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

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)

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 Nieuwkoop 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 104 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 as 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

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

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

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 tur-bulentpipe flow with a drag-reducing polymer (partially hydrolyzed polyacrylamide) dissolved inwater. The Reynolds number based on the mean velocity, the pipe diameter and the local viscosity atthe wall is approximately 10000. We have used polymer solutions with three different concentrationswhich have been chosen such that maximum drag reduction occurs. The amount of drag reductionfound is 60 70%. Our experimental results are compared with results obtained with water and witha very dilute solution which exhibits only a small amount of drag reduction.We have focused on the observation of turbulence statistics (mean velocities and turbulenceintensities) and on the various contributions to the total shear stress. The latter c! onsists of a turbulent,a solvent (viscous) and a polymeric part. The polymers are found to contribute significantly to thetotal stress. With respect to the mean velocity profile we find a thickening of the buffer layer and anincrease in the slope of the logarithmic profile. With respect to the turbulence statistics we find for thestreamwise velocity fluctuations an increase of the root mean square at low polymer concentrationbut a return to values comparable to those for water at higher concentrations. The root mean square ofthe normal velocity fluctuations shows a strong decrease. Also the Reynolds (turbulent) shear stressand the correlation coefficient between the streamwise and the normal components are drasticallyreduced over the entire pipe diameter. In all cases the Reynolds stress stays definitely non-zero atmaximum drag reduction. The consequence of the drop of the Reynolds stress is a large polymerstress, which can be 60% of t! he total stress. The kinetic-energy balance of the mean flow shows alarge transfer of energy directly to the polymers instead of the route by turbulence. The kinetic energyof the turbulence suggests a possibly negative polymeric dissipation of turbulent energy

Numerical simulation of a viscoelastic fluid with anisotropic heat conduction

For the nonisothermal flow of a viscoelastic fluid we have taken into account temperature dependency of the relaxation times and the viscosities in the constitutive equation for the stress. In the energy equation the heat flux is specified by Fourier's law, where anisotropic heat conduction has been taken into account. Furthermore one has to specify which part of the stress work is dissipated and which part is stored as elastic energy. The equations are solved with a finite element method for the balance equations and a streamline integration method for the constitutive equation. The influence of the Deborah number, the Péclet number and the cooling temperature are examined in a flow through a 4 to 1 contraction