High Weissenberg Number Problem Research Articles

Numerical simulation of heat transfer enhancement by elastic turbulence in a curvy channel

Although many investigations on elastic turbulence have been conducted in recent years, two major research topics still call for in-depth mechanistic investigations. Specifically, one is heat transfer performance affected by elastic turbulence; the other is so-called high Weissenberg number problem (HWNP) in numerical simulation of viscoelastic fluid flow. Taking these two topics into account simultaneously, the coupled problem becomes heat transfer characteristic of viscoelastic fluid in elastic turbulence at high Weissenberg number (Wi) and very low Reynolds number (Re). In this work, we implement numerical simulations by embedding log-conformation reformulation algorithm into the open-source software OpenFOAM. The heat transfer process of viscoelastic fluid flow in a three-dimensional (3D) curvy channel is simulated over a wide range of Wi. For the first time, significant heat transfer enhancement induced by elastic turbulence in a curvy channel at high Wi was identified numerically. When Wi is above the critical value of O(1), the heat transfer performance is found to be dramatically improved by elastic turbulence and then approaches a saturation. From the transient analysis of flow motions in the axial and cross sections, it can be seen that the flow twists and wiggles in the curvy channel and the field synergy effect of viscoelastic fluid flow becomes more intensive than that of Newtonian fluid flow. These effects give rise to the extremely irregular flow motions in the cross section and consequently lead to heat transfer enhancement.

Vortex behavior of the Oldroyd-B fluid in the 4-1 planar contraction simulated with the streamfunction–log-conformation formulation

In this paper, we present numerical solutions of the Oldroyd-B fluid flowing through a 4:1 planar contraction, for Weissenberg numbers (Wi) up to 20. The incompressible viscoelastic flows are simulated with the streamfunction–log-conformation methodology. The log-conformation representation guarantees by construction the positive-definiteness of the conformation tensor, which circumvents the appearance of the high Weissenberg number problem. In addition, the streamfunction flow formulation removes the pressure variable from the governing equations and automatically satisfies the mass conservation. Thus, the streamfunction–log-conformation reformulation is beneficial for the accuracy and stability of the numerical algorithm. The resulting governing equations are solved with a high-resolution finite-volume method.Our numerical results for the reattachment length and the intensity of the recirculation vortices produced at the contraction plane are in excellent agreement with the benchmark solutions, available in the literature for Weissenberg numbers up to 3. For highly elastic flows, our results agree qualitatively well with the data of Afonso et al. (2011) [53]. Our simulations predict the reduction of the vortex size with increasing Wi, up to Wi≈5. Moreover, we observe a periodic third vortex growth and annihilation regime for Wi ≥ 15. The periodic vortex growth and annihilation is correlated with the accumulation of elastic strain in the cavity upstream of the contraction. This elastic instability is viewed as a mechanism that releases the elastic energy accumulated in the Oldroyd-B fluid at the fringe of the recirculation vortices. The dimensionless period of the third vortex annihilation appears to be independent on the Weissenberg number.

Numerical study of driven flows of shear thinning viscoelastic fluids in rectangular cavities

In this work, we present a numerical study of flow of shear thinning viscoelastic fluids in rectangular lid driven cavities for a wide range of aspect ratios (depth to width ratio) varying from 1/16 to 4. In particular, the effect of elasticity, inertia, model parameters and polymer concentration on flow features in rectangular driven cavity has been studied for two shear thinning viscoelastic fluids, namely, Giesekus and linear PTT. We perform numerical simulations using the symmetric square root representation of the conformation tensor to stabilize the numerical scheme against the high Weissenberg number problem. The variation in flow structures associated with merging and splitting of elongated vortices in shallow cavities and coalescence of corner eddies to yield a second primary vortex in deep cavities with respect to the variation in flow parameters is discussed. We discuss the effect of the dominant eigenvalues and the corresponding eigenvectors on the location of the primary eddy in the cavity. We also demonstrate, by performing numerical simulations for shallow and deep cavities, that where the Deborah number (based on convective time scale) characterizes the elastic behaviour of the fluid in deep cavities, Weissenberg number (based on shear rate) should be used for shallow cavities.

Visualization of polymer relaxation in viscoelastic turbulent micro-channel flow.

In micro-channels, the flow of viscous liquids e.g. water, is laminar due to the low Reynolds number in miniaturized dimensions. An aqueous solution becomes viscoelastic with a minute amount of polymer additives; its flow behavior can become drastically different and turbulent. However, the molecules are typically invisible. Here we have developed a novel visualization technique to examine the extension and relaxation of polymer molecules at high flow velocities in a viscoelastic turbulent flow. Using high speed videography to observe the fluorescein labeled molecules, we show that viscoelastic turbulence is caused by the sporadic, non-uniform release of energy by the polymer molecules. This developed technique allows the examination of a viscoelastic liquid at the molecular level, and demonstrates the inhomogeneity of viscoelastic liquids as a result of molecular aggregation. It paves the way for a deeper understanding of viscoelastic turbulence, and could provide some insights on the high Weissenberg number problem. In addition, the technique may serve as a useful tool for the investigations of polymer drag reduction.

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.

A front-tracking method for computational modeling of viscoelastic two-phase flow systems

A front-tracking method is developed for direct numerical simulations of viscoelastic two-phase systems in which one or both phases could be viscoelastic. One set of governing equations is written for the whole computational domain and different phases are treated as a single fluid with variable material and rheological properties. The interface is tracked explicitly using a Lagrangian grid while the flow equations are solved on a fixed Eulerian grid. The surface tension is computed at the interface using the Lagrangian grid and included into the momentum equations as a body force. The Oldroyd-B, FENE-CR and FENE-MCR models are employed to model the viscoelasticity. The viscoelastic model equations are solved fully coupled with the flow equations within the front-tracking framework. A fifth-order WENO scheme is used to approximate the convective terms in the viscoelastic model equations and second-order central differences are used for all other spatial derivatives. A log-conformation method is employed to alleviate the high Weissenberg number problem (HWNP) and found to be stable and very robust for a wide range of Weissenberg numbers. The method has been first validated for various benchmark single-phase and two-phase viscoelastic flow problems. Then it has been applied to study motion and deformation of viscoelastic two-phase systems in a pressure-driven flow through a capillary tube with a sudden contraction and expansion. The method has been demonstrated to be grid convergent with second-order spatial accuracy for all the cases considered in this paper.

Robust simulations of viscoelastic flows at high Weissenberg numbers with the streamfunction/log-conformation formulation

A new streamfunction/log-conformation formulation of incompressible viscoelastic flows is presented. The log-conformation representation guaranties the positive-definiteness of the conformation tensor and obviates the high Weissenberg number problem. The streamfunction is defined as a vector potential of the velocity field, and provides a pressureless formulation of the conservation laws, which automatically enforces the incompressibility. The resulting numerical method is free from velocity–pressure decoupling errors, and can achieve stable calculations for large Courant numbers, which improve the robustness and the efficiency of the solver. The two-dimensional flow of an Oldroyd-B fluid inside the lid-driven cavity is simulated for a large range of Weissenberg numbers. The numerical results demonstrate the second-order accuracy of our scheme, and our solutions are in good agreement with the available data from the literature for Weissenberg number 3 and below. Finally, the simulations at higher Weissenberg numbers 5 and 10 reveal a structural mechanism that sustains quasi-periodic elastic instabilities arising at the upstream corner of the moving lid.

Log-conformation representation of hiperbolic conservation laws with source term

The objective of this work is to study, through a simple equation, the statement that the numerical instability associated to the high Weissenberg number in equations with source term can be resolved by the use of the so called logarithmic conformation representation. We will focus on hyperbolic conservation laws, but more specifically on the advection equation with a source term. The source term imposes a necessity of an elastic balance, as well as the CFL convective balance for stability. Will be seen that the representation of such equation by the log-conformation removes the restriction of stability inherent to the elastic balance pointed out by Fattal & Kupferman [3] as the cause of the high Weissenberg number problem (HWNP).

Log-Conformation Representation of Hiperbolic Conservation Laws with Source Term

<pre>The objective of this work is to study, through a simpler equation, the statement that the numerical instability associated to the high number of Weissenberg in equations with source term can be resolved by the use of the so called logarithmic representation conformation. We will focus on hyperbolic conservation laws, but more specifically on the advection equation with source term. The source term imposes a necessity of an elastic balance, as well as the CFL convective balance for stability. We will see that the representation of such equation by log-conformation removes the restriction of stability inherent to the elastic balance pointed out by [3] as the cause of high Weissenberg number problem (HWNP).</pre>

Numerical simulation of polymer flows using non-conforming finite elements

In this paper, we are interested in the simulation of polymer flows for high-Weissenberg numbers. The high-Weissenberg number problem (HWNP) is one of the main difficulties encountered for the numerical simulation of such flows. We develop a numerical approach for two non-linear models: the affine Phan-Thien and Tanner model and the Giesekus model. We consider the 2D case and triangular and quadrilateral meshes. The velocity and the pressure are approximated by non-conforming finite elements while the stress tensor is approximated by P0 totally discontinuous finite elements. We have considered three popular test-cases: a simple channel, a 4:1 abrupt contraction and a cylinder. Comparisons with analytical solutions and experiences are performed, illustrating the good behavior of our code. Moreover, for the Oldroyd-B model, we have performed comparisons of drag values with data given in the literature. We have been able to obtain simulations for large values of Weissenberg number (Wi>21 for the 4:1 contraction), our approach gives a realistic description of polymer flows.

Application of the log-conformation tensor to three-dimensional time-dependent free surface flows

The numerical simulation of flows of highly elastic fluids has been the subject of intense research over the past decades with important industrial applications. Therefore, many efforts have been made to improve the convergence capabilities of the numerical methods employed to simulate viscoelastic fluid flows. An important contribution for the solution of the High-Weissenberg Number Problem has been presented by Fattal and Kupferman [J. Non-Newton. Fluid. Mech. 123 (2004) 281–285] who developed the matrix-logarithm of the conformation tensor technique, henceforth called log-conformation tensor. Its advantage is a better approximation of the large growth of the stress tensor that occur in some regions of the flow and it is doubly beneficial in that it ensures physically correct stress fields, allowing converged computations at high Weissenberg number flows. In this work we investigate the application of the log-conformation tensor to three-dimensional unsteady free surface flows. The log-conformation tensor formulation was applied to solve the Upper-Convected Maxwell (UCM) constitutive equation while the momentum equation was solved using a finite difference Marker-and-Cell type method. The resulting developed code is validated by comparing the log-conformation results with the analytic solution for fully developed pipe flows. To illustrate the stability of the log-conformation tensor approach in solving three-dimensional free surface flows, results from the simulation of the extrudate swell and jet buckling phenomena of UCM fluids at high Weissenberg numbers are presented.

The kernel-conformation constitutive laws

This work presents a generic kernel-conformation tensor transformation for a large class of differential constitutive models, in which several matrix kernel-transformation families can be applied to the conformation tensor evolution equation. The discretization of this generic analytical framework generates numerical results consistent with the standard conformation tensor discretization at low Weissenberg numbers ( Wi). The numerical efficiency of High Weissenberg Number Problems is ultimately related with both the specific kernel function used in the matrix transformation, but is also related to the existence of mathematical singularities introduced either by the physical description of the flow or by the characteristics of the adopted constitutive equation. We also show that the log-conformation [R. Fattal, R. Kupferman, J. Non-Newtonian Fluid Mech., 123(2–3) (2004) 281–285] and the square-root-conformation [N. Balci, B. Thomases, M. Renardy, C.R. Doering, J. Non-Newtonian Fluid Mech., 166(11) (2011) 546–553] approaches are two relevant particular cases of the kernel-conformation tensor transformation. Simulations for the benchmark confined cylinder flow of an Oldroyd-B fluid provide a preliminary assessment of the merits of some tested transformation functions.

Continuum and multi-scale simulation of mixed kinematics polymeric flows with stagnation points: Closure approximation and the high Weissenberg number problem

Abstract It is a well known fact that the upper We limit encountered in continuum level viscoelastic flow simulations with typical constitutive equations for dilute polymeric solutions that predict bounded extensional viscosities in geometries with internal stagnation points on solid surfaces is a strong function of the strain hardening nature of the fluid. Specifically, the upper We limit decreases as the level of strain hardening of the fluid is increased. To provide further insight into the high We limitations in continuum level computations of this class of flows, we have performed extensive continuum and multiscale flow simulations in two benchmark flow problems, namely sedimentation of sphere in a tube and flow past a cylinder in a channel, utilizing the FENE-P (continuum and Brownian Configuration Fields (BCF)) and Giesekus (continuum) constitutive equations as well as the FENE (BCF) dumbbell micromechanical model. Extremely large stress gradients in the axial normal stress along the plane of symmetry in the wake of the cylinder and sphere are observed in FENE-P and Giesekus predictions using both multiscale and continuum numerical techniques at We of O(1) where the numerical simulations begin to breakdown for significantly strain hardening fluids, i.e., b > 300, α < 0.005. The existence of very large localized polymeric stresses and stress gradients is shown to be the consequence of significant over prediction of macromolecular extension by the closed form constitutive equations in the extensionally dominated region of the flow. The inability of the aforementioned constitutive equations to accurately describe the flow microstructure coupling downstream of stagnation points in complex kinematics flows should motivate use of regularization techniques for the polymeric stress or utilization of more sophisticated constitutive equations in flows with strong straining components.

Viscoelastic flow analysis using the software OpenFOAM and differential constitutive equations

Abstract Viscoelastic fluids are of great importance in many industrial sectors, such as in food and synthetic polymers industries. The rheological response of viscoelastic fluids is quite complex, including combination of viscous and elastic effects and non-linear phenomena. This work presents a numerical methodology based on the split-stress tensor approach and the concept of equilibrium stress tensor to treat high Weissenberg number problems using any differential constitutive equations. The proposed methodology was implemented in a new computational fluid dynamics (CFD) tool and consists of a viscoelastic fluid module included in the OpenFOAM, a flexible open source CFD package. Oldroyd-B/UCM, Giesekus, Phan-Thien–Tanner (PTT), Finitely Extensible Nonlinear Elastic (FENE-P and FENE-CR), and Pom–Pom based constitutive equations were implemented, in single and multimode forms. The proposed methodology was evaluated by comparing its predictions with experimental and numerical data from the literature for the analysis of a planar 4:1 contraction flow, showing to be stable and efficient.

The log-conformation tensor approach in the finite-volume method framework

The log-conformation formulation, proposed by Fattal and Kupferman [J. Non-Newt. Fluid Mech. 123 (2004) 281], has helped to provide further insights into the High-Weissenberg Number Problem. In this work, we investigate the performance of the log-conformation formulation in the Finite Volume Method (FVM) framework for creeping flows of viscoelastic fluids in steady and unsteady flows around a confined cylinder. The Oldroyd-B and Phan-Thien–Tanner (PTT) constitutive equations were used to assess the effect of different rheological behaviour on the flow patterns and solution stability. The calculation of the polymer stress contribution is carried out with both the standard technique and with the log-conformation methodology. For all test cases, up to the critical conditions when both methods converge to a steady solution, the use of the log-conformation technique provides solutions with similar accuracy as the standard approach. In terms of stability the log-conformation formulation is found to be significantly more robust, and solutions could be obtained at higher Deborah number flows.

Computation of viscoelastic fluid flows using continuation methods

The numerical simulation of viscoelastic fluid flow becomes more difficult as a physical parameter, the Weissenberg number, increases. Specifically, at a Weissenberg number larger than a critical value, the iterative nonlinear solver fails to converge, a phenomenon known as the high Weissenberg number problem. In this work we describe the application and implementation of continuation methods to the nonlinear Johnson–Segalman model for steady-state viscoelastic flows. Simple, natural, and pseudo-arclength continuation approaches in Weissenberg number are investigated for a discontinuous Galerkin finite element discretization of the equations. Computations are performed for a benchmark contraction flow and, several aspects of the performance of the continuation methods including high Weissenberg number limits, are discussed.

Toward a solution to the high Weissenberg number problem

AbstractWe present results of a numerical study of high Weissenberg number flows using the method of Trebotich, Colella and Miller, J. Comput. Phys. vol. 205 (2005). We consider benchmark viscoelastic flow of an Oldroyd‐B fluid in four to one sudden contractions for We = 1, 10, Re = 1. We conclude that (i) underlying wave behavior in the solution for flows at critical and supercritical elastic Mach number prevents the solution from reaching a steady‐state; and (ii) high resolution is needed to resolve structure in the solution downstream from the re‐entrant corner in order to determine convergence. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

A simple method for simulating general viscoelastic fluid flows with an alternate log-conformation formulation

The log-conformation formulation has alleviated the long-standing high Weissenberg number problem associated with the viscoelastic fluid flows [R. Fattal, R. Kupferman, Constitutive laws for the matrix-logarithm of the conformation tensor, J. Non-Newtonian Fluid Mech. 123 (2004) 281–285]. This formulation ensures that solutions of viscoelastic flow problems are physically admissible, and it is able to capture sharp elastic stress layers. However, the implementations presented in literature thus far require changing the evolution equation for the conformation tensor into an equation for its logarithm, and are based on loosely coupled (partitioned) solution procedures [M.A. Hulsen, et al., Flow of viscoelastic fluids past a cylinder at high Weissenberg number: stabilized simulations using matrix logarithms, J. Non-Newtonian Fluid Mech. 127 (2005) 27–39]. A simple alternate form of the log-conformation formulation is presented in this article, and an implementation is demonstrated in the DEVSS-TG/SUPG finite element method. Besides its straightforward implementation, the new log-conformation formulation can be used to solve all the governing equations (continuity, conservation of momentum and constitutive equation) in a strongly coupled way by Newton’s method. The method can be applied to any conformation tensor model. The flows of Oldroyd-B and Larson-type fluids are tested in the benchmark problem of a flow past a cylinder in a channel. The accuracy of the method is assessed by comparing solutions with published results. The benefits of this new implementation and the pending issues are discussed.

Re-entrant corner flows of upper convected Maxwell fluids: the small and high Weissenberg number limits

We discuss here the steady planar flow of the upper convected Maxwell fluid at re-entrant corners in the singular limits of small and large Weissenberg number. The Weissenberg number is a parameter representing the dimensionless relaxation time and hence the elasticity of the fluid. Its value determines the strength of the fluid memory and thus the influence of elastic effects over viscosity. The small Weissenberg limit is that in which the elastic effects are small and the fluid's memory is weak. It is an extremely singular limit in which the behaviour of a Newtonian fluid is obtained in a main core region away from the corner and walls. Elastic effects are confined to boundary layers at the walls and core regions nearer to the corner. The actual asymptotic structure comprises a complicated four-region structure. The other limit of interest is the large Weissenberg limit (or high Weissenberg number problem) in which the elastic effects now dominate in the main regions of the flow. We explain how the transition in solution from Weissenberg order 1 flows to high Weissenberg flows is achieved, with the singularity in the stress field at the corner remaining the same but its effects now extending over larger length-scales. Implicit in this analysis is the absence of a lip vortex. We also show (for the main core region) that there is a small reduction in the velocity field at the corner and walls where it becomes smoother. This high Weissenberg number limit has a six-region local asymptotic structure and comment is made on its relevance to the case in which a lip vortex is present.

New formulations, positivity preserving discretizations and stability analysis for non-Newtonian flow models

We propose a class of new discretization schemes for solving the rate-type non-Newtonian constitutive equations. The so-called conformation tensor has been known to be symmetric and positive definite in a large class of constitutive equations. Preserving such a positivity property on the discrete level is believed to be crucially important but difficult. High Weissenberg number problems on numerical instabilities have been often associated with this issue. In this paper, we present various discretization schemes that preserve the positive-definiteness of the conformation tensor regardless of the time and spatial resolutions. Moreover, the robustness of the algorithm has been also demonstrated by the stability analysis using the discrete analogue of energy estimates. New schemes presented in this paper are constructed based upon the newly discovered relationship between the rate-type constitutive equations and the symmetric matrix Riccati differential equations.