Binary-fluid turbulence: Signatures of multifractal droplet dynamics and dissipation reduction

Novel turbulence and coarsening arrest in active-scalar fluids

We uncover a new type of turbulence - activity-induced homogeneous and isotropic turbulence - in a model that has been employed to investigate motility-induced phase separation (MIPS) in a system of microswimmers. The active Cahn-Hilliard-Navier-Stokes (CHNS) equations, also called active model H, provide a natural theoretical framework for our study. In this CHNS model, a single scalar order parameter ϕ, positive (negative) in regions of high (low) microswimmer density, is coupled with the velocity field u. The activity of the microswimmers is governed by an activity parameter ζ that is positive for extensile swimmers and negative for contractile swimmers. With extensile swimmers, this system undergoes complete phase separation, which is similar to that in binary-fluid mixtures. By carrying out pseudospectral direct numerical simulations (DNSs), we show, for the first time, that (a) this model develops an emergent nonequilibrium, but statistically steady, state (NESS) of active turbulence, for the case of contractile swimmers, if ζ is sufficiently large and negative, and (b) this turbulence arrests the phase separation. We quantify this suppression by showing how the coarsening-arrest length scale does not grow indefinitely, with time t, but saturates at a finite value at large times. We characterise the statistical properties of this active-scalar turbulence by employing energy spectra and fluxes and the spectrum of ϕ. For sufficiently high Reynolds numbers, the energy spectrum (k) displays an inertial range, with a power-law dependence on the wavenumber k. We demonstrate that, in this range, the flux Π(k) assumes a nearly constant, negative value, which indicates that the system shows an inverse cascade of energy, even though energy injection occurs over a wide range of wavenumbers in our active-CHNS model.

Multifractal Conductance Fluctuations in High-Mobility Graphene in the Integer Quantum Hall Regime

We present the first experimental evidence for the multifractality of a transport property at a topological phase transition. In particular, we show that conductance fluctuations display multifractality at the integer quantum Hall plateau-to-plateau transitions in high-mobility mesoscopic graphene devices. The multifractality gets rapidly suppressed as the chemical potential moves away from these critical points. Our combination of experimental study and multifractal analysis provides a novel method for probing the criticality of wave functions at phase transitions in mesoscopic systems, and quantum criticality in several condensed-matter systems.

Exact relations for energy transfer in simple and active binary fluid turbulence

Inertial range energy transfer in three-dimensional fully developed binary fluid turbulence is studied under the assumption of statistical homogeneity. Using two-point statistics, exact relations corresponding to the energy cascade are derived in terms of (i) two-point increments and (ii) two-point correlators. Despite having some apparent resemblances, the exact relation in binary fluid turbulence is found to be different from that of the incompressible magnetohydrodynamic turbulence [H. Politano and A. Pouquet, Geophys. Res. Lett. 25, 273 (1998)]0094-827610.1029/97GL03642. Besides the usual direct cascade of energy, under certain situations, an inverse cascade of energy is also speculated depending upon the strength of the activity parameter and the interplay between the two-point increments of the fluid velocity and the composition gradient fields. An alternative form of the exact relation is also derived in terms of the "upsilon" variables and a subsequent phenomenology is proposed predicting a k^{-3/2} law for the turbulent energy spectrum.

Formation and evolution of target patterns in Cahn-Hilliard flows

We study the evolution of the concentration field in a single eddy in the two-dimensional (2D) Cahn-Hilliard system to better understand scalar mixing processes in that system. This study extends investigations of the classic studies of flux expulsion in 2D magnetohydrodynamics and homogenization of potential vorticity in 2D fluids. Simulation results show that there are three stages in the evolution: (A) formation of a "jelly roll" pattern, for which the concentration field is constant along spirals; (B) a change in isoconcentration contour topology; and (C) formation of a target pattern, for which the isoconcentration contours follow concentric annuli. In the final target pattern stage, the isoconcentration bands align with stream lines. The results indicate that the target pattern is a metastable state. The band merger process continues on a time scale exponentially long relative to the eddy turnover time. The band merger process resembles step merger in drift-ZF staircases; this is characteristic of the long-time evolution of phase-separated patterns described by the Cahn-Hilliard equation.

Regularity criterion for solutions of the three-dimensional Cahn-Hilliard-Navier-Stokes equations and associated computations

We consider the three-dimensional (3D) Cahn-Hilliard equations coupled to, and driven by, the forced, incompressible 3D Navier-Stokes equations. The combination, known as the Cahn-Hilliard-Navier-Stokes (CHNS) equations, is used in statistical mechanics to model the motion of a binary fluid. The potential development of singularities (blow-up) in the contours of the order parameter ϕ is an open problem. To address this we have proved a theorem that closely mimics the Beale-Kato-Majda theorem for the 3D incompressible Euler equations [J. T. Beale, T. Kato, and A. J. Majda, Commun. Math. Phys. 94, 61 (1984)CMPHAY0010-361610.1007/BF01212349]. By taking an L^{∞} norm of the energy of the full binary system, designated as E_{∞}, we have shown that ∫_{0}^{t}E_{∞}(τ)dτ governs the regularity of solutions of the full 3D system. Our direct numerical simulations (DNSs) of the 3D CHNS equations for (a) a gravity-driven Rayleigh Taylor instability and (b) a constant-energy-injection forcing, with 128^{3} to 512^{3} collocation points and over the duration of our DNSs confirm that E_{∞} remains bounded as far as our computations allow.