Additive equivalence in turbulent drag reduction by flexible and rodlike polymers

Orientational order and topological defects in a dilute solutions of rodlike polymers at low Reynolds number

The relationship between the polymer orientation and the chaotic flow, in a dilute solution of rigid rodlike polymers at low Reynolds number, is investigated by means of direct numerical simulations. It is found that the rods tend to align with the velocity field in order to minimize the friction with the solvent fluid, while regions of rotational disorder are related to strong vorticity gradients, and therefore to the chaotic flow. The "turbulent-like" behavior of the system is therefore associated with the emergence and interaction of topological defects of the mean director field, similarly to active nematic turbulence. The analysis has been carried out in both two and three spatial dimensions.

Turbulent viscosity profile of drag reducing rod-like polymers

Recent theories of drag reduction in wall turbulence assumed that the presence of the polymer leads to an effective viscosity, which increases linearly with the distance from the wall. Such a linear viscosity profile reduces the Reynolds stress (i.e., the momentum flux to the wall), which leads to drag reduction. For the usual flexible polymers employed in drag reduction, the effective viscosity is however a strongly non-linear effect that is difficult to quantify. We therefore investigate the turbulent drag reduction characteristics of a stiff rod-like polymer for which any effective viscosity changes are only due to the orientation of the polymers. The results show that close to the walls the polymers orient and the viscosity is low, whereas in the bulk the polymers are randomly oriented and the effective viscosity is high. This indeed leads to a reduction of the Reynolds stress and hence to a drag reduction.

Emergence of chaos in a viscous solution of rods

It is shown that the addition of small amounts of microscopic rods in a viscous fluid at low Reynolds number causes a significant increase of the flow resistance. Numerical simulations of the dynamics of the solution reveal that this phenomenon is associated to a transition from laminar to chaotic flow. Polymer stresses give rise to flow instabilities which, in turn, perturb the alignment of the rods. This coupled dynamics results in the activation of a wide range of scales, which enhances the mixing efficiency of viscous flows.

Rotational relaxation time as unifying time scale for polymer and fiber drag reduction

Using hybrid direct numerical simulation plus Langevin dynamics, a comparison is performed between polymer and fiber stress tensors in turbulent flow. The stress tensors are found to be similar, suggesting a common drag reducing mechanism in the onset regime for both flexible polymers and rigid fibers. Since fibers do not have an elastic backbone, this must be a viscous effect. Analysis of the viscosity tensor reveals that all terms are negligible, except the off-diagonal shear viscosity associated with rotation. Based on this analysis, we identify the rotational orientation time as the unifying time scale setting a new time criterion for drag reduction by both flexible polymers and rigid fibers.

Comparison of theory and direct numerical simulations of drag reduction by rodlike polymers in turbulent channel flows

Numerical simulations of turbulent channel flows, with or without additives, are limited in the extent of the Reynolds number (Re) and Deborah number (De). The comparison of such simulations to theories of drag reduction, which are usually derived for asymptotically high Re and De, calls for some care. In this paper we present a study of drag reduction by rodlike polymers in a turbulent channel flow using direct numerical simulation and illustrate how these numerical results should be related to the recently developed theory.

Turbulent drag reduction by flexible and rodlike polymers: Crossover effects at small concentrations

Drag reduction by polymers is bounded between two universal asymptotes, the von Kármán log law of the law and the maximum drag reduction (MDR) asymptote. It is theoretically understood why the MDR asymptote is universal, independent of whether the polymers are flexible or rodlike. The crossover behavior from the Newtonian von Kármán log law to the MDR is, however, not universal, showing different characteristics for flexible and rodlike polymers. In this paper we provide a theory for this crossover phenomenology.

Drag reduction by compressible bubbles

Drag reduction by bubbles in stationary turbulent flows is sensitive to the compressibility of the bubbles. Without this dynamical effect the bubbles only renormalize the fluid density and viscosity, an effect that by itself can only lead to a small percentage of drag reduction. We show in this paper that the dynamics of bubbles and their effect on the compressibility of the mixture can lead to a much higher drag reduction.

Phenomenology of wall-bounded Newtonian turbulence

We construct a simple analytic model for wall-bounded turbulence, containing only four adjustable parameters. Two of these parameters are responsible for the viscous dissipation of the components of the Reynolds stress tensor. The other two parameters control the nonlinear relaxation of these objects. The model offers an analytic description of the profiles of the mean velocity and the correlation functions of velocity fluctuations in the entire boundary region, from the viscous sublayer, through the buffer layer, and further into the log-law turbulent region. In particular, the model predicts a very simple distribution of the turbulent kinetic energy in the log-law region between the velocity components: the streamwise component contains a half of the total energy whereas the wall-normal and cross-stream components contain a quarter each. In addition, the model predicts a very simple relation between the von Kármán slope k and the turbulent velocity in the log-law region v+ (in wall units): v+=6k. These predictions are in excellent agreement with direct numerical simulation data and with recent laboratory experiments.

Identification and calculation of the universal asymptote for drag reduction by polymers in wall bounded turbulence

Drag reduction by polymers in wall turbulence is bounded from above by a universal maximal drag reduction (MDR) velocity profile that is a log law, estimated experimentally by Virk as V+(y+) approximately 11.7logy+ - 17. Here V+(y+) and y+ are the mean streamwise velocity and the distance from the wall in "wall" units. In this Letter we propose that this MDR profile is an edge solution of the Navier-Stokes equations (with an effective viscosity profile) beyond which no turbulent solutions exist. This insight rationalizes the universality of the MDR and provides a maximum principle which allows an ab initio calculation of the parameters in this law without any viscoelastic experimental input.