Ma Martien Hulsen

Simulation of viscoelastic flows using Brownian configuration fields

Abstract 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.

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

In this paper we report on (two-component) LDV experiments in a fully developed turbulent 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 stream wise 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.

Brownian configuration fields and variance reduced CONNFFESSIT

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.

A sufficient condition for a positive definite configuration tensor in differential models

Abstract Analogous to the Giesekus theory, it is possible to identify a positive definite (configuration) tensor in other theories for differential models. For a rather general class of differential models, a simple condition is given which is sufficient to prove that such tensors are positive definite.

On the selection of parameters in the FENE-P model

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.

Instationary Eulerian viscoelastic flow simulations using time separable Rivlin-Sawyers constitutive equations

Abstract A time dependent method for solving integral constitutive equations of the Rivlin–Sawyers type is introduced. The deformation history is represented by a finite number of deformation fields. Using these fields the stress integral is approximated as a finite sum. When the flow evolves the deformation fields are convected and deformed. The approach presented in this paper is the first Eulerian method that can handle integral equations in a time dependent way. The method is validated by using the upper-convected Maxwell (UCM) benchmark of a sphere moving in a tube. We show that the method converges with mesh and time step refinement and that the results are accurate, comparable to the results obtained with the differential equivalent of the UCM model. To demonstrate that complicated linear spectra are easily incorporated, results of a Rouse model simulation of 100 modes are presented. We also compare results on a falling sphere problem to the results obtained by a Lagrangian method as reported by Rasmussen and Hassager [H.K. Rasmussen, O. Hassager, On the sedimentation velocity of spheres in a polymeric liquid, Chem. Eng. Sci. 51 (1996) 1431–1440]. The model being employed is the PSM model, for which no differential equivalent exists.

Simulation of the Doi–Edwards model in complex flow

The Doi–Edwards model is simulated in startup of two-dimensional complex flow. The flow geometry is that of a cylinder confined between two parallel plates. Two different, new simulation methods are used. The first is based on the configuration field approach. The second method is a new method which is introduced in this paper. We refer to this approach as the deformation field method. Theoretically the methods are equivalent. It will be shown that the deformation field approach is very efficient. Furthermore, the method opens up possibilities of studying extensions of the Doi–Edwards model which include tube-stretch and convected constraint release.

Numerical simulation of contraction flows using a multi-mode Giesekus model

Abstract The paper presents a finite-element scheme combined with a streamline integration scheme to solve viscoelastic fluid flows with multi-mode differential models. The numerical scheme is applied to the flow through axisymmetrical contractions. An eight-mode Giesekus model, fitted for an LDPE melt, has been used. Results are given for the size of the vortices in front of the contraction. These compare well with experimental values up to very high Deborah numbers. Results for entrance correction and vortex intensities are also presented. The vortex intensity appears to have a maximum as a function of the Deborah number. The velocity profiles in the contraction are over-developed and the velocity on the axis relative to the mean velocity reaches a maximum at the start of the vortex. It is argued that the size of the vortex depends on the ratio of elongational properties versus shear properties. This is further supported by a parametric study on a one-mode Giesekus model.

A new approach to the deformation fields method for solving complex flows using integral constitutive equations

Abstract In this paper, we present a new approach to the deformation fields method that has recently been introduced to solve integral type models in complex flows (E.A.J.F. Peters, M.A. Hulsen, B.H.A.A. van den Brule, J. Non-Newtonian Fluid Mechanics 89 (2000) 209–228). The new approach is based on a change of the reference time of the fields from an absolute time to a time relative to the current time. This basically removes most, if not all, stability and accuracy problems that exist in the original method compared to using differential models. Also the new implementation is much more flexible with respect to the time integral discretisation, opening the way to adaptive refinement. The new implementation has been tested for two problems: the standard 2:1 benchmark of a the flow around a sphere using a UCM model and the flow around a sphere in a different geometry using a PSM model.

On the modelling of a PIB/PB Boger fluid in extensional flow

In this paper we study the transient elongational viscosity of a PIB:PB Boger fluid (fluid B) by performing both experiments and model calculations. The experimental results have been obtained by using a filament stretching device was described by van Nienwkoop and Muller von Czernicki (J. Non-Newtonian Fluid Mech. 67 (1996) 105‐123). We have been able to obtain large strains, up to a Hencky strain of 8. A plateau for the Trouton ratio of close to 10 4 is found while approaching a strain of 8. The plateau value appears to be independent of the strain rate history. The predictions using various multi-mode constitutive models (Giesekus, FENE-P, Hinch, FENE) show that the FENE model is the only model which predicts values for the Trouton ratio that are very close to the experiments for the whole range from strain 1 up to 8. The predictions of the models are computed with the actual strain rate o; (t )a s determined during the experiments in the middle of the filament. During relaxation, both the FENE-P and the FENE model perform well. In order to be able to make predictions without carrying out the relatively expensive FENE calculations, we have developed a new closed form constitutive equation. The model is based on the dumbbell theory in which the connector force used leads to a viscous stress term. The predictions of the model in transient extensional flows are very good and comparable to the FENE model.