The elucidation of the universal scaling properties of equal-time and timedependent correlation f... more The elucidation of the universal scaling properties of equal-time and timedependent correlation functions in the vicinity of a critical point was one of the most important achievements of statistical mechanics over the past forty years. The analogous systematization of the power laws and associated exponents that govern the behaviour of structure functions in a turbulent fluid, or in a passive-scalar advected by such a fluid, is a major challenge in the areas of nonequilibrium statistical mechanics, fluid mechanics, and nonlinear dynamics. We develop here the systematics of the multiscaling of time-dependent structure functions for the case of decaying fluid and passive-scalar turbulence. The nature of multiscaling of time-dependent structure functions has been examined recently but only for the case of statistically steady turbulence. Does it have an analogue in the case of decaying turbulence, since timedependent structure functions must, in this case, depend on the origin of time t 0 at which we start our measurements? This question has not been addressed hitherto. We show here how to answer it in decaying fluid and passive-scalar turbulence. In particular, we propose suitable normalisations of time-dependent structure functions that eliminate their dependence on t 0 ; we demonstrate this analytically for the Kraichnan version of the passive-scalar problem and its shellmodel analogue and numerically for the GOY shell model for fluids and a shell-model version of the advection-diffusion equation. In these models we then analyse the normalised time-dependent structure functions for the case of decaying turbulence like their statistically steady counterparts . This requires a generalisation of the multifractal formalism[2] that finally yields the same bridge relations between dynamic and equal-time multiscaling exponents as for statistically steady turbulence . Studies show that, if dynamic
We use the mean-bacterial-velocity model to investigate the irreversibility of two-dimensional (2... more We use the mean-bacterial-velocity model to investigate the irreversibility of two-dimensional (2D) bacterial turbulence and to compare it with its 2D fluid-turbulence counterpart. We carry out extensive direct numerical simulations of Lagrangian tracer particles that are advected by the velocity field in this model. Our work uncovers an important, qualitative way in which irreversibility in bacterial turbulence is different from its fluid-turbulence counterpart: For large positive (or large but negative) values of the friction (or activity) parameter, the probability distribution functions of energy increments, along tracer trajectories, or the power are positively skewed; so irreversibility in bacterial turbulence can lead, on average, to particles gaining energy faster than they lose it, which is the exact opposite of what is observed for tracers in 2D fluid turbulence.
We use pseudospectral direct numerical simulations (DNSs) to solve the three-dimensional (3D) Hal... more We use pseudospectral direct numerical simulations (DNSs) to solve the three-dimensional (3D) Hall-Vinen-Bekharevich-Khalatnikov (HVBK) model of superfluid Helium. We then explore the statistical properties of inertial particles, in both coflow and counterflow superfluid turbulence (ST) in the 3D HVBK system; particle motion is governed by a generalization of the Maxey-Riley-Gatignol equations. We first characterize the anisotropy of counterflow ST by showing that there exist large vortical columns. The light particles show confined motion as they are attracted towards these columns and they form large clusters; by contrast, heavy particles are expelled from these vortical regions. We characterise the statistics of such inertial particles in 3D HVBK ST: (1) The mean angle Θ(τ), between particle positions, separated by the time lag τ , exhibits two different scaling regions in (a) dissipation and (b) inertial ranges, for different values of the parameters in our model; in particular, the value of Θ(τ), at large τ , depends on the magnitude of Uns. (2) The irreversibility of 3D HVBK turbulence is quantified by computing the statistics of energy increments for inertial particles. (3) The probability distribution function (PDF) of energy increments is of direct relevance to recent experimental studies of irreversibility in superfluid turbulence; we find, in agreement with these experiments, that, for counterflow ST, the skewness of this PDF is less pronounced than its counterparts for coflow ST or for classical-fluid turbulence. I.
We study the statistical properties of orientation and rotation dynamics of elliptical tracer par... more We study the statistical properties of orientation and rotation dynamics of elliptical tracer particles in two-dimensional, homogeneous and isotropic turbulence by direct numerical simulations. We consider both the cases in which the turbulent flow is generated by forcing at large and intermediate length scales. We show that the two cases are qualitatively different. For large-scale forcing, the spatial distribution of particle orientations forms large-scale structures, which are absent for intermediate-scale forcing. The alignment with the local directions of the flow is much weaker in the latter case than in the former. For intermediate-scale forcing, the statistics of rotation rates depends weakly on the Reynolds number and on the aspect ratio of particles. In contrast with what is observed in three-dimensional turbulence, in two dimensions the mean-square rotation rate increases as the aspect ratio increases.
We use pseudospectral direct numerical simulations (DNSs) to solve the three-dimensional (3D) Hal... more We use pseudospectral direct numerical simulations (DNSs) to solve the three-dimensional (3D) Hall-Vinen-Bekharevich-Khalatnikov (HVBK) model of superfluid Helium. We then explore the statistical properties of inertial particles, in both coflow and counterflow superfluid turbulence (ST) in the 3D HVBK system; particle motion is governed by a generalization of the Maxey-Riley-Gatignol equations. We first characterize the anisotropy of counterflow ST by showing that there exist large vortical columns. The light particles show confined motion as they are attracted towards these columns and they form large clusters; by contrast, heavy particles are expelled from these vortical regions. We characterise the statistics of such inertial particles in 3D HVBK ST: (1) The mean angle Θ(τ), between particle positions, separated by the time lag τ , exhibits two different scaling regions in (a) dissipation and (b) inertial ranges, for different values of the parameters in our model; in particular, the value of Θ(τ), at large τ , depends on the magnitude of Uns. (2) The irreversibility of 3D HVBK turbulence is quantified by computing the statistics of energy increments for inertial particles. (3) The probability distribution function (PDF) of energy increments is of direct relevance to recent experimental studies of irreversibility in superfluid turbulence; we find, in agreement with these experiments, that, for counterflow ST, the skewness of this PDF is less pronounced than its counterparts for coflow ST or for classical-fluid turbulence. I.
Disorganized electrical activity in the heart, which is often referred to as electrical turbulenc... more Disorganized electrical activity in the heart, which is often referred to as electrical turbulence, leads to sudden cardiac death. However, to what extent can this electrical turbulence be viewed as classical, fluid turbulence, which is considered as a central problem of great importance in modern physics? In this paper we examine, for the first time, the statistical properties of electrical turbulence in two-and three-dimensional generic models of cardiac tissue by using approaches employed in studies of classical turbulence. In particular, we investigate, via extensive direct numerical simulations, the statistical properties of spiral-and scroll-wave turbulence in two-and three-dimensional excitable media by using the Panfilov and the Aliev-Panfilov mathematical models for cardiac tissue. We use very large simulation domains, and perform state-of-the-art simulations on graphics processing units (GPUs), so that we can compare the statistical properties of spiral-and scroll-wave electrical turbulence with the statistical properties of homogeneous and isotropic two-and threedimensional fluid turbulence. We show that, once electrical-wave turbulence has been initiated, there is a forward cascade, in which spirals or scrolls form, interact, and break to yield a turbulent state that is statistically steady and, far away from boundaries, is statistically homogeneous and isotropic. For the transmembrane potential V and the slow recovery variable g, which define our models for cardiac tissue, we define EV (k) and Eg(k), the electrical-wave, spectral analogs of the fluid energy spectrum E(k) in fluid turbulence; and we show that EV (k) and Eg(k) are spread out over several decades in k. Thus, as in fluid turbulence, spiral-and scroll-wave turbulence involves a wide range of spatial scales. Furthermore, EV (k) and Eg(k) show approximate power laws, in some range of k; unlike fluid-turbulence, the exponents for these power laws cannot, so far, be determined as accurately as their fluid-turbulence counterparts. There are diffusive terms in the equations we consider, but they do not dissipate spiral or scroll waves completely because of the excitability of the medium, so no external forcing is required to maintain these states of spiral-or scroll-wave turbulence (this is unlike fluid turbulence that requires external forcing to reach a statistically steady state). We show that for spiral-or scroll-wave turbulence the dimensionless ratio L/λ is a convenient control parameter like the Reynolds number for fluid turbulence, where L is the linear size of the simulation domain and λ the wavelength of a plane wave in the excitable medium. We calculate several other statistical properties for spiral-and scroll-wave turbulence and, by comparing them with their fluid-turbulence counterparts, we show that, although spiral-and scroll-wave turbulence have some statistical properties like those of fluid turbulence, overall these types of turbulence are special and differ in important ways from fluid turbulence.
We use pseudospectral direct numerical simulations to solve the three-dimensional (3D) Hall–Vinen... more We use pseudospectral direct numerical simulations to solve the three-dimensional (3D) Hall–Vinen–Bekharevich–Khalatnikov (HVBK) model of superfluid helium. We then explore the statistical properties of inertial particles, in both coflow and counterflow superfluid turbulence (ST) in the 3D HVBK system; particle motion is governed by a generalization of the Maxey–Riley–Gatignol equations. We first characterize the anisotropy of counterflow ST by showing that there exist large vortical columns. The light particles show confined motion as they are attracted toward these columns, and they form large clusters; by contrast, heavy particles are expelled from these vortical regions. We characterize the statistics of such inertial particles in 3D HVBK ST: (1) The mean angle [Formula: see text] between particle positions, separated by the time lag τ, exhibits two different scaling regions in (a) dissipation and (b) inertial ranges, for different values of the parameters in our model; in parti...
We carry out extensive direct numerical simulations (DNSs) to investigate the interaction of acti... more We carry out extensive direct numerical simulations (DNSs) to investigate the interaction of active particles and fields in the two-dimensional (2D) Gross-Pitaevskii (GP) superfluid, in both simple and turbulent flows. The particles are active in the sense that they affect the superfluid even as they are affected by it. We tune the mass of the particles, which is an important control parameter. At the one-particle level, we show how light, neutral, and heavy particles move in the superfluid, when a constant external force acts on them; in particular, beyond a critical velocity, at which a vortex-antivortex pair is emitted, particle motion can be periodic or chaotic. We demonstrate that the interaction of a particle with vortices leads to dynamics that depends sensitively on the particle characteristics. We also demonstrate that assemblies of particles and vortices can have rich, and often turbulent spatiotemporal evolution. In particular, we consider the dynamics of the following illustrative initial configurations: (a) one particle placed in front of a translating vortex-antivortex pair; (b) two particles placed in front of a translating vortex-antivortex pair; (c) a single particle moving in the presence of counter-rotating vortex clusters; and (d) four particles in the presence of counter-rotating vortex clusters. We compare our work with earlier studies and examine its implications for recent experimental studies in superfluid Helium and Bose-Einstein condensates.
We present an overview of the statistical properties of turbulence in two-dimensional (2D) fluids... more 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.
We present the first direct-numerical-simulation study of the statistical properties of twodimens... more We present the first direct-numerical-simulation study of the statistical properties of twodimensional superfluid turbulence in the Hall-Vinen-Bekharevich-Khalatnikov two-fluid model. We show that both normal-fluid and superfluid energy spectra can exhibit two power-law regimes, the first associated with an inverse cascade of energy and the second with the forward cascade of enstrophy. We quantify the mutual-friction-induced alignment of normal and superfluid velocities by obtaining probability distribution functions of the angle between them and the ratio of their moduli. Our study leads to specific suggestions for experiments.
Particles of low velocity, travelling without dissipation in a superfluid, can interact and emit ... more Particles of low velocity, travelling without dissipation in a superfluid, can interact and emit sound when they collide. We propose a minimal model in which the equations of motion of the particles, including a short-range repulsive force, are self-consistently coupled with the Gross-Pitaevskii equation. We use this model to demonstrate the existence of an effective superfluid-mediated attractive interaction between the particles; and we study numerically the collisional dynamics of particles as a function of their incident kinetic energy and the length-scale of the repulsive force. We find a transition from almost elastic to completely inelastic (sticking) collisions as the parameters are tuned. We find that aggregation and clustering result from this sticking transition in multi-particle systems.
We obtain the probability distribution functions (PDFs) of the time that a Lagrangian tracer or a... more We obtain the probability distribution functions (PDFs) of the time that a Lagrangian tracer or a heavy inertial particle spends in vortical or strain-dominated regions of a turbulent flow, by carrying out direct numerical simulations of such particles advected by statistically steady, homogeneous, and isotropic turbulence in the forced, three-dimensional, incompressible Navier-Stokes equation. We use the two invariants, Q and R, of the velocity-gradient tensor to distinguish between vortical and strain-dominated regions of the flow and partition the Q-R plane into four different regions depending on the topology of the flow; out of these four regions two correspond to vorticity-dominated regions of the flow and two correspond to strain-dominated ones. We obtain Q and R along the trajectories of tracers and heavy inertial particles and find out the time t_{pers} for which they remain in one of the four regions of the Q-R plane. We find that the PDFs of t_{pers} display exponentially...
We study two-dimensional (2D) binary-fluid turbulence by carrying out an extensive direct numeric... more 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 parameterϕ, and (b) the formation of an inverse-energy-cascade regime in the energy spectrumE(k), in which energy cascades towards wave numberskthat are smaller than the energy-injection scalekin jin the turbulent fluid. We show that the Cahn-Hilliard-Navier-Stokes coupling leads to an arrest of phase separation at a length scaleLc, which we evaluate fromS(k), the spectrum of the fluctuations ofϕ. We demonstrate that (a)Lc ~ LH, the Hinze scale that follows from balancing inertial and interfacial-tension forces, and (b)Lcis independent, within error bars, of the diffusivityD...
We study the challenging problem of the advection of an active, deformable, finite-size droplet b... more 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 wavenumber k is large; this enhancement leads to dissipation reduction.
A lattice-gas model is constructed for oil-water-surfactant mixtures. The phase diagram of this m... more A lattice-gas model is constructed for oil-water-surfactant mixtures. The phase diagram of this model is obtained by using mean-field theory and Monte Carlo simulations aided by lowtemperature expansions. Microstructures, structure factors, and mean droplet lifetimes are also determined in some phases. Both two and three dimensions are studied, the former in more detail than the latter. It is shown that it is natural to interpret the paramagnetic phase in our model as a microemulsion. Our model is found to exhibit various properties that are in qualitative agreement with experimental observations of oil-water-surfactant mixtures: (1) two-and three-phase coexistence occurs between oil-rich, water-rich, and microemulsion phases along first-order phase boundaries or a triple line in certain regions of the phase diagram of our model; (2) the triple line, which ends in a tricritical point, is short and this leads to low oil-microemulsion and water-microemulsion interfacial tensions; (3) microstructures (including bicontinuous ones in three dimensions) and structure factors are similar to some experimental ones; (4) droplets in our microemulsion phase are long lived like their experimental counterparts; (5) long-lived, metastable phases, including long-period, lamellar, and glasslike phases, appear at low temperatures. The limitations of our model are discussed. Our study is compared with other studies of models of oil-water-surfactant mixtures.
We present a detailed study of the statistical steady states of a model for CO oxidation on Pt͑11... more We present a detailed study of the statistical steady states of a model for CO oxidation on Pt͑110͒ proposed by Bär and co-workers. We show that the stability diagram of this model depends sensitively on the boundary conditions. We elucidate several novel properties of a state with meandering spirals ͑M͒ briefly mentioned by Bär and co-workers. ͑1͒ We show that, with periodic boundary conditions, M is the state MP, a binary mixture displaying a coexistence of quasiperiodically rotating spirals and chaotically moving pointlike defects. We show that the transition from MP to the turbulent state T1 is continuous; the transition line marks the locus where the two phases cease to be distinct. ͑2͒ With Neumann boundary conditions M is the state MN, a single quasiperiodically rotating spiral. We show that the MN-T1 transition is discontinuous or first order. We also characterize the transitions from MP and MN to the state S, which has quasiperiodically rotating spirals. We also propose qualitative mechanisms for these transitions.
We present a detailed direct numerical simulation of statistically steady, homogeneous, isotropic... more We present a detailed direct numerical simulation of statistically steady, homogeneous, isotropic, two-dimensional magnetohydrodynamic (2D MHD) turbulence. Our study concentrates on the inverse cascade of the magnetic vector potential. We examine the dependence of the statistical properties of such turbulence on dissipation and friction coefficients. We extend earlier work significantly by calculating fluid and magnetic spectra, probability distribution functions (PDFs) of the velocity, magnetic, vorticity, current, stream-function, and magnetic-vector-potential fields and their increments. We quantify the deviations of these PDFs from Gaussian ones by computing their flatnesses and hyperflatnesses. We also present PDFs of the Okubo-Weiss parameter, which distinguishes between vortical and extensional flow regions, and its magnetic analog. We show that the hyperflatnesses of PDFs of the increments of the stream-function and the magnetic vector potential exhibit significant scale dependence and we examine the implication of this for the multiscaling of structure functions. We compare our results with those of earlier studies.
We generalize the driven diffusive lattice gas model by using a combination of Kawasaki and Glaub... more We generalize the driven diffusive lattice gas model by using a combination of Kawasaki and Glauber dynamics. We find via Monte Carlo simulations and perturbation studies that the simplest possible generalization of the equivalence of the canonical and grand-canonical ensembles, which holds in equilibrium, does not apply for this class of nonequilibrium systems.
Physical review. E, Statistical, nonlinear, and soft matter physics, 2014
We study the statistical properties of orientation and rotation dynamics of elliptical tracer par... more We study the statistical properties of orientation and rotation dynamics of elliptical tracer particles in two-dimensional, homogeneous, and isotropic turbulence by direct numerical simulations. We consider both the cases in which the turbulent flow is generated by forcing at large and intermediate length scales. We show that the two cases are qualitatively different. For large-scale forcing, the spatial distribution of particle orientations forms large-scale structures, which are absent for intermediate-scale forcing. The alignment with the local directions of the flow is much weaker in the latter case than in the former. For intermediate-scale forcing, the statistics of rotation rates depends weakly on the Reynolds number and on the aspect ratio of particles. In contrast with what is observed in three-dimensional turbulence, in two dimensions the mean-square rotation rate increases as the aspect ratio increases.
We carry out an extensive numerical study of the dynamics of spiral waves of electrical activatio... more We carry out an extensive numerical study of the dynamics of spiral waves of electrical activation, in the presence of periodic deformation (PD) in two-dimensional simulation domains, in the biophysically realistic mathematical models of human ventricular tissue due to (a) ten-Tusscher and Panfilov (the TP06 model) and (b) ten-Tusscher, Noble, Noble, and Panfilov (the TNNP04 model). We first consider simulations in cable-type domains, in which we calculate the conduction velocity θ and the wavelength λ of a plane wave; we show that PD leads to a periodic, spatial modulation of θ and a temporally periodic modulation of λ; both these modulations depend on the amplitude and frequency of the PD. We then examine three types of initial conditions for both TP06 and TNNP04 models and show that the imposition of PD leads to a rich variety of spatiotemporal patterns in the transmembrane potential including states with a single rotating spiral (RS) wave, a spiral-turbulence (ST) state with a s...
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Papers by Rahul Pandit