High Weissenberg Number Problem Research Articles

Development of an optimal adaptive finite element stabiliser for the simulation of complex flows

An optimal adaptive multiscale finite element method(AMsFEM) for numerical solutions of flow problems modelled by the Oldroyd B model is developed. Complex flows experience instabilities due to a phenomena known as the high Weissenberg number problem. The stabilisers are terms in-cooperated into the variational formulation when applying the finite element method. For the selected few stabilisers, numerical experiments are performed to study the convergence of the solutions. These demonstrate that adaptive strategies reduce the computational load of flow simulation. A best performing combination of choice of stabiliser and adaptive strategy is suggested.

Finite Volume Computational Analysis of the Heat Transfer Characteristic in a Double-Cylinder Counter-Flow Heat Exchanger with Viscoelastic Fluids

This work presents a computational analysis of the heat-exchange characteristics in a double-cylinder (also known as a double-pipe) geometrical arrangement. The heat-exchange is from a hotter viscoelastic fluid flowing in the core (inner) cylinder to a cooler Newtonian fluid flowing in the shell (outer) annulus. For optimal heat-exchange characteristics, the core and shell fluid flow in opposite directions, the so-called counter-flow arrangement.The mathematical modelling of the given problem reduces to a system of nonlinear coupled Partial Differential Equations (PDEs). Specifically, the rheological behaviour of the core fluid is governed by the Giesekus viscoelastic constitutive model. The governing system of coupled nonlinear PDEs is intractable to analytic treatment and hence is solved numerically using Finite Volume Methods (FVM). The FVM numerical methodology is implemented via the open-source software package OpenFOAM. The numerical methods are stabilized, specifically to address numerical instabilities arising from the High Weissenberg Number Problem (HWNP), via a combination of the Discrete Elastic Viscous Stress Splitting (DEVSS) technique and the Log-Conformation Reformulation (LCR) methodology. The DEVSS and LCR stabilization techniques are integrated into the relevant viscoelastic fluid solvers. The novelties of the study center around the simulation and analysis of the optimal heat-exchange characteristics between the heated Giesekus fluid and the coolant Newtonian fluid within a double-pipe counter-flow arrangement. Existing studies in the literature have either focused exclusively on Newtonian fluids and/or on rectangular geometries. The existing OpenFOAM solvers have also largely focused on non-isothermal viscoelastic flows. The relevant OpenFOAM solvers are modified for the present purposes by incorporating the energy equation for viscoelastic fluid flow. The flow characteristics are presented qualitatively (graphically) via the fluid pressure, temperature, velocity, and the polymer-stress components as well as the related normal stress differences. The results illustrate the required decrease in the core fluid temperature in the longitudinal direction due to the cooling effects of the shell fluid, whose temperature predictably increases in the counter-flow direction.

Numerical verification of sharp corner behavior for Giesekus and Phan-Thien–Tanner fluids

We verify numerically the theoretical stress singularities for two viscoelastic models that occur at sharp corners. The models considered are the Giesekus and Phan-Thien–Tanner (PTT), both of which are shear thinning and are able to capture realistic polymer behaviors. The theoretical asymptotic behavior of these two models at sharp corners has previously been found to involve an integrable solvent and polymer elastic stress singularity, along with narrow elastic stress boundary layers at the walls of the corner. We demonstrate here the validity of these theoretical results through numerical simulation of the classical contraction flow and analyzing the 270° corner. Numerical results are presented, verifying both the solvent and polymer stress singularities, as well as the dominant terms in the constitutive equations supporting the elastic boundary layer structures. For comparison at Weissenberg order one, we consider both the Cartesian stress formulation and the alternative natural stress formulation of the viscoelastic constitutive equations. Numerically, it is shown that the natural stress formulation gives increased accuracy and convergence behavior at the stress singularity and, moreover, encounters no upper Weissenberg number limitation in the global flow simulation for sufficiently large solvent viscosity fraction. The numerical simulations with the Cartesian stress formulation cannot reach such high Weissenberg numbers and run into convergence failure associated with the so-called high Weissenberg number problem.

Numerical simulation of viscoelastic & thixo-viscoelastoplastic complex flows at highly non-linear regimes

<h2>Abstract</h2> The High Weissenberg Number Problem (HWNP), which is the ceiling posed by the degree of non-linearity a numerical scheme can resolve in the approximation of the solution to a complex flow, is one of the main challenges to the Computational Rheology community (Walters & Webster, 2003). Research on this facet of non-Newtonian Fluids Mechanics has been traditionally focused on increasing resolution and capability to computationally capture non-linear phenomena in complex flow (Walters & Webster, 2003). In this talk, a proposal is presented to increase the critical Weissenberg number <i>Wi</i>_crit a numerical scheme can attain, via the generally applicable ABS-f and the VGR corrections reported previously (López-Aguilar et al., 2015), and their use to predict some experimental and numerical signatures in complex flow. The ABS-f correction acts upon stress-invariants used to promote non-linear features in conventional differential-type constitutive equations for viscoelastic fluids, helping in problem-regularisation and physically consistent material-property calculation in complex deformations (López-Aguilar et al., 2015). The VGR correction deals with proper velocity-gradient estimation, used to impose conservation-of-mass discretely over the flow-domain and to consistently specify velocity-gradient components at boundary symmetry lines (López-Aguilar et al., 2015). Here, predictions with an advanced hybrid finite-element/volume algorithm on the benchmark circular 4:1:4 contraction-expansion flow of concentrated wormlike micellar solutions prove notably extended in their <i>Wi</i>-span, where <i>Wi</i>_crit-adjustment is reported over three orders-of-magnitude from uncorrected variants, and with some cases presenting no limitation (López-Aguilar et al., 2015); (López-Aguilar et al., 2016a). Finally, use of these computational tools is exemplified in various contraction-expansion flow settings through the prediction of key experimental features for Boger fluids (López-Aguilar & Tamaddon-Jahromi, 2020; Webster et al., 2019), thixo-viscoelastoplasticity in sharp-cornered geometries (López-Aguilar et al., 2018) and shear-banding in contraction-expansion flow of highly concentrated wormlike micellar solutions (López-Aguilar et al., 2017a).

Stabilized finite element methods for a fully-implicit logarithmic reformulation of the Oldroyd-B constitutive law

Logarithmic conformation reformulations for viscoelastic constitutive laws have alleviated the high Weissenberg number problem, and the exploration of highly elastic flows became possible. However, stabilized formulations for logarithmic conformation reformulations in the context of finite element methods have not yet been widely studied. We present stabilized formulations for the logarithmic reformulation of the Oldroyd-B model by Saramito (2014) based on the Variational Multiscale framework and Galerkin/Least-Squares method. The reformulation allows the use of Newton’s method due to its fully-implicit nature for solving the steady-state problem directly. The proposed stabilization methods cure instabilities of convection and compatibility while preserving a three-field problem. The spatial accuracy of the formulations is assessed with the four-roll periodic box, revealing comparable accuracy between the methods. The formulations are validated with benchmark flows past a cylinder and through a 4:1 contraction. We found an excellent agreement to benchmark results in the literature. The algebraic sub-grid scale method is highly robust, indicated by the high limiting Weissenberg numbers in the benchmark flows.

Numerical simulation of the swirling flow of a finitely extensible non-linear elastic Peterlin fluid

Viscoelastic fluids have been shown to undergo instabilities even at very low Reynolds numbers, and these instabilities can give rise to a phenomenon called elastic turbulence. This phenomenon, observed experimentally in viscoelastic polymer solutions, is driven by the strong coupling between the fluid velocity and the elasticity of the flow. To explore the emergence of these instabilities in a viscoelastic flow, we have chosen to explore, by means of direct numerical simulations, a particular case called von Kármán swirling flow. The simulations employ the finitely extensible nonlinear Peterlin model to represent the dynamics of a dilute polymer solution. Notably, a log-conformation technique is used to solve the governing equations. This method is useful in overcoming the high Weissenberg number problem. The results obtained from the simulations were generally in good agreement with experiments. The torque on the top plate was decomposed into Newtonian and polymeric components, and it was found that the polymeric component was dominant. In addition, flow visualizations revealed that a toroidal vortex was strongly correlated with the distribution of the stresses on the rotating plate.

PEGAFEM-V: A new petrov-galerkin finite element method for free surface viscoelastic flows

The recently proposed finite element (FE) formulation for viscoelastic flows that allows the use of equal order linear interpolants for all variables and simultaneously does not suffer from the high Weissenberg number problem, is extended to free surface flows. The coupling of this Petrov-Galerkin stabilized FE formulation with the quasi-elliptic mesh generator allows us to obtain stable numerical solutions in highly deformed meshes and for very high values of the Weissenberg number (Wi). We present benchmark solutions in three free surface flows: the axisymmetric filament stretching, the elastocapillary-driven formation of bead-on-a-string, and the 2-dimensional, planar extrudate swell flow. In all cases, we attain converged solutions for values of Wi that have never been reached before by FE. The accuracy and robustness of the proposed numerical scheme are illustrated by achieving mesh and time step convergence under extreme mesh deformation conditions such as the bead-on-a-string (BOAS) formation during filament stretching. The formulation is enriched further with a discontinuity capturing scheme that enhances the quality of the solution around singularities dramatically. Finally, our simulations reveal for the first time the existence of lip-vortices in the steady planar extrudate-swell flow of Oldroyd-B fluids, which converge with mesh refinement.

Numerical and experimental investigation of abrasive flow machining of branching channels

Abrasive flow machining (AFM) is a key internal polishing technology applied to workpieces with difficult-to-reach areas. With the complex viscoelastic responses of abrasive media, the media flow is difficult to predict when the channel geometry is complex. It follows that the process outcome is also difficult to predict. In the present works, we proposed to use both experimental observations and computational fluid dynamics (CFD) simulations to investigate the responses of abrasive media in different branching channels during AFM process. We conducted measurement of media pressure and velocity during AFM for branching channels of different shapes and diameters. These data were used to develop and validate the present numerical model. The CFD simulations were conducted using the viscoelastic solver in OpenFOAM extension version, and we implemented log-conformation tensor representation (LCR) to stabilize the solver for high Weissenberg number problem (HWNP). The benchmark of the present simulations with experiments shows good agreement indicating that the present model is validated. The free-slip phenomenon of the abrasive media on the workpiece was uncovered from experimental observation of media flow in transparent channels and a sudden pressure drop (instead of a linear pressure drop) occurring across the connection between the workpiece and the fixture. Analysis of the numerical data indicates that the first normal stress difference (N1) plays an important role in the mass flow rate distributions of abrasive viscoelastic media in branching channels. Degradation of media properties was also studied and analyzed, suggesting that the flow pattern distribution in the workpieces could be influenced by media property degradation.

Solution of transient viscoelastic flow problems approximated by a term-by-term VMS stabilized finite element formulation using time-dependent subgrid-scales

Some finite element stabilized formulations for transient viscoelastic flow problems are presented in this paper. These are based on the Variational Multiscale (VMS) method, following the approach introduced in Castillo and Codina et al. (2019), for the Navier–Stokes problem, the main feature of the method being that the time derivative term in the subgrid-scales is not neglected. The main advantage of considering time-dependent sub-grid scales is that stable solutions for anisotropic space–time discretizations are obtained; however other benefits related with elastic problems are found along this study. Additionally, a split term-by-term stabilization method is discussed and redesigned, where only the momentum equation is approached using a term-by-term methodology, and which turns out to be much more efficient than other residual-based formulations. The proposed methods are designed for the standard and logarithmic formulations in order to deal with high Weissenberg number problems in addition to anisotropic space–time discretizations, ensuring stability in all cases. The proposed formulations are validated in several benchmarks such as the flow over a cylinder problem and the lid-driven cavity problem, obtaining stable and accurate results. A comparison between formulations and stabilization techniques is done to demonstrate the efficiency of time-dependent sub-grid scales and the term-by-term methodologies.

An immersed boundary-lattice Boltzmann method for fluid-structure interaction problems involving viscoelastic fluids and complex geometries

An immersed boundary-lattice Boltzmann method (IB-LBM) for fluid-structure interaction (FSI) problems involving viscoelastic fluids and complex geometries is presented in this paper. In this method, the fluid dynamics and the constitutive equations of viscoelastic fluids are both solved using the lattice Boltzmann method. In order to enhance numerical stability in solving the constitutive equations, an artificial damping is introduced which does not affect the numerical results if the damping effect is much smaller than the relaxation and the convective effects. The structural dynamics including 2D and 3D capsules, 2D and 3D rigid particles and flags, are solved by the finite difference method (2D capsules, 2D and 3D rigid particles and flags) and the finite element method (3D capsules). The interaction between the solid structure and the fluid is enforced by an immersed boundary method. The overall framework of this method is very simple, enabling modelling FSI problems involving viscoelastic fluids and the inertia of both fluids and structures. It is very efficient for FSI problems involving high Weissenberg numbers flows, large deformations and complicated geometries without any preconditioner. This work uses IB-LBM to solve for the first time, flows involving viscoelastic fluids coupled with non-massless deforming structures. The method is also capable of solving very high Weissenberg number problems, as demonstrated by simulations of flexible particle flows at Wi=100. The present method and models are validated by several cases including a 2D rigid particle migration in a Giesekus Couette flow, a spherical particle rotation in an Oldroyd-B shear flow, a spherical particle settling in a FENE-CR fluid, 2D and 3D capsules deformation in a Newtonian shear flow, and a 3D flag flapping in a Newtonian free stream. In addition, the present method is also applied to simulate the deformation of 2D and 3D capsules in an Oldroyd-B shear flow, a 3D flag flapping in an Oldroyd-B free stream, and elastic capsule movement in a contraction-expansion channel filled with an Oldroyd-B fluid. Deformation of the capsules decreases with the increase of the Weissenberg number and the capsules experience monotonically increasing deformation when the Weissenberg number is above a critical value which is respectively 10 for 2D and 2 for 3D simulations. Viscoelasticity of the Oldroyd-B fluid hinders the flapping motion of the 3D flag. For elastic capsules passing through a periodic contraction-expansion channel, the capsules mix up after a short-term evolution, and then migrate to the bottom of the channel and almost follow two steady trajectories after a long-term evolution. The validations and applications provide extensive data which may be used to expand the currently limited database available for FSI benchmark studies.

A numerical approach for non-Newtonian two-phase flows using a conservative level-set method

A finite-volume based conservative level-set method is presented to numerically solve the non-Newtonian multiphase flow problems. One set of governing equations is written for the whole domain, and different phases are treated with variable material and rheological properties. Main challenging areas of numerical simulation of multiphase non-Newtonian fluids, including tracking of the interface, mass conservation of the phases, small timestep problems encountered by non-Newtonian fluids, numerical instabilities regarding the high Weissenberg Number Problem (HWNP), instabilities encouraged by low solvent to polymer viscosity ratio in viscoelastic fluids and instabilities encountered by surface tensions are addressed and proper numerical treatments are provided in the proposed method. The numerical method is validated for different types of non-Newtonian fluids, e.g. shear-thinning, shear-thickening and viscoelastic fluids using structured and unstructured meshes. The proposed numerical solver is capable of readily adopting different constitutive models for viscoelastic fluids to different stabilization approaches. The constitutive equation is solved fully coupled with the flow equations. The method is validated for non-Newtonian single-phase flows against the analytical solution of start-up Poiseuille flow and the numerical solutions of well-known lid-driven Cavity problem. For multiphase flows, impact of a viscoelastic droplet problem, non-Newtonian droplet passing through a contraction-expansion, and Newtonian/non-Newtonian drop deformation suspended in Newtonian/non-Newtonian matrix imposed to shear flow are solved, and the results are compared with the related analytical, numerical and experimental data.

Viscoelastic fluid flow simulation using the contravariant deformation formulation

The new formulation for conformation-tensor based viscoelastic fluid models, written in terms of the contravariant deformation tensor as introduced by Hütter et al. (2018), has been approached numerically. It has been implemented in a finite element method framework with the DEVSS-G/SUPG method for stabilisation. A stress-implicit as well as a stress-explicit formulation is presented that allows to integrate the non-linear flow equations, with or without a solvent contribution, forward in time with second-order accuracy. The time-dependent stability in shear flow is maintained using the contravariant deformation, which is tested by solving a perturbed planar Couette flow of an upper-convected Maxwell (UCM) fluid. The stability and accuracy of our implementation has furthermore been tested by solving the flow around a cylinder with confining walls. The new formulation turns out to be more stable as compared to the formulation in terms of the conformation tensor. It enables to simulate the flow of a Giesekus fluid with non-linear parameter α=0.01 and viscosity ratio ν=ηs/η0=0.59 way beyond the Weissenberg number at which the High Weissenberg Number Problem manifests itself using the standard formulation. The stability appears to be similar to the log-conformation representation. The computed stress profiles up to a Weissenberg number of Wi=0.6 for an Oldroyd-B fluid with viscosity ratio ν=0.59, compare well with benchmark results from other studies, including other finite element, finite volume and spectral element methods.

A new finite element formulation for viscoelastic flows: Circumventing simultaneously the LBB condition and the high-Weissenberg number problem

In this paper, we propose a new, fully consistent and highly stable finite element formulation for the simulation of viscoelastic flows. In our method, we have implemented equal order interpolants for all variables and a combination of classical finite element stabilization techniques (PSPG/DEVSS-TG/SUPG) with the log-conformation representation of the constitutive equation that has allowed us to obtain numerically stable solutions at high Weissenberg numbers. The validity of the presented FEM framework is testified by comparing the numerical results of our method to those of the literature in three benchmark tests: the 2-dimensional flows of a viscoelastic fluid in a square lid-driven cavity and past a cylinder in a channel, and the 3-dimensional flow in a cubic lid-driven cavity. We consider both direct steady-state and transient calculations using the Oldroyd-B and linear PTT models. In all cases, we can reach, and in some cases surpass, the maximum Weissenberg number values attainable by mixed finite element methods, but at a considerably lower computational cost and programming effort. In addition, we perform mesh-convergence tests illustrating that the proposed method is convergent and features almost 2nd order accuracy in space.

Resonance in laminar pipe flow of non-linear viscoelastic fluids

Resonance phenomenon is investigated in pulsating flows in straight circular tubes for the class of constitutively non-linear affine viscoelastic fluids represented by the Johnson–Segalman constitutive model. The relaxation response of non-linear viscoelastic fluids associated with the natural frequencies of oscillation that arise as a consequence of the structure of the constitutive equation is explored. An analytical solution, which circumvents the high-Weissenberg number problem, which plagues numerical solutions of highly viscoelastic fluids, based on an asymptotic expansion in terms of a small material parameter is developed. At the lowest order of the asymptotic expansion the velocity field of the Maxwell fluid in round tubes is recovered, therefore providing validation of the computations. The analytical solution allows insights into the high Weissenberg number behavior of highly elastic fluids and thus avoids the difficulties the numerical solutions of high Weissenberg number problems are fraught with such as instability and lack of convergence. The analysis reveals that the forcing frequency associated with the pressure gradient generates a sequence of resonances of rapidly decaying intensity with increasing forcing frequency for a fixed value of the Weissenberg number. The intensity of the resonance is a rapidly increasing function of the increasing Weissenberg numbers, in particular at the first natural frequency. The latter is a decreasing function of the increasing elasticity of the fluid that is of the increasing Weissenberg numbers. The intensity of the resonance at the first natural frequency is a rapidly increasing function of the increasing elasticity. We show the existence of a limit point for the enhancement in energy savings with increasing elasticity. The effect of resonance on the flow rate, maximum amplitude of the average velocity, the mean flow rate and the oscillating velocity field is explored.

An extended volume of fluid method and its application to single bubbles rising in a viscoelastic liquid

An extended volume of fluid method is developed for two-phase direct numerical simulations of systems with one viscoelastic and one Newtonian phase. A complete set of governing equations is derived by conditional volume averaging the local instantaneous bulk equations and interface jump conditions. The homogeneous mixture model is applied for the closure of the volume-averaged equations. An additional interfacial stress term arises in this volume-averaged formulation which requires special treatment in the finite-volume discretization on a general unstructured mesh. A novel numerical scheme is proposed for the second-order accurate finite-volume discretization of the interface stress term. We demonstrate that this scheme allows for a consistent treatment of the interface stress and the surface tension force in the pressure equation of the segregated solution approach. Because of the high Weissenberg number problem, an appropriate stabilization approach is applied to the constitutive equation of the viscoelastic phase to increase the robustness of the method at higher fluid elasticity. Direct numerical simulations of the transient motion of a bubble rising in a quiescent viscoelastic fluid are performed for the purpose of experimental code validation. The well-known jump discontinuity in the terminal bubble rise velocity when the bubble volume exceeds a critical value is captured by the method. The formulation of the interfacial stress together with the novel scheme for its discretization is found crucial for the quantitatively correct prediction of the jump discontinuity in the terminal bubble rise velocity.

Numerical simulation of the planar extrudate swell of pseudoplastic and viscoelastic fluids with the streamfunction and the VOF methods

We present an Eulerian free-surface flow solver for incompressible pseudoplastic and viscoelastic non-Newtonian fluids. The free-surface flow solver is based on the streamfunction flow formulation and the volume-of-fluid method. The streamfunction solver computes the vector potential of a solenoidal velocity field, which ensures by construction the mass conservation of the solution, and removes the pressure unknown. Pseudoplastic liquids are modelled with a Carreau model. The viscoelastic fluids are governed by differential constitutive models reformulated with the log-conformation approach, in order to preserve the positive-definiteness of the conformation tensor, and to circumvent the high Weissenberg number problem. The volume fraction of the fluid is advected with a geometric conservative unsplit scheme that preserves a sharp interface representation. For the sake of comparison, we also implemented an algebraic advection scheme for the liquid volume fraction. The proposed numerical method is tested by simulating the planar extrudate swell with the Carreau, Oldroyd-B and Giesekus constitutive models. The swell ratio of the extrudates are compared with the data available in the literature, as well as with numerical simulations performed with the open-source rheoTool toolbox in OpenFOAM®. While the simulations of the generalized Newtonian fluids achieved mesh independence for all the methods tested, the flow simulations of the viscoelastic fluids are more sensitive to mesh refinement and the choice of numerical scheme. Moreover, the simulations of Oldroyd-B fluid flows above a critical Weissenberg number are prone to artificial surface instabilities. These numerical artifacts are due to discretization errors within the Eulerian surface-capturing method. However, the numerical issues arise from the stress singularity at the die exit corner, and the unphysical predictions of the Oldroyd-B model in the skin layer of the extrudate after the die exit, where large extensional deformations occur.

Prediction of Flow Effect on Crystal Growth of Semi-Crystalline Polymers Using a Multi-Scale Phase-Field Approach.

A multi-scale phase-field approach, which couples the mesoscopic crystallization with the microscopic orientation of chain segments and macroscopic viscoelastic melt flow, is proposed to study how the crystal growth of semi-crystalline polymers is affected by flows. To make the simulation feasible, we divide the problem into three parts. In the first part, a finitely extensible nonlinear elastic (FENE) dumbbell model is used to simulate the flow induced molecular structure. In the second part, formulas for estimating the density, orientation and aspect ratio of nuclei upon the oriented molecular structure are derived. Finally, in the third part, a massive mathematical model that couples the phase-field, temperature field, flow field and orientation field is established to model the crystal growth with melt flow. Two-dimensional simulations are carried out for predicting the flow effect on the crystal growth of isotactic polystyrene under a plane Poiseuille flow. In solving the model, a semi-analytical method is adopted to avoid the numerical difficult of a “high Weissenberg number problem” in the first part, and an efficient fractional step method is used to reduce the computing complexity in the third part. The simulation results demonstrate that flow strongly affects the morphology of single crystal but does not bring a significant influence on the holistic morphology of bulk crystallization.

Simulations of viscoelastic fluids using a coupled lattice Boltzmann method: Transition states of elastic instabilities

Elastic instabilities could happen in viscoelastic flows as the Weissenberg number is enlarged, and this phenomenon makes the numerical simulation of viscoelastic fluids more difficult. In this study, we introduce a coupled lattice Boltzmann method to solve the equations of viscoelastic fluids, which has a great capability of simulating the high Weissenberg number problem. Different from some traditional methods, two kinds of distribution functions are defined respectively for the evolution of the momentum and stress tensor equations. We mainly aim to investigate some key factors of the symmetry-breaking transition induced by elastic instability of viscoelastic fluids using this numerical coupled lattice Boltzmann method. In the results, we firstly find that the ratio of kinematical viscosity has an important influence on the transition of the elastic instability; the transition between the single stationary and cycling dominant vortex can be controlled via changing the ratio of kinematical viscosity in a periodic extensional flow. Finally, we can also observe a new transition state of instability for the flow showing the banded structure at higher Weissenberg number.

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

SummaryA 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.

Effect of polymer additives on heat transport and large-scale circulation in turbulent Rayleigh-Bénard convection.

The present paper presents direct numerical simulations of Rayleigh-Bénard convection (RBC) in an enclosed cell filled with the polymer solution in order to investigate the viscoelastic effect on the characteristics of heat transport and large-scale circulation (LSC) of RBC. To overcome the difficulties in numerically solving a high Weissenberg number (Wi) problem of viscoelastic fluid flow with strong elastic effect, the log-conformation reformulation method was implemented. Numerical results showed that the addition of polymers reduced the heat flux and the amount of heat transfer reduction (HTR) behaves nonmonotonically, which firstly increases but then decreases with Wi. The maximum HTR reaches around 8.7% at the critical Wi. The nonmonotonic behavior of HTR as a function of Wi was then corroborated with the modifications of the period of LSC and turbulent energy as well as viscous boundary layer thickness. Finally, a standard turbulent kinetic energy (TKE) budget analysis was done for the whole domain, the boundary layer region, and the bulk region. It showed that the role change of elastic stress contributions to TKE is mainly responsible for this nonmonotonic behavior of HTR.