Data reduction in viscoelastic turbulent channel flows based on extended Karhunen–Loeve analysis

A previously attempted data reduction in turbulent viscoelastic channel flow [J. Non-Newton. Fluid Mech., 160 (2009) 55–63] used projections of the numerical velocity fields onto the most energetic, large scale, Karhunen–Loeve (K–L) modes of the fluctuating kinetic energy. However, the conformation field could not be adequately reproduced from the reduced velocity data when those were used in integrating the constitutive model (Giesekus) in a post-processing step. Here we investigate three different data reduction approaches in order to introduce small-scale information. Simultaneously, we also develop a novel formulation, which extends the K–L decomposition to more general objective functions. First, we use as a new objective function a weighted average of the pseudodissipation and the fluctuating kinetic energy. Second, we use the enstrophy. Third, we use the standard K–L approach, but this time in the reconstruction stage of the conformation tensor, we compensate for the loss of information of the flow deformation by suitably rescaling the Weissenberg number. It is shown that, whereas the first two methods fail to give any improvement over the classical K–L approach, the conformation field can be reconstructed fairly accurately using the third.     

An improved algorithm for simulating three-dimensional, viscoelastic turbulence

We present a new finite-difference formulation to update the conformation tensor in dumbbell models (e.g., Oldroyd-B, FENE-P, Giesekus) that guarantees positive eigenvalues of the tensor (i.e., the tensor remains positive definite) and prevents over-extension for finite-extensible models. The formulation is a generalization of the second-order, central difference scheme developed by Kurganov and Tadmor [A. Kurganov, E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection–diffusion equations, J. Comput. Phys. 160 (2000) 241–282] that guarantees a scalar field remains everywhere positive. We have extended the algorithm to guarantee a tensor field remains everywhere positive definite following an update. Extensive testing of the algorithm shows that the volume average of the conformation tensor is conserved. Furthermore, volume averages of the conformation tensor in homogeneous turbulent shear flow made over the Eulerian grid are in quantitative agreement with Lagrangian averages made over fluid particles moving throughout the domain, highlighting the accuracy of the treatment of the convective terms.     

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.     

On the Eigenvalues of the Fourth-Order Constitutive Tensor and Loss of Strong Ellipticity in Elastoplasticity

The eigenvalues of the fourth-order constitutive tangent modulus and the corresponding acoustic tensors are analyzed. Explicit expressions of the eigenvalues are made for the nonsymmetric tangent modulus tensor, and in the case of the deviatoric associative rule for the symmetric part of the tangent modulus and its acoustic tensor. In this context, a rate independent infinitesimal elastoplastic model is considered. The expressions of the plastic hardening modulus are summarized for the different local stability criteria (loss of second order work positiveness, loss of ellipticity, and loss of strong ellipticity). The critical hardening modulus and orientation are discussed in detail in the case of loss of ellipticity and loss of strong ellipticity. This analysis is based on the geometric method and linear, isotropic elasticity and deviatoric associative flow rule. In particular, the critical orientation for the loss of strong ellipticity and the classical shear band localization are compared.     

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.     

Stabilized mixed three‐field formulation for a generalized incompressible Oldroyd‐B model

This paper presents a generalization of the incompressible Oldroyd‐B model based on a thermodynamic framework within which the fluid can be viewed to exist in multiple natural configurations. The response of the fluid is viewed as a combination of an elastic component and a dissipative component. The dissipative component leads to the evolution of the underlying natural configurations, while the response from the natural configuration to the current configuration is considered elastic and therefore non‐dissipative. For an incompressible fluid, it is necessary that both the elastic behavior as well as the dissipative behavior is isochoric. This is achieved by ensuring that the determinant of the stretch tensor associated with the elastic response meets the constraint that its determinant is unity. A new stabilized mixed method is developed for the velocity, pressure and the kinematic tensor fields. Analytical models for fine scale fields are derived via the solution of the fine‐scale equations facilitated by the Variational Multiscale framework that are then variationally embedded in the coarse‐scale variational equations. The resulting method inherits the attributes of the classical SUPG and GLS methods, while a significant new contribution is that the form of the stabilization tensors is explicitly derived. A family of linear and quadratic tetrahedral and hexahedral elements is developed with equal‐order interpolations for the various fields. Numerical tests are presented that validate the incompressibility of the elastic stretch tensor, show optimal L₂ convergence for the conformation tensor field, and present stable response for high Weissenberg number flows. Copyright © 2016 John Wiley & Sons, Ltd.     

Effects of temperature and thermo-mechanical couplings on material instabilities and strain localization of inelastic materials

A general framework for rate-independent, small-strain, thermoinelastic material behaviour is presented, which includes thermo-plasticity and -damage as particular cases. In this context, strain localization and the development of material instabilities are investigated to highlight the roles of thermal effects and thermomechanical couplings. During a loading process, it is shown that two conditions play the essential roles and correspond to the singularity of the isothermal and the adiabatic acoustic tensors. Under quasi-static conditions, strain localization (in a classical sense) may occur when either of these two conditions is met. It involves a jump in temperature rate in the latter case, whereas temperature rate remains continuous in the former, but a discontinuity in the spatial derivatives of the heat flux must occur. This is consistent with the condition of stationarity of acceleration waves, which are shown to be homothermal and propagate with a velocity related to the eigenvalues of the isothermal acoustic tensor. A linear perturbation analysis further clarifies the above findings. In particular, for a quasi-static path of an infinite medium, failure of positive definiteness of either of the acoustic tensors corresponds to bifurcations in wave-like modes of arbitrary wave-length and infinite rate of growth. Under dynamic conditions, unbounded growth of perturbations is associated only to the short wavelength regime and corresponds to divergence growth or flutter phenomena relative to the isothermal acoustic tensor.     

A two-phase solver for complex fluids: Studies of the Weissenberg effect

In this work a new two-phase solver is presented and described, with a particular interest in the solution of highly elastic flows of viscoelastic fluids. The proposed code is based on a combination of classical Volume-of-Fluid and Continuum Surface Force methods, along with a generic kernel-conformation tensor transformation to represent the rheological characteristics of the (multi)-fluid phases. Benchmark test problems are solved in order to assess the numerical accuracy of distinct levels of physical complexities, such as the interface representation, the influence of advection schemes, the influence of surface tension and the role of fluid rheology. In order to demonstrate the new features and capabilities of the solver in simulating of complex fluids in transient regime, we have performed a set of simulations for the problem of a rotating rod inserted into a container with a viscoelastic fluid, known as the Weissenberg or Rod-Climbing effect. Firstly, our results are compared with numerical and experimental data from the literature for low angular velocities. Secondly, we have presented results obtained for high angular velocities (high elasticity) using the Oldroyd-B model which displayed very elevated climbing heights. Furthermore, above a critical value for the angular velocity, it was observed the onset of elastic instabilities driven by the combination of elastic stresses, interfacial curvature and secondary flows, that to the authors best knowledge, were not yet reported in the literature.     

Stability of plane poiseuille and couette flow of a Maxwell fluid

The classical Orr-Sommerfeld analysis is extended to a Maxwell fluid in fully developed Poiseuille flow between two flat plates and Couette flow between two flat plates. For the Poiseuille flow problem eigenmodes that are anti-symmetric in position are considered to augment the literature results for the symmetric eigenmodes. A shooting method with a stiff integrator, orthonormalization, and Newton-Raphson iterations on the eigenvalue are used to find the eigenvalues. The most dangerous mode is the anti-symmetric one, and both symmetric and anti-symmetric modes are more dangerous when the wave number and the Weissenberg number are large. No unstable eigenvalues are found.     

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.     

Numerical simulation of the fountain flow instability in injection molding

For the first time, the viscoelastic flow front instability is studied in the full non-linear regime by numerical simulation. A two-component viscoelastic numerical model is developed which can predict fountain flow behavior in a two-dimensional cavity. The eXtended Pom-Pom (XPP) viscoelastic model is used. The levelset method is used for modeling the two-component flow of polymer and gas. The difficulties arising from the three-phase contact point modeling are addressed, and solved by treating the wall as an interface and the gas as a compressible fluid with a low viscosity. The resulting set of equations is solved in a decoupled way using a finite element formulation. Since the model for the polymer does not contain a solvent viscosity, the time discretized evolution equation for the conformation tensor is substituted into the momentum balance in order to obtain a Stokes like equation for computing the velocity and pressure at the new time level. Weissenberg numbers range from 0.1 to 10. The simulations reveal a symmetric fountain flow for Wi = 0.1–5. For Wi = 10 however, an oscillating motion of the fountain flow is found with a spatial period of three times the channel height, which corresponds to experimental observations.     

Simulation of viscoelastic fluids in a 2D abrupt contraction by spectral element method

This study presents the vortex structure and numerical instability increase occurring when the level of elasticity is enhanced in inertial flows in planar contraction configuration for finitely extensible nonlinear elastic model by Peterlin (FENE‐P) fluid 1 . The re‐entrant corner effect on corner vortices is also considered. The calculations are performed using extended matrix logarithm formulation described in a previous paper: A. Jafari et al. A new extended matrix logarithm formulation for the simulation of viscoelastic fluids by spectral elements. Computer & Fluids 2010; 39 (9):1425–1438. In that reference, the proposed algorithm has been tested for simple geometry such as Poiseuille flow. In this study, we are interested in the capability of this algorithm for more complex geometry. This formulation helps to reach higher values of the Weissenberg number when compared with the classical one. Copyright © 2015 John Wiley & Sons, Ltd.     

Finite deformations of Ogden's materials under impact loading

The present paper is devoted to the modeling of finite deformations of a hyperelastic body described by Ogden's model under contact/impact conditions. Frictional contact problems are solved by means of the bi-potential method. The first order algorithm is applied to integrate the equation of motion. The total Lagrangian formulation is adopted to describe the geometrically non-linear behavior. For the finite element implementation, the explicit expression of the tangent operator is derived including the case of repeated eigenvalues. A numerical example is given to illustrate efficiency and accuracy of the method.     

The effects of resolution and noise on kinematic features of fine-scale turbulence

The effect of spatial resolution and experimental noise on the kinematic fine-scale features in shear flow turbulence is investigated by means of comparing numerical and experimental data. A direct numerical simulation (DNS) of a nominally two-dimensional planar mixing layer is mean filtered onto a uniform Cartesian grid at four different, progressively coarser, spatial resolutions. Spatial gradients are then calculated using a simple second-order scheme that is commonly used in experimental studies in order to make direct comparisons between the numerical and previously obtained experimental data. As expected, consistent with other studies, it is found that reduction of spatial resolution greatly reduces the frequency of high magnitude velocity gradients and thereby reduces the intermittency of the scalar analogues to strain (dissipation) and rotation (enstrophy). There is also an increase in the distances over which dissipation and enstrophy are spatially coherent in physical space as the resolution is coarsened, although these distances remain a constant number of grid points, suggesting that the data follow the applied filter. This reduction of intermittency is also observed in the eigenvalues of the strain-rate tensor as spatial resolution is reduced. The quantity with which these eigenvalues is normalised is shown to be extremely important as fine-scale quantities, such as the Kolmogorov length scale, are showed to change with different spatial resolution. This leads to a slight change in the modal values for these eigenvalues when normalised by the local Kolmogorov scale, which is not observed when they are normalised by large-scale, resolution-independent quantities. The interaction between strain and rotation is examined by means of the joint probability density function (pdf) between the second and third invariants of the characteristic equation of the velocity gradient tensor, Q and R respectively and by the alignments between the eigenvectors of the strain-rate tensor and the vorticity vector. Gaussian noise is shown to increase the divergence error of a dataset and subsequently affect both the Q–R joint pdf and the magnitude of the alignment cosines. The experimental datasets are showed to behave qualitatively similarly to the numerical datasets to which Gaussian noise has been added, confirming the importance of understanding the limitations of coarsely resolved, noisy experimental data.     

A theoretical and computational framework for anisotropic continuum damage mechanics at large strains

The main objective of this work is the formulation and algorithmic treatment of anisotropic continuum damage mechanics at large strains. Based on the concept of a fictitious, isotropic, undamaged configuration an additional linear tangent map is introduced which allows the interpretation as a damage deformation gradient. Then, the corresponding Finger tensor – denoted as damage metric – constructs a second order, internal variable. Due to the principle of strain energy equivalence with respect to the fictitious, effective space and the standard reference configuration, the free energy function can be computed via push-forward operations within the nominal setting. Referring to the framework of standard dissipative materials, associated evolution equations are constructed which substantially affect the anisotropic nature of the damage formulation. The numerical integration of these ordinary differential equations is highlighted whereby two different schemes and higher order methods are taken into account. Finally, some numerical examples demonstrate the applicability of the proposed framework.     

Viscoelastic flow analysis using the software OpenFOAM and differential constitutive equations

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.     

On interface equilibrium and inclusion problems

The condition for interface equilibrium, in which the Eshelby's energymomentum tensor is recognized as a generalization of the chemical potential in Gibbs' classical results, is used to evaluate the stability of the shape of inclusions in an infinite body with Eshelby's results for infinitesimal transformation strains.     

On the Standing Wave Problem in Deep Water

We present a new formulation of the classical two-dimensional standing wave problem which makes transparent the (seemingly mysterious) elimination of the quadratic terms made in [6]. Despite the presence of infinitely many resonances, corresponding to an infinite dimensional kernel of the linearized operator, we solve the infinite dimensional bifurcation equation by uncoupling the critical modes up to cubic order, via a Lyapunov--Schmidt like process. This is done without using a normalization of the cubic order terms as in [6], where the computation contains a mistake, although the conclusion was in the end correct. Then we give all possible bifurcating formal solutions, as powers series of the amplitude (as in [6]), with an arbitrary number, possibly infinite, of dominant modes.     

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.