstream solutions compare well quantitatively with predictions of boundary layer analysis.

Numerical simulation of viscoelastic flows through a planar contraction

In this study, three nonlinear rheological models consisting of the Giesekus, the FENE-P, and the Phan-Thien-Tanner model are used to simulate the flow of a viscoelastic fluid through a planar 4:1 contraction. Both stress and velocity fields are examined at different sections of the flow and the predictions of the numerical simulations are compared with the experimental results of L.M. Quinzani, R.C. Armstrong adn R.A. Brown, J. Non-Newtonian Fluid Mech., 52 (1994) 1–36. Overall, the numerical simulations allow a description of the essential features of the flow, and reproduce much of the experimental results with good accuracy. Excellent qualitative agreement between the numerical results and the experimental observations is reported. However, the agreement remains semi-quantitative especially for the first normal stress difference around the entry section of the flow. This is to be expected in view that the simulations were limited to only one-mode models.

Numerical simulation of viscoelastic flow around a cylinder using an integral constitutive equation

Numerical simulations have been undertaken for the flow of a 5 wt. % polyisobutylene (PIB) solution in tetradecane (C14) around a cylinder placed between parallel plates. The PIB solution has been the subject of an extensive rheological characterization and the flow geometry corresponds to the one used in a previous study for laser Doppler anemometry and flow-induced birefringence measurements. The constitutive equation used is an integral-type K-BKZ model with a relaxation spectrum, which fits well experimental data for the shear and elongational viscosities and the normal stresses as measured in shear flow. Stable numerical solutions have been obtained and used to compare the K-BKZ model predictions with the experimental data reported recently by Baaijens. The velocity and shear stress values predicted by the simulation are in excellent agreement with the experimental ones. Very good agreement is also obtained for the predictions of the first normal stress difference both before and after the cylinder, in sharp contrast with previous experimental results, which had shown a much slower relaxation of the stresses. The drag force exerted by the fluid on the cylinder has been calculated and reported to be a decreasing function of the Deborah number De.

Time-dependent numerical simulation of viscoelastic flow and stability

The 4×4 mixed finite-element method is extended in order to calculate time-dependent viscoelastic flow. The basic algorithm uses a fully implicit technique with a predictor-corrector control of the time step for monitoring the accuracy. Excellent agreement is found with analytical test cases. Several decoupled schemes are also developed with a view to reducing the cost of the time-dependent calculations. However, none of them reaches that goal on actual flow problems because of a drastic reduction of the time step. The time-dependent algorithm is used for verifying the stability of steady-state viscoelastic flow solutions. The approach consists of using a steady-state solution as a set of initial conditions for a time-dependent algorithm with the hope that, in case of instability, the system will lead to a stable solution. More precisely, we consider the flow of an Oldroyd-B fluid through a four-to-one contraction. Starting from a steady-state flow, we impose a pressure impulse in the entry section and calculate the approach toward another steady-state. For planar contractions, we do not constrain the flow to be symmetric while, for circular contractions, the flow is endowed with swirling capabilities. It is found that the stable or unstable character of the flow depends upon the mesh as well as upon the method of discretization.

Numerical simulation of contraction flow for viscoelastic fluids at high Weissenberg number by the streamline‐upwind finite element method

Numerical simulation of viscoelastic flow at high Weissenberg number (We) was carried out by the streamline‐upwind finite element method with the subelements for stress components proposed by Marchal and Crochet. This method and the Galerkin finite element method were applied to the stick‐slip flow with the singular point in order to examine the effectiveness of this method. The Oldroyd‐B model was used as a constitutive equation. This model incorporates strong elasticity and numerical solutions sometimes diverge. When the Galerkin finite element method was used, the numerical solutions had oscillation and the loss of convergence occurred at a relatively low value of We. On the other hand, when the streamline‐upwind method was used with the stress subelements, the oscillation of numerical solutions did not appear and the solution could be obtained up to a high value of We. Using the latter method, the calculation of the tapered contraction flow was carried out with the Giesekus model as a constitutive equation. The limit values of We were not encountered and we could calculate even at We≳100 using various values of model parameters. It was concluded that the streamline‐upwind finite element method with the stress subelements was very useful to simulate the contraction flow up to high values of We.

Numerical simulation of viscoelastic flow in two‐dimensional L‐shaped channels

Numerical simulation of the flow of a White–Metzner type liquid in two‐dimensional L‐shaped channels has been carried out using the finite difference technique. Results are compared with experimental data obtained in previous work. Numerical predictions are in agreement with the experiment for velocity and pressure distributions. Further considerations are given to streamlines and stress component profiles. A streamline shift toward the outer wall accompanied by reverse flow occurs in the upstream vicinity of the re‐entrant corner. It is shown that the fluid is subjected to stretching near the re‐entrant corner, where reversal of flow occurs. Effects of inertia and elasticity are addressed by presenting the size of the reverse flow region. Elasticity has the effect of increasing its size, which is opposite to an inertial effect.

Numerical Simulation of Contraction Flow for Viscoelastic Fluids at High Weissenberg Number by the Streamline-Upwind Finite Element Method

Numerical simulation of viscoelastic flow at high Weissenberg number (We) was carried out by the streamline-upwind finite element method with the sub-elements for stress components proposed by Marchal and Crochet. This method and the Galerkin finite element method were applied to the stick-slip flow with the singular point in order to examine the effectiveness of this method. The Oldroyd-B model was used as a constitutive equation. This model incorporates strong elasticity and numerical solutions sometimes diverge. When the Galerkin finite element method was used, the numerical solutions had oscillation and the loss of convergence occurred at relatively low value of We. On the other hand, when the streamline-upwind method was used with the stress sub-elements, the oscillation of numerical solutions did not appear and the solution could be obtained up to a high value ofWe. Using the latter method, the calculation of the tapered contraction flow was carried out with the Giesekus model as a constitutive equation. The limit values of We were not encountered and we could calculate even at We> 100 using various values of model parameters. It was concluded that the streamline-upwind finite element method with the stress sub-elements was very useful to simulate the contraction flow up to high values of We.

Numerical simulation of viscoelastic flow past a cylinder

The flow of an upper-convected Maxwell fluid past a circular cylinder is simulated numerically using the algorithn SIMPLER, which is based on a finite volume discretization on a staggered grid of the governing equations and an iterative solution to the nonlinearly coupled equations. The effect of the viscoelasticity of the fluid on the flow is examined. The drag force on the cylinder and the heat transfer from the cylinder to the surrounding fluid are calculated and compared with those obtained in experiments. One feature seen in the experiments is the existence of a critical velocity across which the variation of the drag and heat transfer changes from a characteristic Newtonian variation to a flat response. This feature is interpreted as a change of type when the critical speed U c of flow becomes greater than the material wave speed c = ▪. The numerical computation is completely consistent with this interpretation. Using an Einstein-type formula for the relaxation and λ = Aø with A independent of ø we find that the experiments follow a scaling law U cø 1 2 =constant, where ø is the mass fraction of polymers in water in the regime of extreme dilution, the drag reduction range, consistent with the interpretation that U c = c.

Numerical simulation of viscoelastic flow through a sudden contraction using a type dependent difference method

Numerical simulation of the flow of upper convected Maxwell fluid through a planar 4:1 contraction has been performed using type dependent difference approximation of the vorticity equation. For creeping flow assumption, the numerical convergence has been achieved up to much higher values of the elasticity parameter than those obtained by conventional finite difference methods. For non-vanishing Reynolds number flow, it is shown that the corner vortices disappear, which is in good qualitative agreement with existing experimental results. In doing so, spatial distributions of stream function, vorticity and stresses are considered in relation to a change of type of vorticity.

Numerical simulations of viscoelastic flow: the effect of mesh size

This paper examines the effects of mesh size and configuration in the numerical simulation of both planar and axisymmetric 4:1 contraction flows. A selection of constitutive models are employed in an attempt to predict certain vortex characteristics observed experimentally by others [15,16] for constant viscosity and shear-thinning viscoelastic liquids. Both finite difference and finite element methods are used, and the dependence of the breakdown of algorithmic convergence on mesh and the mode of stress decomposition is noted.