Computing viscoelastic fluid flow problems at low cost

doi:10.1016/0377-0257(92)85004-G

Journal of Non-Newtonian Fluid Mechanics

Volume 45, Issue 2, November 1992, Pages 209-229

https://doi.org/10.1016/0377-0257(92)85004-G Get rights and content

Abstract

A numerical method for solving viscoelastic fluid flow problems is presented. Based on the Lesaint-Raviart method, it allows computations at moderate values of the Weissenberg number in a very efficient manner. The components of the stress tensor are computed on an element-by-element basis, reducing substantially the storage requirements with respect to other methods.

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Cited by (28)

Metric-based anisotropic mesh adaptation for viscoelastic flows
2023, Computers and Mathematics with Applications
The computation of flows of viscoelastic fluids is expensive compared to standard fluids. Besides degrees-of-freedom for velocity and pressure, the viscoelastic stresses introduce an additional unknown to the system. While adaptive meshing techniques are commonly used for simulations of standard fluids, their application is seldom explored in the context of viscoelastic fluids. In this work, we apply an anisotropic mesh adaptation algorithm to solve the viscoelastic flow problem. Our flow solver is based on a logarithmic reformulation of the viscoelastic constitutive laws, which greatly reduces the stringent stability requirements on the mesh size and allows adaptive meshing in this context. For discretization, we use a stabilized finite element method. We present numerical results for a flow past a cylinder. The adaptive method produces near mesh-independent results up to Weissenberg number $Wi = 0.7$ . Our results indicate that with an anisotropic refinement strategy, fewer mesh elements are needed to maintain the same error level as with an isotropic refinement strategy. In a stick-slip transition flow, we demonstrate the capability of the adaptive method to produce optimal meshes even in the presence of a singularity in the solution.
Testing viscoelastic numerical schemes using the Oldroyd-B fluid in Newtonian kinematics
2020, Applied Mathematics and Computation
Citation Excerpt :
It is worth noting that the assumption of fixed Newtonian kinematics for Oldroyd-B fluids has previously been applied for asymptotic and computational studies of other benchmark problems, for instance, the flow of a viscoelastic fluid around a cylinder [22,23]. It has also been used in early numerical work for the stick-slip problem [24,25], before the asymptotic classification of the problem. Further, for other viscoelastic models, such as Phan-Thien–Tanner (PTT) [26,27] and Giesekus [28], a Newtonian velocity field predominates at stress singularities for sharp corners [29–31] and stick-slip [32,33].
We focus here on using a Newtonian velocity field to evaluate numerical schemes for two different formulations of viscoelastic flow. The two distinct formulations we consider, correspond to either using a fixed basis for the elastic stress or one that uses the flow directions or streamlines. The former is the traditional Cartesian stress formulation, whilst the later may be referred to as the natural stress formulation of the equations. We choose the Oldroyd-B fluid and three benchmarks in computational rheology: the 4:1 contraction flow, the stick-slip and cross-slot problems. In the context of the contraction flow, fixing the kinematics as Newtonian, actually gives a larger stress singularity at the re-entrant corner, the matched asymptotics of which are presented here. Numerical results for temporal and spatial convergence of the two formulations are compared first in this decoupled velocity and elastic stress situation, to assess the performance of the two approaches. This may be regarded as an intermediate test case before proceeding to the far more difficult fully coupled velocity and stress situation. We also present comparison results between numerics and asymptotics for the stick-slip problem. Finally, the natural stress formulation is used to investigate the cross-slot problem, again in a Newtonian velocity field.
Stick-slip singularity of the Giesekus fluid
2015, Journal of Non-Newtonian Fluid Mechanics
Citation Excerpt :
For viscoelastic fluids, there is a paucity of analytical results and the question of well-posedness for these problems is an open issue. Further, numerical simulation tends to be problematic, see for example Lipscombe et al. [20] and Fortin et al. [10] for difficulties encountered in earlier numerical work. This has been attributed to the highly singular stresses encountered.
The local asymptotic behaviour at the stick-slip singularity is determined for the Giesekus fluid in the presence of a solvent viscosity. In planar steady flow, the method of matched asymptotic expansions is used to show that it comprises a three region structure. Specifically, an outer or core region that links boundary layers at the rigid stick and free slip surfaces. In the outer region, the velocity field is shown to be Newtonian at leading order, with solvent stresses dominating the polymer stresses. In terms of the radial distance r from the singularity at the join of the stick and slip surfaces, the velocity field vanishes as $O (r^{\frac{1}{2}})$ . Consequently, the singular velocity gradients and solvent stresses are of $O (r^{- \frac{1}{2}})$ with the less singular polymer stresses being shown to be $O (r^{- \frac{5}{16}})$ . The solvent and polymer stresses become comparable near the rigid stick and free slip surfaces, where boundary layers are required. These are of thickness $O (r^{\frac{5}{4}})$ at the rigid stick surface and thickness $O (r^{\frac{17}{14}})$ at the free slip surface. Solutions are constructed for both stick-slip and slip-stick flow regimes. These asymptotic results do not hold for the Oldroyd-B model nor for the case when the solvent viscosity is absent.
How accurate is your solution? Error indicators for viscoelastic flow calculations
2000, Journal of Non-Newtonian Fluid Mechanics
Citation Excerpt :
In [22] the non-linear system of Eqs. (22)–(24) was solved using Newton’s method with a finite difference approximation to the Jacobian and a preconditioned GMRES iterative solver [35], as explained in [7,23].
The present work is an extension of earlier results of Owens [R.G. Owens, Comput. Methods Appl. Mech. Eng. 164 (1998) 375–395] and is an attempt to provide some theoretical undergirding to the question of appropriate error indicators for numerical solutions of flows of viscoelastic fluids having a differential constitutive equation. In particular, it is shown that the local elemental residual for the elastic stresses only accounts for the so-called stress cell error and as a consequence is, in general, an inadequate measure of the local error. An improved error indicator is then proposed which takes account of the transmitted error. Acknowledging that the stress errors form the major contribution to our error indicator for sufficiently large Deborah numbers we have devised a method of computing the error indicator on an element-by-element basis. Numerical results are presented to show how the approximate error and the ‘exact’ error obtained by calculating the difference between the numerical solution and a reference calculation decrease with mesh refinement. As an illustration of the use of our error indicator we proceed to describe an adaptive spectral element method for flow of an Oldroyd-B fluid past a sphere in a tube. The use of consistent upwinding is shown to result in greater accuracy and stability than is possible with a Galerkin approach. The results are compared with those in the literature.
Decoupled approach for the problem of viscoelastic fluid flow of PTT model. I: Continuous stresses
2000, Computer Methods in Applied Mechanics and Engineering
We present in this work a numerical analysis of a decoupled approach with continuous stresses for the viscoelastic fluid flows obeying the PTT model. The Streamline Upwind Petrov–Galerkin is used to treat the transport part of the equations. This model introduces supplementary difficulties linked to the non-linearities of the exponential type present in the constitutive equation. In exchange, it seems to be more realistic and gives best numerical results. The method is tested on the classical benchmark problem: Stick–Slip flow. Numerical results show that our method is remarkably stable and no upper Weissenberg limit number was encountered for reasonable values of ϵ, a material-depending parameter.
Mixed finite element methods for viscoelastic flow analysis: A review
1998, Journal of Non-Newtonian Fluid Mechanics
The progress made during the past decade in the application of mixed finite element methods to solve viscoelastic flow problems using differential constitutive equations is reviewed. The algorithmic developments are discussed in detail. Starting with the classical mixed formulation, the elastic viscous stress splitting (EVSS) method as well as the related discrete EVSS and the so-called EVSS-G method are discussed among others. Furthermore, stabilization techniques such as the streamline upwind Petrov–Galerkin (SUPG) and the discontinuous Galerkin (DG) are reviewed. The performance of the numerical schemes for both smooth and non-smooth benchmark problems is discussed. Finally, the capabilities of viscoelastic flow solvers to predict experimental observations are reviewed.

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Computing viscoelastic fluid flow problems at low cost

Abstract

J. Non-Newtonian Fluid Mech.

J. Non-Newtonian Fluid Mech.

J. Non-Newtonian Fluid Mech.

J. Non-Newtonian Fluid Mech.

J. Non-Newtonian Fluid Mech.