On the Rapid Increase of Intermittency in the Near-Dissipation Range of Fully Developed Turbulence

Intermittency, measured as log(F(r)/3), where F(r) is the flatness of velocity increments at scale r, is found to rapidly increase as viscous effects intensify, and eventually saturate at very small scales. This feature defines a finite intermediate range of scales between the inertial and dissipation ranges, that we shall call near-dissipation range. It is argued that intermittency is multiplied by a universal factor, independent of the Reynolds number Re, throughout the near-dissipation range. The (logarithmic) extension of the near-dissipation range varies as \sqrt(log Re). As a consequence, scaling properties of velocity increments in the near-dissipation range strongly depend on the Reynolds number.Comment: 7 pages, 7 figures, to appear in EPJ

A Stochastic Representation of the Local Structure of Turbulence

Based on the mechanics of the Euler equation at short time, we show that a Recent Fluid Deformation (RFD) closure for the vorticity field, neglecting the early stage of advection of fluid particles, allows to build a 3D incompressible velocity field that shares many properties with empirical turbulence, such as the teardrop shape of the R-Q plane. Unfortunately, non gaussianity is weak (i.e. no intermittency) and vorticity gets preferentially aligned with the wrong eigenvector of the deformation. We then show that slightly modifying the former vectorial field in order to impose the long range correlated nature of turbulence allows to reproduce the main properties of stationary flows. Doing so, we end up with a realistic incompressible, skewed and intermittent velocity field that reproduces the main characteristics of 3D turbulence in the inertial range, including correct vorticity alignment properties.Comment: 6 pages, 3 figures, final version, published

Orientation dynamics of small, triaxial-ellipsoidal particles in isotropic turbulence

The orientation dynamics of small anisotropic tracer particles in turbulent flows is studied using direct numerical simulation (DNS) and results are compared with Lagrangian stochastic models. Generalizing earlier analysis for axisymmetric ellipsoidal particles (Parsa et al. 2012), we measure the orientation statistics and rotation rates of general, triaxial ellipsoidal tracer particles using Lagrangian tracking in DNS of isotropic turbulence. Triaxial ellipsoids that are very long in one direction, very thin in another, and of intermediate size in the third direction exhibit reduced rotation rates that are similar to those of rods in the ellipsoid's longest direction, while exhibiting increased rotation rates that are similar to those of axisymmetric discs in the thinnest direction. DNS results differ significantly from the case when the particle orientations are assumed to be statistically independent from the velocity gradient tensor. They are also different from predictions of a Gaussian process for the velocity gradient tensor, which does not provide realistic preferred vorticity-strain-rate tensor alignments. DNS results are also compared with a stochastic model for the velocity gradient tensor based on the recent fluid deformation approximation (RFDA). Unlike the Gaussian model, the stochastic model accurately predicts the reduction in rotation rate in the longest direction of triaxial ellipsoids since this direction aligns with the flow's vorticity, with its rotation perpendicular to the vorticity being reduced. For disc-like particles, or in directions perpendicular to the longest direction in triaxial particles, the model predicts {noticeably} smaller rotation rates than those observed in DNS, a behavior that can be understood based on the probability of vorticity orientation with the most contracting strain-rate eigen-direction in the model.Comment: 24 pages, 15 color figures, references added, published versio

Multi-scale model of gradient evolution in turbulent flows

A multi-scale model for the evolution of the velocity gradient tensor in fully developed turbulence is proposed. The model is based on a coupling between a ``Restricted Euler'' dynamics [{\it P. Vieillefosse, Physica A, {\bf 14}, 150 (1984)}] which describes gradient self-stretching, and a deterministic cascade model which allows for energy exchange between different scales. We show that inclusion of the cascade process is sufficient to regularize the well-known finite time singularity of the Restricted Euler dynamics. At the same time, the model retains topological and geometrical features of real turbulent flows: these include the alignment between vorticity and the intermediate eigenvector of the strain-rate tensor and the typical teardrop shape of the joint probability density between the two invariants, $R-Q$, of the gradient tensor. The model also possesses skewed, non-Gaussian longitudinal gradient fluctuations and the correct scaling of energy dissipation as a function of Reynolds number. Derivative flatness coefficients are in good agreement with experimental data.Comment: 4 pages, 4 figure

Reynolds number effect on the velocity increment skewness in isotropic turbulence

Second and third order longitudinal structure functions and wavenumber spectra of isotropic turbulence are computed using the EDQNM model and compared to results of the multifractal formalism. At the highest Reynolds number available in windtunnel experiments, $R_\lambda=2500$, both the multifractal model and EDQNM give power-law corrections to the inertial range scaling of the velocity increment skewness. For EDQNM, this correction is a finite Reynolds number effect, whereas for the multifractal formalism it is an intermittency correction that persists at any high Reynolds number. Furthermore, the two approaches yield realistic behavior of second and third order statistics of the velocity fluctuations in the dissipative and near-dissipative ranges. Similarities and differences are highlighted, in particular the Reynolds number dependence

Local and nonlocal pressure Hessian effects in real and synthetic fluid turbulence

The Lagrangian dynamics of the velocity gradient tensor A in isotropic and homogeneous turbulence depend on the joint action of the self-streching term and the pressure Hessian. Existing closures for pressure effects in terms of A are unable to reproduce one important statistical role played by the anisotropic part of the pressure Hessian, namely the redistribution of the probabilities towards enstrophy production dominated regions. As a step towards elucidating the required properties of closures, we study several synthetic velocity fields and how well they reproduce anisotropic pressure effects. It is found that synthetic (i) Gaussian, (ii) Multifractal and (iii) Minimal Turnover Lagrangian Map (MTLM) incompressible velocity fields reproduce many features of real pressure fields that are obtained from numerical simulations of the Navier Stokes equations, including the redistribution towards enstrophy-production regions. The synthetic fields include both spatially local, and nonlocal, anisotropic pressure effects. However, we show that the local effects appear to be the most important ones: by assuming that the pressure Hessian is local in space, an expression in terms of the Hessian of the second invariant Q of the velocity gradient tensor can be obtained. This term is found to be well correlated with the true pressure Hessian both in terms of eigenvalue magnitudes and eigenvector alignments.Comment: 10 pages, 4 figures, minor changes, final version, published in Phys. Fluid

Gaussian multiplicative Chaos for symmetric isotropic matrices

Motivated by isotropic fully developed turbulence, we define a theory of symmetric matrix valued isotropic Gaussian multiplicative chaos. Our construction extends the scalar theory developed by J.P. Kahane in 1985

Matrix exponential-based closures for the turbulent subgrid-scale stress tensor

Two approaches for closing the turbulence subgrid-scale stress tensor in terms of matrix exponentials are introduced and compared. The first approach is based on a formal solution of the stress transport equation in which the production terms can be integrated exactly in terms of matrix exponentials. This formal solution of the subgrid-scale stress transport equation is shown to be useful to explore special cases, such as the response to constant velocity gradient, but neglecting pressure-strain correlations and diffusion effects. The second approach is based on an Eulerian-Lagrangian change of variables, combined with the assumption of isotropy for the conditionally averaged Lagrangian velocity gradient tensor and with the recent fluid deformation approximation. It is shown that both approaches lead to the same basic closure in which the stress tensor is expressed as the matrix exponential of the resolved velocity gradient tensor multiplied by its transpose. Short-time expansions of the matrix exponentials are shown to provide an eddy-viscosity term and particular quadratic terms, and thus allow a reinterpretation of traditional eddy-viscosity and nonlinear stress closures. The basic feasibility of the matrix-exponential closure is illustrated by implementing it successfully in large eddy simulation of forced isotropic turbulence. The matrix-exponential closure employs the drastic approximation of entirely omitting the pressure-strain correlation and other nonlinear scrambling terms. But unlike eddy-viscosity closures, the matrix exponential approach provides a simple and local closure that can be derived directly from the stress transport equation with the production term, and using physically motivated assumptions about Lagrangian decorrelation and upstream isotropy