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

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

Formal Verification of Nonlinear Inequalities with Taylor Interval Approximations

We present a formal tool for verification of multivariate nonlinear inequalities. Our verification method is based on interval arithmetic with Taylor approximations. Our tool is implemented in the HOL Light proof assistant and it is capable to verify multivariate nonlinear polynomial and non-polynomial inequalities on rectangular domains. One of the main features of our work is an efficient implementation of the verification procedure which can prove non-trivial high-dimensional inequalities in several seconds. We developed the verification tool as a part of the Flyspeck project (a formal proof of the Kepler conjecture). The Flyspeck project includes about 1000 nonlinear inequalities. We successfully tested our method on more than 100 Flyspeck inequalities and estimated that the formal verification procedure is about 3000 times slower than an informal verification method implemented in C++. We also describe future work and prospective optimizations for our method.Comment: 15 page

Metalibm: A Mathematical Functions Code Generator

International audienceThere are several different libraries with code for mathematical functions such as exp, log, sin, cos, etc. They provide only one implementation for each function. As there is a link between accuracy and performance, that approach is not optimal. Sometimes there is a need to rewrite a function's implementation with the respect to a particular specification. In this paper we present a code generator for parametrized implementations of mathematical functions. We discuss the benefits of code generation for mathematical libraries and present how to implement mathematical functions. We also explain how the mathematical functions are usually implemented and generalize this idea for the case of arbitrary function with implementation parameters. Our code generator produces C code for parametrized functions within a known scheme: range reduction (domain splitting), polynomial approximation and reconstruction. This approach can be expanded to generate code for black-box functions, e.g. defined only by differential equations

Lagrangian dynamics and statistical geometric structure of turbulence

The local statistical and geometric structure of three-dimensional turbulent flow can be described by properties of the velocity gradient tensor. A stochastic model is developed for the Lagrangian time evolution of this tensor, in which the exact nonlinear self-stretching term accounts for the development of well-known non-Gaussian statistics and geometric alignment trends. The non-local pressure and viscous effects are accounted for by a closure that models the material deformation history of fluid elements. The resulting stochastic system reproduces many statistical and geometric trends observed in numerical and experimental 3D turbulent flows, including anomalous relative scaling.Comment: 5 pages, 5 figures, final version, publishe

Fine-scale statistics of temperature and its derivatives in convective turbulence

We study the fine-scale statistics of temperature and its derivatives in turbulent Rayleigh-Benard convection. Direct numerical simulations are carried out in a cylindrical cell with unit aspect ratio filled with a fluid with Prandtl number equal to 0.7 for Rayleigh numbers between 10^7 and 10^9. The probability density function of the temperature or its fluctuations is found to be always non-Gaussian. The asymmetry and strength of deviations from the Gaussian distribution are quantified as a function of the cell height. The deviations of the temperature fluctuations from the local isotropy, as measured by the skewness of the vertical derivative of the temperature fluctuations, decrease in the bulk, but increase in the thermal boundary layer for growing Rayleigh number, respectively. Similar to the passive scalar mixing, the probability density function of the thermal dissipation rate deviates significantly from a log-normal distribution. The distribution is fitted well by a stretched exponential form. The tails become more extended with increasing Rayleigh number which displays an increasing degree of small-scale intermittency of the thermal dissipation field for both the bulk and the thermal boundary layer. We find that the thermal dissipation rate due to the temperature fluctuations is not only dominant in the bulk of the convection cell, but also yields a significant contribution to the total thermal dissipation in the thermal boundary layer. This is in contrast to the ansatz used in scaling theories and can explain the differences in the scaling of the total thermal dissipation rate with respect to the Rayleigh number.Comment: 22 pages and 15 figure

Intermittency of velocity time increments in turbulence

We analyze the statistics of turbulent velocity fluctuations in the time domain. Three cases are computed numerically and compared: (i) the time traces of Lagrangian fluid particles in a (3D) turbulent flow (referred to as the "dynamic" case); (ii) the time evolution of tracers advected by a frozen turbulent field (the "static" case), and (iii) the evolution in time of the velocity recorded at a fixed location in an evolving Eulerian velocity field, as it would be measured by a local probe (referred to as the "virtual probe" case). We observe that the static case and the virtual probe cases share many properties with Eulerian velocity statistics. The dynamic (Lagrangian) case is clearly different; it bears the signature of the global dynamics of the flow.Comment: 5 pages, 3 figures, to appear in PR

Lagrangian Velocity Statistics in Turbulent Flows: Effects of Dissipation

We use the multifractal formalism to describe the effects of dissipation on Lagrangian velocity statistics in turbulent flows. We analyze high Reynolds number experiments and direct numerical simulation (DNS) data. We show that this approach reproduces the shape evolution of velocity increment probability density functions (PDF) from Gaussian to stretched exponentials as the time lag decreases from integral to dissipative time scales. A quantitative understanding of the departure from scaling exhibited by the magnitude cumulants, early in the inertial range, is obtained with a free parameter function D(h) which plays the role of the singularity spectrum in the asymptotic limit of infinite Reynolds number. We observe that numerical and experimental data are accurately described by a unique quadratic D(h) spectrum which is found to extend from $h_{min} \approx 0.18$ to $h_{max} \approx 1$, as the signature of the highly intermittent nature of Lagrangian velocity fluctuations.Comment: 5 pages, 3 figures, to appear in PR

Fully developed turbulence and the multifractal conjecture

We review the Parisi-Frisch MultiFractal formalism for Navier--Stokes turbulence with particular emphasis on the issue of statistical fluctuations of the dissipative scale. We do it for both Eulerian and Lagrangian Turbulence. We also show new results concerning the application of the formalism to the case of Shell Models for turbulence. The latter case will allow us to discuss the issue of Reynolds number dependence and the role played by vorticity and vortex filaments in real turbulent flows.Comment: Special Issue dedicated to E. Brezin and G. Paris

On the origin of intermittency in wave turbulence

Using standard signal processing tools, we experimentally report that intermittency of wave turbulence on the surface of a fluid occurs even when two typical large-scale coherent structures (gravity wave breakings and bursts of capillary waves on steep gravity waves) are not taken into account. We also show that intermittency depends on the power injected into the waves. The dependence of the power-law exponent of the gravity-wave spectrum on the forcing amplitude cannot be also ascribed to these coherent structures. Statistics of these both events are studied.Comment: To be published in EP