An adaptive remeshing strategy for viscoelastic fluid flow simulations

In the last few years, we have developed a fairly general adaptive finite element solution procedure which can be applied to a large variety of problems. In this paper, this strategy is briefly recalled and applied to the solution of two-dimensional viscoelastic fluid flow problems. A log-conformation formulation recently introduced by Fattal and Kupferman [R. Fattal, R. Kupferman, Time-dependent simulation of viscoelastic flows at high Weissenberg number using the log-conformation representation, J. Non-Newtonian Fluid Mech. 126 (2005) 23-37] was implemented in order to improve the convergence properties of the numerical scheme. We confirm some results obtained in Hulsen, Fattal and Kupferman [M. Hulsen, R. Fattal, R. Kupferman, Flow of viscoelastic fluids past a cylinder at high Weissenberg number: stabilized simulations using matrix logarithm, J. Non-Newtonian Fluid Mech. 127 (2005) 27-39] and in some instances, we show that mesh adaptation allows to almost automatically reproduce accurate results obtained on very fine structured meshes.     

A stable unstructured finite volume method for parallel large-scale viscoelastic fluid flow calculations

A new stable unstructured finite volume method is presented for parallel large-scale simulation of viscoelastic fluid flows. The numerical method is based on the side-centered finite volume method where the velocity vector components are defined at the mid-point of each cell face, while the pressure term and the extra stress tensor are defined at element centroids. The present arrangement of the primitive variables leads to a stable numerical scheme and it does not require any ad-hoc modifications in order to enhance the pressure–velocity–stress coupling. The log-conformation representation proposed in [R. Fattal, R. Kupferman, Constitutive laws for the matrix–logarithm of the conformation tensor, J. Non-Newtonian Fluid Mech. 123 (2004) 281–285] has been implemented in order improve the limiting Weissenberg numbers in the proposed finite volume method. The time stepping algorithm used decouples the calculation of the polymeric stress by solution of a hyperbolic constitutive equation from the evolution of the velocity and pressure fields by solution of a generalized Stokes problem. The resulting algebraic linear systems are solved using the FGMRES(m) Krylov iterative method with the restricted additive Schwarz preconditioner for the extra stress tensor and the geometric non-nested multilevel preconditioner for the Stokes system. The implementation of the preconditioned iterative solvers is based on the PETSc library for improving the efficiency of the parallel code. The present numerical algorithm is validated for the Kovasznay flow, the flow of an Oldroyd-B fluid past a confined circular cylinder in a channel and the three-dimensional flow of an Oldroyd-B fluid around a rigid sphere falling in a cylindrical tube. Parallel large-scale calculations are presented up to 523,094 quadrilateral elements in two-dimension and 1,190,376 hexahedral elements in three-dimension.     

模拟微可压粘弹性流体的WCCBS_SU方法

针对微可压缩粘弹性流动问题,发展了微可压缩流的WCCBS方法,详细推导了基于Oldroyd-B本构模型的WCCBS_SU方法的求解过程。在流场微可压的条件下,分别对平面Poiseuille流和4：1粘弹性收缩流进行了数值模拟。Poiseuille流在不同We数下数值结果与解析解的比较,验证了本文方法具有较高的精度和较好...     

黏弹流体流动的数值模拟研究进展

综述了黏弹流体流动数值模拟的研究进展,突出介绍近十年来有限元法在黏弹流体流动数值模拟研究中取得的成果,通过动量方程的适当变形和本构方程离散权函数的合理选择,可以显著增强数值计算的稳定性。得到较高Weissenberg数下的解,同时文中对黏弹流体流动数值模拟中本构方程的应用、非等温情况和三维空间下的研究进行了介绍。     

积分型Maxwell流体挤出胀大的数值模拟

应用Luo提出的基于常规有限单元的应力计算方法和范毓润回避奇点的方法,模拟了积分型Maxwell黏弹流体的挤出胀大流动,计算得到了Weissenberg数达1．0下的合理结果。     

Coaxial-disk flow of an Oldroyd-B fluid: exact solution and stability

This paper reports an exact solution for the coaxial disk flow of an Oldroyd-B fluid. The flow is approximately generated by the parallel-plate viscometer. Asymptotic and numerical solutions are reported showing that there is a critical Weissenberg number based on the angular velocity and the Maxwellian relaxation time, above which the flow is unstable. A linearized stability analysis for the basic inertialess flow confirms this numerical instability and yields the critical Weissenberg number.     

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(10⁴)). 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.     

Property preserving reformulation of constitutive laws for the conformation tensor

The challenge for computational rheologists is to develop efficient and stable numerical schemes in order to obtain accurate numerical solutions for the governing equations at values of practical interest of the Weissenberg number. This study presents a new approach to preserve the symmetric positive definiteness of the conformation tensor and to bound the magnitude of its eigenvalues. The idea behind this transformation lies with the matrix logarithm formulation. Under the logarithmic transformation, the eigenvalue spectrum of the new conformation tensor varies from infinite positive to infinite negative. However reconstructing the classical formulation from unbounded eigenvalues does not achieve meaningful results. This enhanced formulation, expressed in terms of the hyperbolic tangent, overcomes the failure of alternative formulations by bounding the magnitude of eigenvalues in a manner that positive definite is always satisfied. In order to evaluate the capability of the hyperbolic tangent formulation, we performed a numerical simulation of FENE-P fluid in a rectangular channel in the context of the finite element method. Under this new transformation, the maximum attainable Weissenberg number increases by 21.4% and 112.5% compared to the standard log-conformation and classical constitutive equation, respectively.     

Plane flow of a fluid of second grade through a contraction

We have studied the flow of a fluid of second grade through three types of plane channels with a contraction. The numerical method makes use of non-symmetric first- and third-order derivatives for maximizing the diagonal terms of the iterative systems. We have examined, in particular, the growth and decay of the corner eddies as a function of the Reynolds and Weissenberg numbers.     

Time-dependent simulation of viscoelastic flows at high Weissenberg number using the log-conformation representation

We present a second-order finite-difference scheme for viscoelastic flows based on a recent reformulation of the constitutive laws as equations for the matrix logarithm of the conformation tensor. We present a simple analysis that clarifies how the passage to logarithmic variables remedies the high-Weissenberg numerical instability. As a stringent test, we simulate an Oldroyd-B fluid in a lid-driven cavity. The scheme is found to be stable at large values of the Weissenberg number. These results support our claim that the high Weissenberg numerical instability may be overcome by the use of logarithmic variables. Remaining issues are rather concerned with accuracy, which degrades with insufficient resolution.     

A finite volume approach for unsteady viscoelastic fluid flows

A finite volume, time‐marching for solving time‐dependent viscoelastic flow in two space dimensions for Oldroyd‐B and Phan Thien–Tanner fluids, is presented. A non‐uniform staggered grid system is used. The conservation and constitutive equations are solved using the finite volume method with an upwind scheme for the viscoelastic stresses and an hybrid scheme for the velocities. To calculate the pressure field, the semi‐implicit method for the pressure linked equation revised method is used. The discretized equations are solved sequentially, using the tridiagonal matrix algorithm solver with under‐relaxation. In both, the full approximation storage multigrid algorithm is used to speed up the convergence rate. Simulations of viscoelastic flows in four‐to‐one abrupt plane contraction are carried out. We will study the behaviour at the entrance corner of the four‐to‐one planar abrupt contraction. Using this solver, we show convergence up to a Weissenberg number We of 20 for the Oldroyd‐B model. No limiting Weissenberg number is observed even though a Phan Thien–Tanner model is used. Several numerical results are presented. Smooth and stable solutions are obtained for high Weissenberg number. Copyright © 2002 John Wiley & Sons, Ltd.     

A numerical stabilization framework for viscoelastic fluid flow using the finite volume method on general unstructured meshes

A robust finite volume method for viscoelastic flow analysis on general unstructured meshes is developed. It is built upon a general‐purpose stabilization framework for high Weissenberg number flows. The numerical framework provides full combinatorial flexibility between different kinds of rheological models on the one hand, and effective stabilization methods on the other hand. A special emphasis is put on the velocity‐stress‐coupling on colocated computational grids. Using special face interpolation techniques, a semi‐implicit stress interpolation correction is proposed to correct the cell‐face interpolation of the stress in the divergence operator of the momentum balance. Investigating the entry‐flow problem of the 4:1 contraction benchmark, we demonstrate that the numerical methods are robust over a wide range of Weissenberg numbers and significantly alleviate the high Weissenberg number problem. The accuracy of the results is evaluated in a detailed mesh convergence study.     

A new finite element scheme using the Lagrangian framework for simulation of viscoelastic fluid flows

A new numerical scheme for simulation of viscoelastic fluid flows was designed, making use of finite element algorithms generally regarded as advantageous for tackling the problem. This includes the Lagrangian approach for the solution of viscoelastic constitutive equation using the co-deformational frame of reference with a possibility of analytically solving the equation along the particles trajectories, which in turn allowed eluding the solution of any system of linear equations for the stress. Then, the full ellipticity of the momentum conservation equation was utilised thanks to a possibility of accurate determination of the stress tensor independently of the velocity field at the current stage of computation. The needed independent stress was calculated at each time step on the basis of the past deformation history, which in turn was determined on the basis of the past velocity fields, all incorporated into a modified Euler time stepping algorithm. Owing to explicit inclusion of the full viscous term from the viscoelastic model into the momentum conservation equation, no stress splitting was necessary. The trajectory feet tracking was done accurately using a semi-analytic solution of the displacement gradient evolution equation and a weak formulation of the kinematics equation, the latter at the expense of solving an extra symmetric system of linear equations.The error expressed in the form of the Sobolev norms was determined using a comparison with available analytical solution for UCM fluid in the transient regime or numerically obtained steady-state stress values for the PTT fluid in Couette flow. The implementation of the PTT fluid model was done by modifying the relative displacement gradient tensor so that a new convective frame was defined.The stability of the algorithm was assessed using the well-known benchmark problem of a sphere sedimenting in a tube with viscoelastic fluid. The stable numerical results were obtained at high Weissenberg numbers, with the limit of convergence Wi=6.6, exceeding any previously reported values. The robustness of the code was proven by simulation of the Weissenberg effect (the rod-climbing phenomenon) with the use of PTT fluid.     

Transient 3D flow of polymer solutions: A Lagrangian computational method

We describe a computational method for the numerical simulation of three-dimensional transient flows of polymer solutions that extends the work of Harlen et al. [O.G. Harlen, J.M. Rallison, P. Szabó, A split Lagrangian–Eulerian method for simulating transient viscoelastic flows, J. Non-Newtonian Fluid Mech. 60 (1995) 81–104]. The method uses a Lagrangian computation of the stress together with an Eulerian computation of the velocity field. Adaptive mesh reconnection based on Delaunay tetrahedra is used to ensure well-shaped elements. Additional shape-quality improvement procedures are developed to improve the algorithm. We validate the method for the benchmark problem of a rigid sphere falling in a cylindrical pipe. Inertia is neglected. We compare results for the axisymmetric case with previous work (using a FENE model), and then consider the off-axis non-axisymmetric case. In the latter case, we find that as the sphere falls, it drifts across the pipe, a phenomenon previously observed in experiments but not fully explained. The physical mechanisms that cause the time-dependent drift are identified, and a simple model based on the normal stresses in the fluid is shown to predict the magnitude of the drift velocity.We also consider a second benchmark problem involving a constriction in an axisymmetric pipe. Numerical difficulties associated with ill-shaped elements near the concave boundary arise for higher Weissenberg numbers. The merits and drawbacks of the new numerical method, and its applicability to various flow geometries are discussed.     

Drop oscillations under simple shear in a highly viscoelastic matrix

Recent two-dimensional numerical simulations and experiments have shown that, when a drop undergoes shear in a viscoelastic matrix liquid, the deformation can undergo an overshoot. I implement a volume-of-fluid algorithm with a paraboloid reconstruction of the interface for the calculation of the surface tension force for three-dimensional direct numerical simulations for a Newtonian drop in an Oldroyd-B liquid near criticalities. Weissenberg numbers up to 1 at viscosity ratio 1 and retardation parameter 0.5 are examined. Critical capillary numbers rise with the Weissenberg number. Just below criticality, drop deformation begins to undergo an overshoot when the Weissenberg number is sufficiently high. The overshoot becomes more pronounced, and at higher matrix Weissenberg numbers, such as 0.8, drop deformation undergoes novel oscillations before settling to a stationary shape. Breakup simulations are also described.     

Extrudate swell through an orifice die

The extrudate swell of a viscoelastic fluid through an orifice die is investigated by using a mixed finite element and a streamline integration method (FESIM), using a version of the K-BKZ model. The free surface calculation is based on a local mass conservation scheme and an approximate numerical treatment for the contact point movement of the free surface. The numerical results show a vortex growth and an increasing swelling ratio with the Weissenberg number. Convergence with mesh refinement is demonstrated, even at a high Weissenberg number of O(587), where the swelling ratio reaches a value of about 360%. In addition, it is found that the effective flow channel at the entrance region next to the orifice die is reduced due to the enhanced vortex growth, which may be a source of flow instability.     

On the stability of the torsional flow of a class of Oldroyd-type fluids

It is shown that the exact solution of the torsional flow of a class of Oldroyd-type fluids is kinematically similar to that for a Newtonian fluid. Furthermore, it is shown by a linearized stability analysis and by numerical integration, that the basic flow is unstable at high Weissenberg numbers. An Oldroyd fluid which has a negative second-normal stress coefficient is found to be more stable than one with zero (or positive) second-normal stress coefficient in this flow.     

Nonlinear galerkin method and subgrid-scale model for two-dimensional turbulent flows

The nonlinear Galerkin method, derived by Marion and Temam from results in dynamical systems theory, is investigated in the framework of numerical simulation of two-dimensional incompressible turbulent flows. We use the nonlinear Galerkin method together with a hyperviscosity subgrid-scale parametrization. We state the theoretical background, define the scheme, and derive some technical improvements for the method. We also report numerical experiments conducted for freely decaying as well as forced turbulence. Our main conclusion is that the performance of the nonlinear Galerkin method is important only if the dissipation range is large.This work was supported by Contract D.R.E.T. 90/1551/A000: Développement d'algorithmes de simulation numérique des équations de Navier-Stokes.     

Flow of viscoelastic fluids between plates rotating about distinct axes

We discuss the flow of BKZ fluids in an orthogonal rheometer. Some analytical results are proved, and numerical solutions are obtained for the Currie model. These solutions show a boundary layer behavior at high Reynolds numbers and the possibility of discontinuous solutions or nonexistence at high Weissenberg numbers.