This article provides a review of recent progress in understanding and predicting polymer drag reduction (DR) in turbulent wall-bounded shear flows. The reduction in turbulent friction losses by the dilute addition of high–molecular weight polymers to flowing liquids has been extensively studied since the phenomenon was first observed over 60 years ago. Although it has long been reasoned that the dynamical interactions between polymers and turbulence are responsible for DR, it was not until recently that progress had been made to begin to elucidate these interactions in detail. These advancements come largely from numerical simulations of viscoelastic turbulent flows and detailed turbulence measurements in flows of dilute polymer solutions using laser-based optical techniques. This review presents a selective overview of the current state of the numerics and experimental techniques and their impact on understanding the mechanics and prediction of polymer DR. It includes a discussion of areas in which our ...

Mechanics and Prediction of Turbulent Drag Reduction with Polymer Additives

We study the elasto-capillary self-thinning and ultimate breakup of three polystyrene-based ideal elastic fluids by measuring the evolution in the filament diameter as slender viscoelastic threads neck and eventually break. We examine the dependence of the transient diameter profile and the time to breakup on the molecular weight, and compare the observations with simple theories for breakup of slender viscoelastic filaments. The evolution of the transient diameter profile predicted by a multimode FENE-P model quantitatively matches the data provided the initial stresses in the filament are taken into account. Finally, we show how the transient uniaxial extensional viscosity of a dilute polymer solution can be estimated from the evolution in the diameter of the necking filament. The resulting “apparent extensional viscosity” profiles are compared with similar results obtained from a filament stretching rheometer. Both transient profiles approach the same value for the steady state extensional viscosity, which increases with molecular weight in agreement with the Rouse–Zimm theory. The apparent discrepancy in the growth rate of the two transient curves can be quantitatively explained by examining the effective stretch rate in each configuration. Filament thinning studies and filament stretching experiments thus form complementary experiments that lead to consistent measures of the transient extensional viscosity of a given test fluid.

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Elasto-capillary thinning and breakup of model elastic liquids

▪ Abstract Filament-stretching rheometers are devices for measuring the extensional viscosity of moderately viscous non-Newtonian fluids such as polymer solutions. In these devices, a cylindrical liquid bridge is initially formed between two circular end-plates. The plates are then moved apart in a prescribed manner such that the fluid sample is subjected to a strong extensional deformation. Asymptotic analysis and numerical computation show that the resulting kinematics closely approximate those of an ideal homogeneous uniaxial elongation. The evolution in the tensile stress (measured mechanically) and the molecular conformation (measured optically) can be followed as functions of the rate of stretching and the total strain imposed. The resulting rheological measurements are a sensitive discriminant of molecularly based constitutive equations proposed for complex fluids. The dynamical response of the elongating filament is also coupled to the extensional rheology of the polymeric test fluid, and this can...

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Filament-stretching rheometry of complex fluids

We show how to transform a large class of differential constitutive models into an equation for the (matrix) logarithm of the conformation tensor. Under this transformation, the extensional components of the deformation field act additively, rather than multiplicatively. This transformation is motivated by numerical evidence that the high Weissenberg number problem may be caused by the failure of polynomial-based approximations to properly represent exponential profiles developed by the conformation tensor. The potential merits of the new formulation are demonstrated for a finitely-extensible fluid in a two-dimensional lid-driven cavity at Weissenberg number W i = 5 .

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Constitutive laws for the matrix-logarithm of the conformation tensor

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 a FEM context using the DEVSS/DG formulation for viscoelastic fluid flow. We present a stability analysis in 1D and identify the failure of the numerical scheme to balance exponential growth as a possible source for numerical instabilities at high Weissenberg numbers. A different derivation of the log-based evolution equation than in [R. Fattal, R. Kupferman, Constitutive laws for the matrix-logarithm of the conformation tensor, J. Non-Newtonian Fluid Mech. 123 (2004) 281–285] is also presented. We show numerical results for the flow around a cylinder for an Oldroyd-B and a Giesekus model. With the log conformation representation, we are able to obtain solutions beyond the limiting Weissenberg numbers in the standard scheme. In particular, for the Giesekus model the improvement is rather dramatic: there does not seem to be a limit for the chosen model parameter (α = 0.01). However, it turns out that although in large parts of the flow the solution converges, we have not been able to obtain convergence in localized regions of the flow. Possible reasons include artefacts of the model and unresolved small scales. More work is necessary, including the use of more refined meshes and/or higher order schemes, before any conclusion can be made on the local convergence problems. © 2005 Elsevier B.V. All rights reserved.

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Flow of viscoelastic fluids past a cylinder at high Weissenberg number : stabilized simulations using matrix logarithms

It is well known that the drag in a turbulent flow of a polymer solution is significantly reduced compared to Newtonian flow. Here we consider this phenomenon by means of a direct numerical simulation of a turbulent channel flow. The polymers are modelled as elastic dumbbells using the FENE-P model. In the computations the polymer model is solved simultaneously with the flow equations, i.e. the polymers are deformed by the flow and in their turn influence the flow structures by exerting a polymer stress. We have studied the results of varying the polymer parameters, such as the maximum extension, the elasticity and the concentration. For the case of highly extensible polymers the results of our simulations are very close to the maximum drag reduction or Virk (1975) asymptote. Our simulation results show that at approximately maximum drag reduction the slope of the mean velocity profile is increased compared to the standard logarithmic profile in turbulent wall flows. For the r.m.s. of the streamwise velocity fluctuations we find initially an increase in magnitude which near maximum drag reduction changes to a decrease. For the velocity fluctuations in the spanwise and wall-normal directions we find a continuous decrease as a function of drag reduction. The Reynolds shear stress is strongly reduced, especially near the wall, and this is compensated by a polymer stress, which at maximum drag reduction amounts to about 40% of the total stress. These results have been compared with LDV experiments of Ptasinski et al. (2001) and the agreement, both qualitatively and quantitatively, is in most cases very good. In addition we have performed an analysis of the turbulent kinetic energy budgets. The main result is a reduction of energy transfer from the streamwise direction, where the production of turbulent kinetic energy takes place, to the other directions. A substantial part of the energy production by the mean flow is transferred directly into elastic energy of the polymers. The turbulent velocity fluctuations also contribute energy to the polymers. The elastic energy of the polymers is subsequently dissipated by polymer relaxation. We have also computed the various contributions to the pressure fluctuations and identified how these change as a function of drag reduction. Finally, we discuss some cross-correlations and various length scales. These simulation results are explained here by two mechanisms. First, as suggested by Lumley (1969) the polymers damp the cross-stream or wall-normal velocity fluctuations and suppress the bursting in the buffer layer. Secondly, the ‘shear sheltering’ mechanism acts to amplify the streamwise fluctuations in the thickened buffer layer, while reducing and decoupling the motions within and above this layer. The expression for the substantial reduction in the wall drag derived by considering the long time scales of the nonlinear fluctuations of this damped shear layer, is shown to be consistent with the experimental data of Virk et al. (1967) and Virk (1975).

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Turbulent channel flow near maximum drag reduction: simulations, experiments and mechanisms

In this paper we present a new approach for calculating viscoelastic flows. The polymer stress is not determined from a closed-form constitutive equation, but from a microscopic model. In this description, we replace the collection of individual polymer molecules by an ensemble of configuration fields, representing the internal degrees of freedom of the polymers. Similar to the motion of real molecules, these configuration fields are convected and deformed by the flow and are subjected to Brownian motion. We incorporated this field description in a finite element calculation. An important advantage of our approach is that the difficulties associated with particle tracking of individual molecules are circumvented. In order to validate our approach and to demonstrate its robustness, we present the results for the start-up of planar flow of an Oldroyd-B fluid past a cylinder between two parallel plates. The results are very promising. We find excellent agreement between the results of the configuration field formulation and those obtained using a closed-form constitutive equation. Moreover, the microscopic method appears to be more robust than the conventional macroscopic technique.

Simulation of viscoelastic flows using Brownian configuration fields

In this paper we report on (two-component) LDV experiments in a fully developed tur- bulent pipe flow with a drag-reducing polymer (partially hydrolyzed polyacrylamide) dissolved in water. The Reynolds number based on the mean velocity, the pipe diameter and the local viscosity at the wall is approximately 10000. We have used polymer solutions with three different concentrations which have been chosen such that maximum drag reduction occurs. The amount of drag reduction found is 60-70%. Our experimental results are compared with results obtained with water and with a very dilute solution which exhibits only a small amount of drag reduction. We have focused on the observation of turbulence statistics (mean velocities and turbulence intensities) and on the various contributions to the total shear stress. The latter consists of a turbulent, a solvent (viscous) and a polymeric part. The polymers are found to contribute significantly to the total stress. With respect to the mean velocity profile we find a thickening of the buffer layer and an increase in the slope of the logarithmic profile. With respect to the turbulence statistics we find for the streamwise velocity fluctuations an increase of the root mean square at low polymer concentration but a return to values comparable to those for water at higher concentrations. The root mean square of the normal velocity fluctuations shows a strong decrease. Also the Reynolds (turbulent) shear stress and the correlation coefficient between the streamwise and the normal components are drastically reduced over the entire pipe diameter. In all cases the Reynolds stress stays definitely non-zero at maximum drag reduction. The consequence of the drop of the Reynolds stress is a large polymer stress, which can be 60% of the total stress. The kinetic-energy balance of the mean flow shows a large transfer of energy directly to the polymers instead of the route by turbulence. The kinetic energy of the turbulence suggests a possibly negative polymeric dissipation of turbulent energy.

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Experiments in Turbulent Pipe Flow with Polymer Additives at Maximum Drag Reduction

Stochastic simulation techniques, such as Brownian dynamics, provide us an extremely powerful tool for solving the usually nonlinear equations describing polymer dynamics in solutions and melts [1]. However, the most challenging problems (e.g. the investigation of the universal behaviour of long polymer chains, or the flow calculation based on stochastic simulation techniques) involve a very large number of degrees of freedom and hence require an enomous amount of computer time. In order to solve such problems on currently available computers it is therefore necessary to develop strategies to drastically suppress the level of the fluctuations in the simulations. The purpose of this note is to show that the recently proposed concept of Brownian configuration fields [2] in viscoelastic flow calculations can be regarded as an extremely powerful extension of variance reduction techniques based on parallel process simulation.

Brownian configuration fields and variance reduced CONNFFESSIT

We compare the FENE and FENE-P models in different flow situations. We start with Brownian dynamics simulations of start-up of shear and uniaxial elongational flow. The FENE-P model predicts the FENE behaviour unsatisfactorily. However, we will show that its performance can be substantially improved. We propose to determine the parameters in the FENE-P model such that important FENE flow characteristics are recovered. The three resulting models (FENE, FENE-P and the proposed FENE-P model) will first be compared in the above-mentioned rheometrical flows. To show that the improvement persists in a complex flow field, we also simulate the flow of the FENE fluid past a cylinder using the Brownian configuration field technique. We then compare the results with those of the proposed FENE-P model and those of the original FENE-P model.

van den Bhaa Ben Brule

Papers

Turbulent channel flow near maximum drag reduction: simulations, experiments and mechanisms

Simulation of viscoelastic flows using Brownian configuration fields

Experiments in Turbulent Pipe Flow with Polymer Additives at Maximum Drag Reduction

Brownian configuration fields and variance reduced CONNFFESSIT

On the selection of parameters in the FENE-P model