The helical decomposition and the instability assumption

Direct numerical simulations show that the triadic transfer function T(k,p,q) peaks sharply when q (or p) is much smaller than k. The triadic transfer function T(k,p,q) gives the rate of energy input into wave number k from all interactions with modes of wave number p and q, where k, p, q form a triangle. This observation was thought to suggest that energy is cascaded downscale through non-local interactions with local transfer and that there was a strong connection between large and small scales. Both suggestions were in contradiction with the classical Kolmogorov picture of the energy cascade. The helical decomposition was found useful in distinguishing between kinematically independent interactions. That analysis has gone beyond the question of non-local interaction with local transfer. In particular, an assumption about the statistical direction of triadic energy transfer in any kinematically independent interaction was introduced (the instability assumption). That assumption is not necessary for the conclusions about non-local interactions with local transfer recalled above. In the case of turbulence under rapid rotation, the instability assumption leads to the prediction that energy is transferred in spectral space from the poles of the rotation axis toward the equator. The instability assumption is thought to be of general validity for any type of triad interactions (e.g. internal waves). The helical decomposition and the instability assumption offer detailed information about the homogeneous statistical dynamics of the Navier-Stokes equations. The objective was to explore the validity of the instability assumption and to study the contributions of the various types of helical interactions to the energy cascade and the subgrid-scale eddy-viscosity. This was done in the context of spectral closures of the Direct Interaction or Quasi-Normal type

Regeneration of near-wall turbulence structures

An examination of the regeneration mechanisms of near-wall turbulence and an attempt to investigate the critical Reynolds number conjecture of Waleffe & Kim is presented. The basis is an extension of the 'minimal channel' approach of Jimenez and Moin which emphasizes the near-wall region and further reduces the complexity of the turbulent flow. Reduction of the flow Reynolds number to the minimum value which will allow turbulence to be sustained has the effect of reducing the ratio of the largest scales to the smallest scales or, equivalently, of causing the near-wall region to fill more of the area between the channel walls. In addition, since each wall may have an active near-wall region, half of the channel is always somewhat redundant. If a plane Couette flow is instead chosen as the base flow, this redundancy is eliminated: the mean shear of a plane Couette flow has a single sign, and at low Reynolds numbers, the two wall regions share a single set of structures. A minimal flow with these modifications possesses, by construction, the strongest constraints which allow sustained turbulence, producing a greatly simplified flow in which the regeneration process can be examined

Publisher’s Note: “Polymer drag reduction in exact coherent structures of plane shear flow” [Phys. Fluids 16, 3470 (2004)]

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/69804/2/PHFLE6-16-12-4761-1.pd

Toward a structural understanding of turbulent drag reduction: nonlinear coherent states in viscoelastic shear flows

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.Comment: 5 pages, 3 figures, published version, Phys. Rev. Lett. 89, 208301 (2002

Polymer drag reduction in exact coherent structures of plane shear flow

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, We ≈ 7We≈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.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/69532/2/PHFLE6-16-9-3470-1.pd