Alain Pumir

A numerical study of the mixing of a passive scalar in three dimensions in the presence of a mean gradient

The mixing of a passive scalar in the presence of a mean gradient is studied in three dimensions by direct numerical simulations. The driving velocity field is either a solution of the three‐dimensional (3‐D) Navier–Stokes equations, at a microscale Reynolds number in between 20 and 70, and with a Prandtl number varying between 1/8 and 1, or a solution of the Euler equation restricted to a shell of wave numbers, which formally corresponds to an infinite Prandtl number. The probability distribution function (PDF) of the scalar gradients parallel and perpendicular to the direction of the mean gradient are studied. The gradients parallel to the mean gradient have a skewness of order 1 in the range of Peclet number considered. The PDFs are sharply peaked and their maxima correspond to a perfect mixing of the scalar. The PDF of the scalar gradient perpendicular to the mean gradient are reasonably well fit by stretched exponentials. Similar properties are observed for the restricted Euler model. In physical spa...

TURBULENCE IN HOMOGENEOUS SHEAR FLOWS

Homogeneous shear flows with an imposed mean velocity U=Syx are studied in a period box of size Lx×Ly×Lz, in the statistically stationary turbulent state. In contrast with unbounded shear flows, the finite size of the system constrains the large‐scale dynamics. The Reynolds number, defined by Re≡SL2y/ν varies in the range 2600⩽Re⩽11300. The total kinetic energy and enstrophy in the volume of numerical integration have large peaks, resulting in fluctuations of kinetic energy of order 30%–50%. The mechanism leading to these fluctuations is very reminiscent of the ‘‘streaks’’ responsible for the violent bursts observed in turbulent boundary layers. The large scale anisotropy of the flow, characterized by the two‐point correlation tensor 〈uiuj〉 depends on the aspect ratio of the system. The probability distribution functions (PDF) of the components of the velocity are found to be close to Gaussian. The physics of the Reynolds stress tensor, uv, is very similar to what is found experimentally in wall bounded ...

A numerical study of pressure fluctuations in three‐dimensional, incompressible, homogeneous, isotropic turbulence

Pressure fluctuations in incompressible turbulence are studied by direct numerical simulations of the three‐dimensional (3‐D) Navier–Stokes equations. The pressure probability distribution function (PDF) is shown to have an exponential tail on the negative side, and to be independent of the Reynolds number for Reλ≲60. At higher Reynolds numbers, the low pressure part of the pressure PDF becomes super exponential. The joint PDFs of strain, vorticity, and pressure (considered pairwise) show a strong dissymmetry between positive and negative pressure fluctuations. The results obtained from the numerical solutions of the Navier–Stokes equations are compared with a Gaussian velocity field. The two statistical ensembles are shown to lead to quantitatively different results.

Small‐scale properties of scalar and velocity differences in three‐dimensional turbulence

Turbulent flows are known to concentrate strong vorticity in vortex tubes, giving rise to large velocity jumps across the tubes. When a passive scalar is advected by the flow, very steep scalar fronts separate well‐mixed regions, and result in large scalar differences. The properties of these large jumps are investigated by studying the probability distribution functions of velocity, scalar differences as a function of the separation between the points, of the Reynolds and of the Prandtl number. Over the range of parameters covered by the direct numerical simulations reported here (20≤Rλ≤90 and 1/32≤Pr≤1), it is found that the widths of the velocity (respectively, the scalar) jumps scale like the Kolmogorov length (respectively, like the Batchelor length). For both the scalar and the velocity, the large differences over small distance become rarer as the Reynolds number increases.

Control of Rotating Waves in Cardiac Muscle: Analysis of the Effect of an Electric Field

The effect of an electric field on rotating waves in cardiac muscle is considered from a theoretical point of view. A model of excitation propagation taking into account the cellular structure of the heart is presented and studied. The application of a direct current electric field along the cardiac tissue is known to induce changes in membrane potential which decay exponentially with distance. Investigation of the model shows that the electric field induces a gradient of potential inside a cell which does not decay with distance, and results in modification of excitation propagation which extends a considerable distance from the electrodes. In two dimensions, it induces a drift of rotating waves. The effect of the electric field on propagation velocity and on rotating waves cannot be obtained in any arbitrary models of cardiac muscle. For an electric field of about 1 V cm-1 and junctional resistances of about 20 MΩ, the change in velocity of propagation can be up to several percent, resulting in a drift velocity of rotating waves of the order of 1 cm s-1. To test these predictions, experiments with cardiac preparations are proposed.

Turbulence and Stochastic Processes

sec:08-1In 1931 the monograph Analytical Methods in Probability Theory appeared, in which A.N. Kolmogorov laid the foundations for the modern theory of Markov processes [1]. According to Gnedenko: “In the history of probability theory it is difficult to find other works that changed the established points of view and basic trends in research work in such a decisive way”. Ten years later, his article on fully developed turbulence provided the framework within which most, if not all, of the subsequent theoretical investigations have been conducted [2] (see e.g. the review by Biferale et al. in this volume [3]. Remarkably, the greatest advances made in the last few years towards a thorough understanding of turbulence developed from the successful marriage between the theory of stochastic processes and the phenomenology of turbulent transport of scalar fields. In this article we will summarize these recent developments which expose the direct link between the intermittency of transported fields and the statistical properties of particle trajectories advected by the turbulent flow (see also [4], and, for a more thorough review, [5]. We also discuss the perspectives of the Lagrangian approach beyond passive scalars, especially for the modeling of hydrodynamic turbulence.

Semiclassical Approach of the “Tetrad Model” of Turbulence

We consider the ‘tetrad model’ of turbulence, which describes the coarse grained velocity derivative tensor as a function of scale with the help of a mean field approach. The model is formulated in terms of a set of stochastic differential equations, and the fundamental object of this model is a tetrahedron (tetrad) of Lagrangian particles, whose characteristic size lies in the inertial range. We present the approximate ‘semiclassical’ method of resolution of the model. Our first numerical results show that the solution correctly reproduces the main features obtained in Direct Numerical Simulations, as well as in experiments (van der Bos et al., 2002).

Formation and Interaction of Intense Vortex Sheets in 3-Dimensional, Incompressible, Hydrodynamics

Various questions related to the physics of inviscid flows are reviewed. The emergence of strong vortex sheets has repeatedly been observed in the simulation of the 3-dimensional equations, with a variety of initial conditions. In the case of axisymmetric Euler flows, the origin of these sheets can be easily understood with the help of an analogy with thermally driven flows. A more general mechanism to explain these sheets is proposed. Questions of singularities are briefly reviewed. Lastly, preliminary results on the connection between the sheets forming in inviscid flows, and the vortex tubes observed in high Reynolds number flows are presented.SommarioSono considerate varie questioni correlate con la fisica dei flussi non-viscosi. La nascita di strati di forti vortici è stata ripetutamente osservata nella simulazione delle equazioni tridimensionali, per diverse condizioni iniziali. Nel caso di flussi di Eulero assialsimmetrici, lorigine di questi piani può essere facilmente compresa con laiuto di una analogia con i flussi guidati termicamente. Viene inoltre proposto un più generale meccanismo per giustificare questi strati e si passano in rassegna brevemente questioni riguardanti le singolarità. Infine, vengono presentati alcuni risultati preliminari sulla connessione tra i piani formantisi in flussi non viscosi ed i vortici tubolari osservati nei flussi ad alti numeri di Reynolds.