Toward a Structural Understanding of Turbulent Drag Reduction: Nonlinear Coherent States in Viscoelastic Shear Flows

Abstract

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...

Polymer drag reduction in exact coherent structures of plane shear flow

“…41 That work provided some structural insight into the mechanism of drag reduction. The polymer molecules become highly elongated as they move through the streamwise streak.…”

Turbulent drag reduction in dam-break flows

“…The main difficulty is clearly finding the coupling of turbulence and polymer dynamics. Recently, promising attempts by means of effective theories (Sreenivasan and White 2000; Stone et al 2002;Boffetta et al 2003aBoffetta et al , 2003bL'vov et al 2003;Benzi et al 2004) and direct numerical simulations (Housiadas and Beris 2003;Min et al 2003aMin et al , 2003bPtasinski et al 2003;De Angelis et al 2003) reproduce important aspects of dilute polymer dynamics, such as the onset of drag reduction, suppressed wall-normal fluctuations, the logarithmic velocity law, and the maximum drag reduction asymptote (Virk 1975).…”

Traveling Waves in Pipe Flow

“…It has been suggested that these structures provide a skeleton for the transition to turbulence and the observed intermittency [17,18]. They clearly dominate various observables in low Reynolds number turbulent flows [10], and are also relevant for an understanding of the effects of non-Newtonian additives [19].The existence of exact coherent states in pipe flow has been an object of speculation for some time [5,6,7]. As we will show here Hagen-Poiseuille flow supports families of travelling waves with structures similar to those observed in other shear flows as well.…”

“…It has been suggested that these structures provide a skeleton for the transition to turbulence and the observed intermittency [17,18]. They clearly dominate various observables in low Reynolds number turbulent flows [10], and are also relevant for an understanding of the effects of non-Newtonian additives [19].…”