A FENE-P k–ε turbulence model for low and intermediate regimes of polymer-induced drag reduction

A new low-Reynolds-number k–ε turbulence model is developed for flows of viscoelastic fluids described by the finitely extensible nonlinear elastic rheological constitutive equation with Peterlin approximation (FENE-P model). The model is validated against direct numerical simulations in the low and intermediate drag reduction (DR) regimes (DR up to 50%). The results obtained represent an improvement over the low DR model of Pinho et al. (2008) [A low Reynolds number k–ε turbulence model for FENE-P viscoelastic fluids, Journal of Non-Newtonian Fluid Mechanics, 154, 89–108]. In extending the range of application to higher values of drag reduction, three main improvements were incorporated: a modified eddy viscosity closure, the inclusion of direct viscoelastic contributions into the transport equations for turbulent kinetic energy (k) and its dissipation rate, and a new closure for the cross-correlations between the fluctuating components of the polymer conformation and rate of strain tensors (NLT_ij). The NLT_ij appears in the Reynolds-averaged evolution equation for the conformation tensor (RACE), which is required to calculate the average polymer stress, and in the viscoelastic stress work in the transport equation of k. It is shown that the predictions of mean velocity, turbulent kinetic energy, its rate of dissipation by the Newtonian solvent, conformation tensor and polymer and Reynolds shear stresses are improved compared to those obtained from the earlier model.     

Numerical simulation studies of the planar entry flow of polymer melts

This paper is concerned with the numerical simulation of planar entry flow using a penalty finite element method and the comparison of predictions with flow visualization and birefringence data for two polymer melts. The Phan-Thien Tanner (PTT) model was fit to the steady state shear and extensional viscosity data and the transient extensional viscosity data of both polystyrene and low-density polyethylene (LDPE) melts to obtain the parameters λ, ξ, and ϵ in this model. Agreement was found between the flow visualization and birefringence data and the predictions of streamlines and stress. With some modification of the constitutive equation, the vortex growth and intensity observed for LDPE could be predicted by the use of the PTT model and the material parameters fit to the rheological properties. Likewise, the flow behavior of polystyrene, in which only small vortices with no growth were observed, was also predicted. Furthermore, it was found that the size and intensity of the vortex could be affected by the parameter ϵ in the PTT model which controls the predictions of the extensional viscosity. Based on these results it seems that accurate simulation of entry flow behavior requires the use of a constitutive equation which is capable of giving realistic preciction's of a fluid's extentional flow properties.     

Elongational flow behavior of polymeric fluids according to the Leonov model

The viscoelastic behavior of polymeric systems based upon the Leonov model has been examined for (i) the stress growth at constant strain rate, (ii) the stress growth at constant speed and (iii) the elastic recovery in elongational flow. The model parameters have been determined from the available rheological data obtained either in steady shear flow (shear viscosity and first normal-stress difference as a function of shear rate) or oscillatory flow (storage and loss moduli as a function of frequency in the linear region) or from extensional flow at very small strain rates (time-dependent elongation viscosity in the linear viscoelastic limit). In addition, the effect of the parameter characterizing the strain-hardening of the material during elongation has also been studied. The estimation of this parameter has been based upon the structural characteristics of the polymer chain which include the critical molecular weight and molecular weight of an independent segment. Five different polymer melts have been considered with varying number of modes (maximum four modes). Resulting predictions are in fair agreement with corresponding experimental data in the literature.     

Drag reduction in the turbulent pipe flow of polymers

The paper concerns an experimental study of the fully developed turbulent pipe flow of several different aqueous polymer solutions: 0.25%, 0.3% and 0.4% carboxymethylcellulose (CMC), 0.2% xanthan gum (XG), a 0.09%/0.09% CMC/XG blend, 0.125% and 0.2% polyacrylamide (PAA). The flow data include friction factor vs. Reynolds number, mean velocity and near-wall shear rate distributions, and axial velocity fluctuation intensity u′ at a fixed radial location as a laminar/turbulent transition indicator. For each fluid we also include measurements of shear viscosity, first normal-stress difference and extensional viscosity. At high shear rates we find that the degree of viscoelasticity increases with concentration (0.3% CMC is an exception) for a given polymer, and in the sequence XG, CMC/XG, CMC, PAA, whilst at low shear rates the ranking changes to CMC, CMC/XG, XG, PAA. The extensional viscosity ranking is XG/CMC, XG, CMC, PAA at high strain rates and the same as that for the viscoelasticity at low shear rates. We find that the observed drag-reduction behaviour is consistent for most part with the viscoelastic and extensional-viscosity behaviour at the low shear and strain rates typical of those occurring in the outer zone of the buffer region.Although laminar/turbulent transition is practically indiscernible from the friction factor vs. Reynolds number plots, particularly for PAA and XG, the u′ level provides a very clear indicator and it is found that the transition delay follows much the same trend with elasticity/extensional viscosity as the drag reduction.     

A Reynolds stress model for turbulent flow of homogeneous polymer solutions

Using a priori analyses of direct numerical simulation (DNS) data, a Reynolds stress model (RSM) is developed to account for the influence of polymer additives on turbulent flow over a wide range of flow conditions. The Finitely Extensible Nonlinear Elastic-Peterlin (FENE-P) rheological constitutive model is utilized to evaluate the polymer contribution to the stress tensor. Thirteen DNS data sets are used to analyze the budgets of elastic stress–velocity gradient correlations as well as Reynolds stress and dissipation transport. Closures are developed in the framework of the RSM model for all the required unknown and non-linear terms. The polymer stresses, velocity profiles, turbulent flow statistics and the percentage of friction drag reduction predicted by the RSM model are in good agreement with present and those obtained from independent DNS data over a wide range of rheological and flow parameters.     

Selection of ethylene-vinyl-acetate properties for modified bitumen with enhanced end-performance

Bitumen modification with ethylene-vinyl acetate (EVA), in a wide range temperatures (between ??30 and 100 °C), has been studied as a function of polymer concentration and EVA characteristics (vinyl acetate (VA) content and melt flow index (MFI)). Viscous flow, dynamic shear (DSR) temperature sweep, and technological tests were conducted to assess binder performance at medium-to-high in-service temperatures. Evaluation of binder low-temperature viscoelastic behavior has been performed using a solid rectangular fixture (SRF) in torsional mode, either in the linear viscoelastic region or under non-linear conditions (by strain breakage tests between ??30 and 0 °C). Further microstructural analysis based on modulated differential scanning calorimetry (MDSC) and optical microscopy was conducted to support rheological and technological results. Hence, total crystalline fraction (related to the VA content and polymer concentration) turned out to be a key parameter to achieve a suitable binder modification at medium-high temperatures. In addition, MFI appears to be an important EVA parameter at low temperatures, as it was found that lower MFI values enhanced resistance to low-temperature cracking.     

A detailed comparison of various FENE dumbbell models

The rheological behaviour of dilute solutions of finitely extensible non-linear elastic (FENE) dumbbells in both steady state and transient shear and simple elongational flow is investigated. Three dumbbell models are compared: the original FENE model with the Warner spring force, which is treated by brownian dynamics simulations, and the FENE-P model based on the Peterlin approximation and the FENE-CR model as suggested by Chilcott and Rallison, which are treated by standard numerical techniques. It is shown that in the linear viscoelastic limit and in steady state flows the behaviour is similar, except for the FENE-CR dumbbell in shear flow, modelling a Boger fluid. In transient flows larger differences appear.     

Influence of dispersion procedure on rheological properties of aqueous solutions of high molecular weight PEO

The linear and nonlinear viscoelastic behaviors of poly(ethylene oxide) (PEO) in aqueous media have been investigated as a function of concentration and molecular weight. A particular interest has been paid to study the effect of turbulent flow under stirring, inducing both shear and elongational stresses, on the rheological behavior of the polymer solutions. The comparison of intrinsic viscosity and viscoelastic properties between shaken and stirred PEO solutions is discussed at the molecular scale in terms of chain scission and aggregation. Results point out that the effect of the mechanical history on the rheological response of PEO solutions depends also on the concentration regime and molecular weight. Indeed, the influence of the dispersion procedure vanishes by decreasing both the concentration and the molecular weight.     

Analysis of polymer drag reduction mechanisms from energy budgets

The transfer of energy in drag reducing viscoelastic flows is analyzed through a sequence of energetic budgets that include the mean and turbulent kinetic energy, and the mean polymeric energy and mean elastic potential energy. Within the context of single-point statistics, this provides a complete picture of the energy exchange between the mean, turbulent and polymeric fields. The analysis utilizes direct simulation data of a fully developed channel flow at a moderately high friction Reynolds number of 1000 and at medium (30%) and high (58%) drag reduction levels using a FENE-P polymeric model.Results show that the primary effect of the interaction between the turbulent and polymeric fields is to transfer energy from the turbulence to the polymer, and that the magnitude of this transfer does not change between the low and high drag reduction flows. This one-way transfer, with an amplitude independent of the drag reduction regime, comes in contradiction with the purely elastic coupling which is implicit within the elastic theory of the polymer drag reduction phenomenon by Tabor and De Gennes (Europhys. Lett. 2, pp. 519–522, 1986).     

Surface tension driven jet break up of strain-hardening polymer solutions

Experimental studies attempting to ascertain the influence of viscoelasticity on the atomization of polymer solution are often hindered by the inability to decouple the effect of shear thinning from the effect of extensional hardening. Here, the influence of viscoelasticity on the jet break up of a series of non-shear-thinning viscoelastic fluids is quantified. Previous characterization using an opposed-nozzle rheometer identified the critical extensional rates for strain hardening of these model fluids. The strain hardening fluids exhibit a beads-on-string structure with reduction or elimination of satellite drops. Capillary instabilities grow on the filaments connecting the spheres and eventually break the filaments up into a string of very small drops about one order of magnitude smaller than the satellite drops formed by a Newtonian fluid with the same shear viscosity, surface tension, and density. These results confirm that strain hardening is the key rheological property in jet break up and that the critical extensional rate of a fluid is pertinent in determining the final characteristics of break up. Results suggest that the opposed-nozzle rheometer does probe extensional behavior in the range of extensional rates that are relevant to jet break up, providing a tool to roughly predict jet break up.     

Parametric study of FENE and FENE-P models in steady and unsteady flow in a circular pipe using CONNFFESSIT approach

The temporal rheological behavior of FENE and FENE-P models is investigated using the flow in a circular, straight tube. The CONNFFESSIT approach is applied to simulate transient Hagen–Poiseuille flow for different pressure drops and model parameters. The material functions for steady and unsteady flow were extracted and compared with the reported values in the literature when it was available. For the case of the FENE model, where there is no analytical solution, the relationship between model parameters and rheological behavior has been discussed and compared with the FENE-P model.     

Extensional rheology of concentrated poly(ethylene oxide) solutions

The shear and extensional rheology of three concentrated poly(ethylene oxide) solutions is examined. Shear theology including steady shear viscosity, normal stress difference and linear viscoelastic material functions all collapse onto master curves independent of concentration and temperature. Extensional flow experiments are performed in fiber spinning and opposed nozzles geometries. The concentration dependence of extensional behavior measured using both techniques is presented. The zero-shear viscosity and apparent extensional viscosities measured with both extensional rheometers exhibit a power law dependence with polymer concentration. Strain hardening in the fiber spinning device is found to be of similar magnitude for all test fluids, irrespective of strain rate. The opposed nozzle device measures an apparent extensional viscosity which is one order of magnitude smaller than the value determined with the fiber spinline device. This could be attributed to errors caused by shear, dynamic pressure, and the relatively small strains developed in the opposed nozzle device. This instrument cannot measure local kinematics or stresses, but averages these values over the non-homogenous flow field. These results show that it is not possible to measure the extensional viscosity of non-Newtonian and shear thinning fluids with this device. Fiber spin-line experiments are coupled with a momentum balance and constitutive model to predict stress growth and diameter profiles. A one-mode Giesekus model accurately captures the plateau values of steady and dynamic shear properties, but fails to capture the gradual shear thinning of viscosity. Giesekus model parameters determined from shear rheology are not capable of quantitatively predicting fiber spinline kinematics. However, model parameters fit to a single spinline experiment accurately predict stress growth behavior for different applied spinline tensions.     

Viscoelastic flow in a two-dimensional collapsible channel

We compute the flow of three viscoelastic fluids (Oldroyd-B, FENE-P, and Owens blood model) in a two-dimensional channel partly bounded by a tensioned membrane, a benchmark geometry for fluid–structure interactions. The predicted flow patterns are compared to those of a Newtonian liquid. We find that computations fail beyond a limiting Weissenberg number. Flow fields and membrane shape differ significantly because of the different degree of shear thinning and molecular extensibility underlying the three different microstructural models.     

Review of the entry flow problem: Experimental and numerical

This paper is concerned with a review of both experimental and numerical studies of axisymmetric and planar entry flows which have been considered as test problems for the numerical simulation of viscoelastic fluids. The test of the method is usually based upon whether the numerical model predicts vortices in the entry corners. However, it is not clear as to whether one should observe vortices for all viscoelastic fluids. Polyacrylamide solutions and Boger fluids exhibit vortices in axisymmetric flow and the size of the vortex does increase with fluid elasticity. However, the vortex is nearly suppressed in planar entry flow. On the other hand, not all polymer melts are found to exhibit vortices in either axisymmetric or planar entry flow. It is our belief that the origin of vortices is not related to the elasticity based on shear flow propertes but to the behavior of the transient extensional viscosity. Certain polymer melts such as low density polyethylene exhibit vortices in both planar and axisymmetric flow along with unbounded stress growth at the start up of extensional flow. It is believed that the constitutive equations used in the numerical simulation must reflect this extensional behavior if vortices are to be predicted. A review of the numerical simulations concerned with entry flow shows that there is considerable doubt about the accuracy of the predictions for most of the studies. Even for those where the numerical solution is thought to be accurate, the magnitude of the stream function associated with the vortices is usually very low. None of the differential models used to date predicts strain hardening extensional viscosity, but those which are thought to predict vortices do rise more rapidly to the steady-state extensional viscosity values with time. It is recommended that the search of test fluids be widened beyond polymer solutions as there may already exist a number of polymer melts which behave similarly to the predictions of existing constitutive equations.     

Rheological characterization of polyethylene terephthalate resins using a multimode Phan-Tien-Tanner constitutive relation

Linear viscoelastic, shear, and extensional rheological characterization of linear and branched Poly(Ethylene Terephthalate) resins (PET) was carried out by means of both a parallel-plate and capillary rheometers. Before loading into the rheometers, the polymer pellets were thoroughly dried at well-characterized conditions long enough to obtain consistent and reproducible results. Continuing polymer degradation and poly-condensation reactions in the relatively open environment of the parallel-plate rheometer were accounted for by correcting the data using material-time super-position. The rheological data obtained were used to fit by nonlinear optimization, the linear relaxation spectrum and nonlinear parameters of a multi-mode Phan-Thien and Tanner (PTT) constitutive relation. It was found that this model can represent rheological data for PET resins very well and as a result may be used in relevant processing flow simulations, i.e. film casting.     

Rheomechanical and morphological study of compatibilized PP/EVOH blends

In order to find the relationships between processibility and properties of the polypropylene/ethylene vinyl alcohol copolymer (PP/EVOH) blends, their rheological behavior, in both shear and extensional flows, was studied and related with mechanical, morphological, and barrier properties of the materials. The nonlinear viscoelastic behavior in shear was also analyzed. The data showed that the rheological parameters (viscosity, loss modulus, storage modulus, extensional viscosity, and Trouton ratio) improved with the addition of low quantities of sodium ionomer copolymer used as compatibilizer. At the same time, the overall properties of the PP/EVOH blends improved as a result of the compatibilizer addition. The morphological analysis showed that the changes in the material properties were related with a more uniform distribution of EVOH particles in the PP matrix. The rheological data obtained allowed us to choose the optimal range for EVOH and ionomer contents, especially in terms of combining good processing characteristics with the good final properties.     

On the skin friction coefficient in viscoelastic wall-bounded flows

Analysis of the skin friction coefficient for wall bounded viscoelastic flows is performed by utilizing available direct numerical simulation (DNS) results for viscoelastic turbulent channel flow. The Oldroyd-B, FENE-P and Giesekus constitutive models are used. First, we analyze the friction coefficient in viscous, viscoelastic and inertial stress contributions, as these arise from suitable momentum balances, for the flow in channels and pipes. Following Fukagata et al. (Phys. Fluids, 14, p. L73, 2002) and Yu et al. (Int. J. Heat. Fluid Flow, 25, p. 961, 2004) these three contributions are evaluated averaging available numerical results, and presented for selected values of flow and rheological parameters. Second, based on DNS results, we develop a universal function for the relative drag reduction as a function of the friction Weissenberg number. This leads to a closed-form approximate expression for the inverse of the square root of the skin friction coefficient for viscoelastic turbulent pipe flow as a function of the friction Reynolds number involving two primary material parameters, and a secondary one which also depends on the flow. The primary parameters are the zero shear-rate elasticity number, El₀, and the limiting value for the drag reduction at high Weissenberg number, LDR, while the secondary one is the relative wall viscosity, μ_w. The predictions reproduce both types A and B of drag reduction, as first introduced by Virk (Nature, 253, p. 109, 1975), corresponding to partially and fully extended polymer molecules, respectively. Comparison of the results for the skin friction coefficient against experimental data shows good agreement for low and moderate drag reduction which is the region covered by the simulations.     

Suppression or enhancement of interfacial instability in two-layer plane Couette flow of FENE-P fluids past a deformable solid layer

The linear stability of two-layer plane Couette flow of FENE-P fluids past a deformable solid layer is analyzed in order to examine the effect of solid deformability on the interfacial instability due to elasticity and viscosity stratification at the two-fluid interface. The solid layer is modeled using both linear viscoelastic and neo-Hookean constitutive equations. The limiting case of two-layer flow of upper-convected Maxwell (UCM) fluids is used as a starting point, and results for the FENE-P case are obtained by numerically continuing the UCM results for the interfacial mode to finite values of the chain extensibility parameter. For the case of two-layer plane Couette flow past a rigid solid surface, our results show that the finite extensibility of the polymer chain significantly alters the neutral stability boundaries of the interfacial instability. In particular, the two-layer Couette flow of FENE-P fluids is found to be unstable in a larger range of nondimensional parameters when compared to two-layer flow of UCM fluids. The presence of the deformable solid layer is shown to completely suppress the interfacial instability in most of the parameter regimes where the interfacial mode is unstable, while it could have a completely destabilizing effect in other parameter regimes even when the interfacial mode is stable in rigid channels. When compared with two-layer UCM flow, the two-layer FENE-P case is found in general to require solid layers with relatively lower shear modulii in order to suppress the interfacial instability. The results from the linear elastic solid model are compared with those obtained using the (more rigorous) neo-Hookean model for the solid, and good agreement is found between the two models for neutral stability curves pertaining to the two-fluid interfacial mode. The present study thus provides an important extension of the earlier analysis of two-layer UCM flow [V. Shankar, Stability of two-layer viscoelastic plane Couette flow past a deformable solid layer: implications of fluid viscosity stratification, J. Non-Newtonian Fluid Mech. 125 (2005) 143–158] to more accurate constitutive models for the fluid and solid layers, and reaffirms the central conclusion of instability suppression in two-layer flows of viscoelastic fluids by soft elastomeric coatings in more realistic settings.     

Observability of viscoelastic fluids

We apply the observability rank condition to study the observability of various viscoelastic fluids under imposed shear or extensional flows. In this paper the observability means the ability of determining the viscoelastic stress from the time history of the observations of the first normal stress difference. We consider four viscoelastic models: the upper convected Maxwell (UCM) model, the Phan–Thien–Tanner (PTT) model, the Johnson–Segalman (JS) model and the Giesekus model. Our study reveals that all of the four models have observability for all stress components almost everywhere under shear flow whereas under extensional flow most of the models have no observability for the shear stress component. More specifically, for UCM and JS models under imposed shear flow, the observations of the first normal stress difference allow the reconstruction of all components of viscoelastic stress. For UCM and JS models under extensional flow, the two normal stress components can be determined from the measurements of the first normal stress difference; the shear stress component does not affect the evolution of the normal stress components and consequently it cannot be extracted from the observations. Under shear flow, the PTT and Giesekus models have observability almost everywhere. That is, all components of the viscoelastic stress can be determined from the observations when the vector formed by the components of viscoelastic stress does not lie on a certain surface. Under extensional flow, the PTT model has observability almost everywhere for normal stress components whereas the Giesekus model has observability almost everywhere for all stress components. We also run simulations using the unscented Kalman filter (UKF) to reconstruct the viscoelastic stress from observations without and with noises. The UKF yields accurate and robust estimates for the viscoelastic stress both in the absence and in the presence of observation noises.     

A study on coherent structures and drag-reduction in the wall turbulence with polymer additives by TRPIV

An experimental measurement was performed using time-resolved particle image velocimetry (TRPIV) to investigate the spatial topological character of coherent structures in wall-bounded turbulence of polymer additive solution. The fully developed near-wall turbulent flow fields with and without polymer additives at the same Reynolds number were measured by TRPIV in a water channel. The comparisons of turbulent statistics confirm that due to viscoelastic structure of long-chain polymers, the wall-normal velocity fluctuation and Reynolds shear stress in the near-wall region are suppressed significantly. Furthermore, it is noted that such a behavior of polymers is closely related to the decease of the motion of the second and forth quadrants, i.e., the ejection and sweep events, in the near-wall region. The spatial topological mode of coherent structures during bursts has been extracted by the new mu-level criteria based on locally averaged velocity structure function. Although the general shapes of coherent structures are unchanged by polymer additives, the fluctuating velocity, velocity gradient, velocity strain rate and vorticity of coherent structures during burst events are suppressed in the polymer additive solution compared with that in water. The results show that due to the polymer additives the occurrence and intensity of coherent structures are suppressed, leading to drag reduction.