Nontrivial steady flows have recently been found that capture the main structures of the turbulent buffer layer. We study the effects of polymer addition on these "exact coherent states" (ECS) in plane Couette flow. Despite the simplicity of the ECS flows, these effects closely mirror those observed experimentally: structures shift to larger length scales, wall-normal fluctuations are suppressed while streamwise ones are enhanced, and drag is reduced. The mechanism underlying these effects is elucidated. These results suggest that the ECS are closely related to buffer layer turbulence.PACS numbers: 83.60. Yz,83.80.Rs,47.20.Ky,47.27.Cn Rheological drag reduction, the suppression by additives of skin friction in turbulent flow, has received much attention since its discovery in 1947 [1,2,3]. For many polymer-solvent systems, the pressure drop measured in the pipe flow of the solution can be 30 − 50% less than for the solvent alone. The central rheological feature of drag-reducing additives is their extensional behavior in solution: for dilute polymer solutions in particular the stresses arising in extensional flow can be orders of magnitude larger than those developed in a shear flow. This fact is well-recognized; nevertheless the mechanism of interaction between polymer stretching and turbulent structure is not well-understood and the goal of the present work is to attempt to shed light on this interaction.A key structural observation from experiments and direct numerical simulations (DNS) of drag-reducing solutions is the modification of the buffer region near the wall [4,5,6,7,8,9,10]. It has long been known that the flow in this region is very structured, containing streamwise vortices that lead to streaks in the streamwise velocity [11]; these structures are thickened in both the wall-normal and spanwise directions during flow of drag reducing solutions [4,5]. Because of its importance in the production and dissipation of turbulent energy [11], any effort to mechanistically understand rheological drag reduction should address this region.To better understand the effect of the polymer on the buffer layer, we wish to study a model flow that has structures similar to those seen in this region but without the full complexities of time-dependent turbulent flows. Fortunately, a family of such flows exists, in the recently-discovered "exact coherent states" (ECS) found by computational bifurcation analysis in plane Couette and plane Poiseuille flows [12,13,14,15,16]. These are three-dimensional, traveling wave flows (hence steady in a traveling reference frame) that appear via saddle-node bifurcations [35] at a Reynolds number somewhat below the transition value seen in experiments [17,18]. The structure of the ECS captures the counter-rotating staggered streamwise vortices that dominate the structure in the buffer region. From the dynamical point of view, there is evidence that these states form a part of the dynamical skeleton of the turbulent flow: i.e., they are saddle points that underlie the strange attract...

Recently discovered traveling-wave solutions to the Navier-Stokes equations in plane shear geometries provide model flows for the study of turbulent drag reduction by polymer additives. These solutions, or "exact coherent states" (ECS), qualitatively capture the dominant structure of the near-wall buffer region of shear turbulence, i.e., counter-rotating pairs of streamwise-aligned vortices flanking a low-speed streak in the streamwise velocity. The optimum length scales for the ECS match well the length scales of the turbulent coherent structures and evidence suggests that the ECS underlie the dynamics of these structures. We study here the effect of viscoelasticity on these states. The changes to the velocity field for the viscoelastic ECS, where the FENE-P model calculates the polymer stress, mirror the modifications seen in experiments of fully turbulent flows of polymer solutions at low to moderate levels of drag reduction: drag is reduced, streamwise velocity fluctuations increase while wall-normal fluctuations decrease, and smaller wavelength structures are suppressed. These modifications to the ECS are due to the suppression of the streamwise vortices. The polymer molecules become highly stretched in the wavy, streamwise streaks, where the flow is predominately elongational, then relax as they move from the streaks into and around the streamwise vortices, where the flow is predominately rotational. This relaxation of the polymer molecules produces a force that directly opposes the fluid motion in the vortices, weakening them. Since the pressure fluctuations have their greatest magnitude (i.e., they are most negative) in the cores of the vortices, a reduction in vortex strength leads to a decrease in the magnitude of the pressure fluctuations. The pressure fluctuations redistribute energy from the streamwise velocity fluctuations to the Reynolds shear stress, so a decrease in their magnitude leads to a reduction in turbulent drag. For the viscoelastic ECS, we also find that after the onset of drag reduction (at Weissenberg number, WeϷ 7) there is a dramatic increase in the critical wall-normal length scale at which the ECS can exist. This sharp increase in length scale mirrors experimental observations and is also consistent with the observed shift to higher Reynolds numbers of the transition to turbulence in polymer solutions.

A Brownian dynamics study of bead-spring-chain polymer dynamics is undertaken in a model flow that captures key features of the buffer region of near-wall turbulence-wavy streamwise vortices superimposed on a mean shear. In this flow and in any Lagrangian chaotic flow, a Hookean dumbbell polymer will stretch indefinitely if and only if the Weissenberg number based on the largest Lyapunov exponent for the velocity field is у 1 2 . In the flow investigated here, this criterion is found to be good predictor of when the stretch of finitely extensible chains approaches its maximum value. The chains become highly stretched in the streamwise streaks and relax as they move into and around the vortex cores, leading to large differences in stress in different regions of the flow. Hydrodynamic and excluded volume interactions between polymer segments have no qualitative effects once results are normalized for the change in relaxation time due to their inclusion. The results from the bead-spring-chain models are used to assess the utility of the simpler FENE-P model. Although the FENE-P model does not capture the hysteresis in stress that is seen with the bead-spring-chain models, it otherwise qualitatively captures the behavior of the bead-spring chains. Most importantly, large polymer stress in the flow is seen at the same spatial positions for both the FENE-P and the more detailed models.

Wormlike micelles provide an opportunity to study the behavior of semiflexible macromolecules in elongational flows. We constructed a microfluidic cross-flow device coupled with fluorescence microscopy to image individual wormlike micelles and measure their dynamics in planar elongational flow. These polymer micelles prove stable in elongational flow and exhibit a sharp transition between regimes where Brownian motion dominates the micellar dynamics and where the micelles stretch with the flow. The coil-stretch transition and micellar relaxation time were identified by examining a distribution of micelle lengths at various flow rates. The relationship between micellar relaxation time and length is consistent with hydrodynamic theory. At higher Weissenberg number, micelle stretching is nearly as rapid as the rate of stretching of the surrounding fluid, yet also results more frequently in sharply folded conformations. In contrast to DNA in extensional flow, these relatively more stiff macromolecules exhibit fewer alignment modes.