Impact of the Peterlin approximation on polymer dynamics in turbulent flows

Josephson-Anderson relation as diagnostic of turbulent drag reduction by polymers

The detailed Josephson-Anderson relation, which equates instantaneously the volume-integrated vorticity flux and the work by pressure drop, has been the key to drag reduction in superconductors and superfluids. We employ a classical version of this relation to investigate the dynamics of polymer drag-reduced channel flows, particularly in the high-extent drag reduction (HDR) regime which is known to exhibit strong space-time intermittency. We show that high drag is not created instantaneously by near-wall coherent vortex structures as assumed in prior works. These predominantly spanwise near-wall vortex structures can produce a net "up-gradient" flux of vorticity toward the wall, which instead reduces instantaneous drag. Increase of wall vorticity and skin friction due to this up-gradient flux occurs after an apparent lag of several advection times, increasing with the Weissenberg number. This increasing lag appears to be due to polymer damping of up-gradient nonlinear vorticity transport that arises from large-scale eddies in the logarithmic layer. The relatively greater polymer damping of down-gradient transport due to small-scale eddies results in lower net vorticity flux and hence lower drag. The Josephson-Anderson relation thus provides an exact tool to diagnose the mechanism of polymer drag reduction in terms of vorticity dynamics and it explains also prior puzzling observations on transient drag reduction, as for centerline-release experiments in pipe flow.

Coexistence of Active and Hydrodynamic Turbulence in Two-Dimensional Active Nematics

In active nematic liquid crystals, activity is able to drive chaotic spatiotemporal flows referred to as active turbulence. Active turbulence has been characterized through theoretical and experimental work as a low Reynolds number phenomenon. We show that, in two dimensions, the active forcing alone is able to trigger hydrodynamic turbulence leading to the coexistence of active and inertial turbulence. This type of flow develops for sufficiently active and extensile flow-aligning nematics. We observe that the combined effect of an extensile nematic and large values of the flow-aligning parameter leads to a broadening of the elastic energy spectrum that promotes a growth of kinetic energy able to trigger an inverse energy cascade.

Drag Reduction in Turbulent Wall-Bounded Flows of Realistic Polymer Solutions

Suspensions of DNA macromolecules (0.8 wppm, 60 kbp), modeled as finitely extensible nonlinear elastic dumbbells coupled to the Newtonian fluid, show drag reduction up to 27% at friction Reynolds number 180, saturating at the previously unachieved Weissenberg number ≃10^{4}. At a large Weissenberg number, the drag reduction is entirely induced by the fully stretched polymers, as confirmed by the extensional viscosity field. The polymer extension is strongly non-Gaussian, in contrast to the assumptions of classical viscoelastic models.

Effect of internal friction on the coil-stretch transition in turbulent flows

A polymer in a turbulent flow undergoes the coil-stretch transition when the Weissenberg number, i.e. the product of the Lyapunov exponent of the flow and the relaxation time of the polymer, surpasses a critical value. The effect of internal friction on the transition is studied by means of Brownian dynamics simulations of the elastic dumbbell model in a homogeneous and isotropic, incompressible, turbulent flow and analytical calculations for a stochastic velocity gradient. The results are explained by adapting the large deviations theory of Balkovsky et al. [Phys. Rev. Lett., 2000, 84, 4765] to an elastic dumbbell with internal viscosity. In turbulent flows, a distinctive feature of the probability distribution of polymer extensions is its power-law behaviour for extensions greater than the equilibrium length and smaller than the contour length. It is shown that although internal friction does not modify the critical Weissenberg number for the coil-stretch transition, it makes the slope of the probability distribution of the extension steeper, thus rendering the transition sharper. Internal friction therefore provides a possible explanation for the steepness of the distribution of polymer extensions observed in experiments at large Weissenberg numbers.

Essentially Entropic Lattice Boltzmann Model

The entropic lattice Boltzmann model (ELBM), a discrete space-time kinetic theory for hydrodynamics, ensures nonlinear stability via the discrete time version of the second law of thermodynamics (the H theorem). Compliance with the H theorem is numerically enforced in this methodology and involves a search for the maximal discrete path length corresponding to the zero dissipation state by iteratively solving a nonlinear equation. We demonstrate that an exact solution for the path length can be obtained by assuming a natural criterion of negative entropy change, thereby reducing the problem to solving an inequality. This inequality is solved by creating a new framework for construction of Padé approximants via quadrature on appropriate convex function. This exact solution also resolves the issue of indeterminacy in case of nonexistence of the entropic involution step. Since our formulation is devoid of complex mathematical library functions, the computational cost is drastically reduced. To illustrate this, we have simulated a model setup of flow over the NACA-0012 airfoil at a Reynolds number of 2.88×10^{6}.

Polymer stretching in the inertial range of turbulence

We study the deformation of flexible polymers whose contour length lies in the inertial range of a homogeneous and isotropic turbulent flow. By using the elastic dumbbell model and a stochastic velocity field with nonsmooth spatial correlations, we obtain the probability density function of the extension as a function of the Weissenberg number and of the scaling exponent of the velocity structure functions. In a spatially rough flow, as in the inertial range of turbulence, the statistics of polymer stretching differs from that observed in laminar flows or in smooth chaotic flows. In particular, the probability distribution of polymer extensions decays as a stretched exponential, and the most probable extension grows as a power law of the Weissenberg number. Furthermore, the ability of the flow to stretch polymers weakens as the flow becomes rougher in space.