Effectiveness of stirring measures in an axisymmetric rotating annulus flow

Lagrangian stirring in a thermally driven rotating annulus is investigated numerically using a Navier-Stokes model and a second order Runge-Kutta integration routine. The stirring properties are investigated using finite scale Lyapunov exponents, Lagrangian coherent structures and a leaking method. The ability of these measures to identify transport barriers, regions of well and poorly stirred flow, and stable and unstable manifolds is investigated, as well as the stirring properties of the annulus flow. It is found that finite scale Lyapunov exponents characterise the stirring properties of flows occurring in the rotating annulus more efficiently than the leaking method or Lagrangian coherent structures. The strength of the stirring varies monotonically with thermal forcing amplitude, but non-monotonically with forcing frequency. The flows investigated are axisymmetric (i.e. two dimensional) and time dependent.     

Multifractal dimension of Lagrangian turbulence

We report experimental measurements of the Lagrangian multifractal dimension spectrum in an intensely turbulent laboratory water flow by the optical tracking of tracer particles. The Legendre transform of the measured spectrum is compared with measurements of the scaling exponents of the Lagrangian velocity structure functions, and excellent agreement between the two measurements is found, in support of the multifractal picture of turbulence. These measurements are compared with three model dimension spectra. When the nonexistence of structure functions of order less than -1 is accounted for, the models are shown to agree well with the measured spectrum.     

Stretching in a model of a turbulent flow

Using a multi-scaled, chaotic flow known as the KS model of turbulence [J.C.H. Fung, J.C.R. Hunt, A. Malik, R.J. Perkins, Kinematic simulation of homogeneous turbulence by unsteady random fourier modes, J. Fluid Mech. 236 (1992) 281-318], we investigate the dependence of Lyapunov exponents on various characteristics of the flow. We show that the KS model yields a power law relation between the Reynolds number and the maximum Lyapunov exponent, which is similar to that for a turbulent flow with the same energy spectrum. Our results show that the Lyapunov exponents are sensitive to the advection of small eddies by large eddies, which can be explained by considering the Lagrangian correlation time of the smallest scales. We also relate the number of stagnation points within a flow to the maximum Lyapunov exponent, and suggest a linear dependence between the two characteristics.     

Lagrangian chaos and Eulerian chaos in shear flow dynamics

Shear flow dynamics described by the two-dimensional incompressible Navier-Stokes equations is studied for a one-dimensional equilibrium vorticity profile having two minima. These lead to two linear Kelvin-Helmholtz instabilities; the resulting nonlinear waves corresponding to the two minima have different phase velocities. The nonlinear behavior is studied as a function of two parameters, the Reynolds number and a parameter lambda specifying the width of the minima in the vorticity profile. For parameters such that the instabilities grow to a sufficient level, there is Lagrangian chaos, leading to mixing of vorticity, i.e., momentum transport, between the chains of vortices or cat's eyes. Lagrangian chaos is quantified by plotting the finite time Lyapunov exponents on a grid of initial points, and by the probability distribution of these exponents. For moderate values of lambda, there is Lagrangian chaos everywhere except near the centers of the vortices and near the boundaries, and there are competing effects of homogenization of vorticity and formation of structures associated with secondary resonances. For smaller values of lambda Lagrangian chaos occurs in the regions in the centers of the vortices, and the Eulerian behavior of the flow undergoes bifurcations leading to Eulerian chaos, as measured by the time series of several Galilean invariant quantities. A discussion of Lagrangian chaos and its relation to Eulerian chaos is given.(c) 2001 American Institute of Physics.     

Relating Lagrangian passive scalar scaling exponents to Eulerian scaling exponents in turbulence

Intermittency is a basic feature of fully developed turbulence, for both velocity and passive scalars. Intermittency is classically characterized by Eulerian scaling exponent of structure functions. The same approach can be used in a Lagrangian framework to characterize the temporal intermittency of the velocity and passive scalar concentration of a an element of fluid advected by a turbulent intermittent field. Here we focus on Lagrangian passive scalar scaling exponents, and discuss their possible links with Eulerian passive scalar and mixed velocity-passive scalar structure functions. We provide different transformations between these scaling exponents, associated to different transformations linking space and time scales. We obtain four new explicit relations. Experimental data are needed to test these predictions for Lagrangian passive scalar scaling exponents.     

Towards a geometrical classification of statistical conservation laws in turbulent advection

The paper revisits the compressible Kraichnan model of turbulent advection in order to derive explicit quantitative relations between scaling exponents and Lagrangian particle configuration geometry.     

Two-dimensional turbulence of dilute polymer solutions

We investigate theoretically and numerically the effect of polymer additives on two-dimensional turbulence by means of a viscoelastic model. We provide compelling evidence that, at vanishingly small concentrations, such that the polymers are passively transported, the probability distribution of polymer elongation has a power law tail: Its slope is related to the statistics of finite-time Lyapunov exponents of the flow, in quantitative agreement with theoretical predictions. We show that at finite concentrations and sufficiently large elasticity the polymers react on the flow with manifold consequences: Velocity fluctuations are drastically depleted, as observed in soap film experiments; the velocity statistics becomes strongly intermittent; the distribution of finite-time Lyapunov exponents shifts to lower values, signaling the reduction of Lagrangian chaos.     

Superstatistical mechanics of tracer-particle motions in turbulence

The Lagrangian stochastic model of Reynolds [Phys. Fluids 15, L1-4 (2003)]] for the accelerations of fluid particles in turbulence is shown to predict precisely the observed Reynolds-number dependency of the distribution of Lagrangian accelerations and the exponents characterizing the observed extended self-similarity scaling of the Lagrangian velocity structure functions. Departures from superstatistics of the log-normal kind are accounted for and their impact upon model predictions is quantified.     

Kraichnan Flow in a Square: An Example of Integrable Chaos

The Kraichnan flow provides an example of a random dynamical system accessible to an exact analysis. We study the evolution of the infinitesimal separation between two Lagrangian trajectories of the flow. Its long-time asymptotics is reflected in the large deviation regime of the statistics of stretching exponents. Whereas in the flow that is isotropic at small scales the distribution of such multiplicative large deviations is Gaussian, this does not have to be the case in the presence of an anisotropy. We analyze in detail the flow in a two-dimensional periodic square where the anisotropy generally persists at small scales. The calculation of the large deviation rate function of the stretching exponents reduces in this case to the study of the ground state energy of an integrable periodic Schrödinger operator of the Lamé type. The underlying integrability permits to explicitly exhibit the non-Gaussianity of the multiplicative large deviations and to analyze the time-scales at which the large deviation regime sets in. In particular, we indicate how the divergence of some of those time scales when the two Lyapunov exponents become close allows a discontinuity of the large deviation rate function in the parameters of the flow. The analysis of the two-dimensional anisotropic flow permits to identify the general scenario for the appearance of multiplicative large deviations together with the restrictions on its applicability.     

Lagrangian Velocity Fluctuations in Fully Developed Turbulence: Scaling, Intermittency, and Dynamics

New aspects of turbulence are uncovered if one considers the flow motion from the perspective of a fluid particle (known as the Lagrangian approach) rather than in terms of a velocity field (the Eulerian viewpoint). Using a new experimental technique, based on the scattering of ultrasound, we have obtained a direct measurement of particle velocities, resolved at all scales, in a fully turbulent flow. We find that the Lagrangian velocity autocorrelation function and the Lagrangian time spectrum are in agreement with the Kolmogorov K41 phenomenology. Intermittency corrections are observed and we give a measurement of the Lagrangian structure function exponents. They are more intermittent than the corresponding Eulerian exponents. We also propose a novel analysis of intermittency in turbulence: our measurement enables us to study it from a dynamical point of view. We thus analyze the Lagrangian velocity fluctuations in the framework of random walks. We find experimentally that the elementary steps in the walk have random uncorrelated directions but a magnitude that displays extremely long-range correlations in time. Theoretically, we study a Langevin equation that incorporates these features and we show that the resulting dynamics accounts for the observed one-point and two-point statistical properties of the Lagrangian velocity fluctuations. Our approach connects the intermittent statistical nature of turbulence to the dynamics of the flow.     

Flow and magnetic structures in a kinematic ABC-dynamo

Dynamo theory describes the magnetic field induced by the rotating, convecting and electrically conducting fluid in a celestial body. The classical ABC-flow model represents fast dynamo action, required to sustain such a magnetic field. In this letter,Lagrangian coherent structures(LCSs) in the ABC-flow are detected through Finite-time Lyapunov exponents(FTLE). The flow skeleton is identified by extracting intersections between repelling and attracting LCSs. For the case A = B = C = 1, the skeleton structures are made up from lines connecting two different types of stagnation points in the ABC-flow. The corresponding kinematic ABC-dynamo problem is solved using a spectral method, and the distribution of cigar-like magnetic structures visualized.Inherent links are found to exist between LCSs in the ABC-flow and induced magnetic structures, which provides insight into the mechanism behind the ABC-dynamo.     

Lyapunov exponents and information dimension of the mass distribution in turbulent compressible flows

Turbulent density fluctuations in isothermal highly compressible turbulent flows are highly clumped and can be quantified by the scaling properties of powers of the mass distribution. This Eulerian quantity can be related to Lagrangian properties of the system given by the Lyapunov exponents of tracer particles advected with the flow. Using highly resolved numerical simulations, we show that the Kaplan-Yorke conjecture holds within numerical uncertainties.     

Small-scale vorticity filaments and structure functions of the developed turbulence

We develop a theory of turbulence based on the Navier-Stokes equation, without using dimensional or phenomenological considerations. A small scale vortex filament is the main element of the theory. The theory allows to obtain the scaling law and to calculate the scaling exponents of Lagrangian and Eulerian velocity structure functions in the inertial range. The obtained results are shown to be in very good agreement with numerical simulations and experimental data. The introduction of stochasticity into the equations and derivation of scaling exponents are discussed in details. A weak dependence on statistical propositions is demonstrated. The relation of the theory to the multifractal model is discussed.     

Avalanche Dynamics in Quenched Random Directed Sandpile Models

We numerically investigate the quenched random directed sandpile models which are local, conservative and Abelian. A local flow balance between the outflow of grains during a single toppling at a site and the total number of grains flowing into the same site plays an important role when all the nearest-neighbouring sites of the above-mentioned site topple for once. The quenched model has the same critical exponents with the Abelian deterministic directed sandpile model when the local flow balance exists, otherwise the critical exponents of this quenched model and the annealed Abelian random directed sandpile model are the same. These results indicate that the presence or absence of this local flow balance determines the universality class of the Abelian directed sandpile model.     

Exponential basis functions in solution of incompressible fluid problems with moving free surfaces

In this paper, a new simple meshless method is presented for the solution of incompressible inviscid fluid flow problems with moving boundaries. A Lagrangian formulation established on pressure, as a potential equation, is employed. In this method, the approximate solution is expressed by a linear combination of exponential basis functions (EBFs), with complex-valued exponents, satisfying the governing equation. Constant coefficients of the solution series are evaluated through point collocation on the domain boundaries via a complex discrete transformation technique. The numerical solution is performed in a time marching approach using an implicit algorithm. In each time step, the governing equation is solved at the beginning and the end of the step, with the aid of an intermediate geometry. The use of EBFs helps to find boundary velocities with high accuracy leading to a precise geometry updating. The developed Lagrangian meshless algorithm is applied to variety of linear and nonlinear benchmark problems. Non-linear sloshing fluids in rigid rectangular two-dimensional basins are particularly addressed.     

Path Lengths in Turbulence

By tracking tracer particles at high speeds and for long times, we study the geometric statistics of Lagrangian trajectories in an intensely turbulent laboratory flow. In particular, we consider the distinction between the displacement of particles from their initial positions and the total distance they travel. The difference of these two quantities shows power-law scaling in the inertial range. By comparing them with simulations of a chaotic but non-turbulent flow and a Lagrangian Stochastic model, we suggest that our results are a signature of turbulence.     

Lagrangian Dispersion in Gaussian Self-Similar Velocity Ensembles

We analyze the Lagrangian flow in a family of simple Gaussian scale-invariant velocity ensembles that exhibit both spatial roughness and temporal correlations. We argue that the behavior of the Lagrangian dispersion of pairs of fluid particles in such models is determined by the scale dependence of the ratio between the correlation time of velocity differences and the eddy turnover time. For a non-trivial scale dependence, the asymptotic regimes of the dispersion at small and large scales are described by the models with either rapidly decorrelating or frozen velocities. In contrast to the decorrelated case, known as the Kraichnan model and exhibiting Lagrangian flows with deterministic or stochastic trajectories, fast separating or trapped together, the frozen model is poorly understood. We examine the pair dispersion behavior in its simplest, one-dimensional version, reinforcing analytic arguments by numerical analysis. The collected information about the pair dispersion statistics in the limiting models allows to partially predict the extent of different phases of the Lagrangian flow in the model with time-correlated velocities.     

Lagrangian chaos and the effect of drag on the enstrophy cascade in two-dimensional turbulence

We investigate the effect of drag force on the enstrophy cascade of two-dimensional Navier-Stokes turbulence. We find a power law decrease of the energy wave number (k) spectrum that is faster than the classical (no-drag) prediction of k(-3). It is shown that the enstrophy cascade with drag can be analyzed by making use of a previous theory for finite lifetime passive scalars advected by a Lagrangian chaotic fluid flow. Using this we relate the power law exponent of the energy wave number spectrum to the distribution of finite time Lyapunov exponents and the drag coefficient.     

Characterization of nondiffusive transport in plasma turbulence via a novel Lagrangian method

A novel method to probe and characterize the nature of the transport of passive scalars carried out by a turbulent flow is introduced. It requires the determination of two exponents which encapsulate the statistical and correlation properties of the component of interest of the Lagrangian velocities of the flow. Numerical simulations of a magnetically confined, near-critical turbulent plasma, known to exhibit superdiffusive radial transport, are used to illustrate the method. It is shown that the method can easily detect the change in the dynamics of the radial transport that takes place after adding to the simulations a (subdominant) diffusive channel of tunable strength.