Nairita Pal

An overview of the statistical properties of two-dimensional turbulence in fluids with particles, conducting fluids, fluids with polymer additives, binary-fluid mixtures, and superfluids

We present an overview of the statistical properties of turbulence in two-dimensional (2D) fluids. After a brief recapitulation of well-known results for statistically homogeneous and isotropic 2D fluid turbulence, we give an overview of recent progress in this field for such 2D turbulence in conducting fluids, fluids with polymer additives, binary-fluid mixtures, and superfluids; we also discuss the statistical properties of particles advected by 2D turbulent fluids.

Two-dimensional Turbulence in Symmetric Binary-Fluid Mixtures: Coarsening Arrest by the Inverse Cascade

We study two-dimensional (2D) binary-fluid turbulence by carrying out an extensive direct numerical simulation (DNS) of the forced, statistically steady turbulence in the coupled Cahn-Hilliard and Navier-Stokes equations. In the absence of any coupling, we choose parameters that lead (a) to spinodal decomposition and domain growth, which is characterized by the spatiotemporal evolution of the Cahn-Hilliard order parameter ϕ, and (b) the formation of an inverse-energy-cascade regime in the energy spectrum E(k), in which energy cascades towards wave numbers k that are smaller than the energy-injection scale kin j in the turbulent fluid. We show that the Cahn-Hilliard-Navier-Stokes coupling leads to an arrest of phase separation at a length scale Lc, which we evaluate from S(k), the spectrum of the fluctuations of ϕ. We demonstrate that (a) Lc ~ LH, the Hinze scale that follows from balancing inertial and interfacial-tension forces, and (b) Lc is independent, within error bars, of the diffusivity D. We elucidate how this coupling modifies E(k) by blocking the inverse energy cascade at a wavenumber kc, which we show is ≃2π/Lc. We compare our work with earlier studies of this problem.

Exotic multifractal conductance fluctuations in graphene

In quantum systems, signatures of multifractality are rare. They have been found only in the multiscaling of eigenfunctions at critical points. Here we demonstrate multifractality in the magnetic field-induced universal conductance fluctuations of the conductance in a quantum condensed matter system, namely, high-mobility single-layer graphene field-effect transistors. This multifractality decreases as the temperature increases or as doping moves the system away from the Dirac point. Our measurements and analysis present evidence for an incipient Anderson-localization near the Dirac point as the most plausible cause for this multifractality. Our experiments suggest that multifractality in the scaling behavior of local eigenfunctions are reflected in macroscopic transport coefficients. We conjecture that an incipient Anderson-localization transition may be the origin of this multifractality. It is possible that multifractality is ubiquitous in transport properties of low-dimensional systems. Indeed, our work suggests that we should look for multifractality in transport in other low-dimensional quantum condensed-matter systems.Multifractality is ubiquitous in classical systems but rare in quantum ones. Here the authors present observations demonstrating that universal conductance fluctuations in high-mobility single-layer graphene field-effect transistors are multifractal and may arise from Anderson-localization.

Binary-Fluid Turbulence: Signatures of Multifractal Droplet Dynamics and Dissipation Reduction

We study the challenging problem of the advection of an active, deformable, finite-size droplet by a turbulent flow via a simulation of the coupled Cahn-Hilliard-Navier-Stokes (CHNS) equations. In these equations, the droplet has a natural two-way coupling to the background fluid. We show that the probability distribution function of the droplet center of mass acceleration components exhibit wide, non-Gaussian tails, which are consistent with the predictions based on pressure spectra. We also show that the droplet deformation displays multifractal dynamics. Our study reveals that the presence of the droplet enhances the energy spectrum E(k), when the wave number k is large; this enhancement leads to dissipation reduction.

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.

Publisher's Note: Regularity criterion for solutions of the three-dimensional Cahn-Hilliard-Navier-Stokes equations and associated computations [Phys. Rev. E 94, 063103 (2016)]

This corrects the article DOI: 10.1103/PhysRevE.94.063103.

The role of BKM-type theorems in 3D Euler, Navier–Stokes and Cahn–Hilliard–Navier–Stokes analysis

Abstract The Beale–Kato–Majda theorem contains a single criterion that controls the behaviour of solutions of the 3 D incompressible Euler equations. Versions of this theorem are discussed in terms of the regularity issues surrounding the 3 D incompressible Euler and Navier–Stokes equations together with a phase-field model for the statistical mechanics of binary mixtures called the 3 D Cahn–Hilliard–Navier–Stokes (CHNS) equations. A theorem of BKM-type is established for the CHNS equations for the full parameter range. Moreover, for this latter set, it is shown that there exists a Reynolds number and a bound on the energy-dissipation rate that, remarkably, reproduces the R e 3 ∕ 4 upper bound on the inverse Kolmogorov length normally associated with the Navier–Stokes equations alone. An alternative length-scale is introduced and discussed, together with a set of pseudo-spectral computations on a 12 8 3 grid.