查看更多>>摘要：We survey recent results in the mathematical literature on the equations of incompressible fluid dynamics, highlighting common themes and how they might contribute to the understanding of some phenomena in the theory of fully developed turbulence. This article is part of the theme issue 'Scaling the turbulence edifice (part 1)'.
查看更多>>摘要：The multifractal model of turbulence (MFM) and the three-dimensional Navier-Stokes equations are blended together by applying the probabilistic scaling arguments of the former to n hierarchy of weak solutions of the latter. This process imposes a lower hound on both the multifractal spectrum C(h), which appears naturally in the Large Deviation formulation of the MFM, and on h the standard scaling parameter. These bounds respectively take the form: (i) C(h) >= 1 - 3h, which is consistent with Kolmogorov's four-firths law; and (ii) h >= 2/3. The Litter significant as it prevents solutions from approaching the Navier-Stokes singular set of Caffarelli, Kohn and Nirenberg. This article is part of the theme issue 'Scaling the turbulence edifice (part 1)'.
查看更多>>摘要：We expose a hidden scaling symmetry of the Navier-Stokes equations in the limit of vanishing viscosity, which stems from dynamical space-time rescaling around suitably defined Lagrangian scaling centres. At a dynamical level, the hidden symmetry projects solutions which differ up to Galilean invariance and global temporal scaling onto the same representative flow. At a statistical level, this projection repairs the scale invariance, which is broken by intermittency in the original formulation. Following previous work by the first author, we here postulate and substantiate with numerics that hidden symmetry statistically holds in the inertial interval of fully developed turbulence. We show that this symmetry accounts for the scale-invariance of a certain class of observables, in particular, the Kolmogorov multipliers. This article is part of the theme issue 'Scaling the turbulence edifice (part 1)'.
查看更多>>摘要：Turbulence is unique in its appeal across physics, mathematics and engineering. And yet a microscopic theory, starting from the basic equations of hydrodynamics, still eludes us. In the last decade or so, new directions at the interface of physics and mathematics have emerged, which strengthens the hope of 'solving' one of the oldest problems in the natural sciences. This two-part theme issue unites these new directions on a common platform emphasizing the underlying complementarity of the physicists' and the mathematicians' approaches to a remarkably challenging problem. This article is part of the theme issue 'Scaling the turbulence edifice (part 1)'.
查看更多>>摘要：We consider the problem of anomalous dissipation for passive scalars advected by an incompressible flow. We review known results on anomalous dissipation from the point of view of the analysis of partial differential equations, and present simple rigorous examples of scalars that admit a Batchelor-type energy spectrum and exhibit anomalous dissipation in the limit of zero scalar diffusivity. This article is part of the theme issue 'Scaling the turbulence edifice (part 1)'.
查看更多>>摘要：In this paper, we consider a simplified model of turbulence for large Reynolds numbers driven by a constant power energy input on large scales. In the statistical stationary regime, the behaviour of the kinetic energy is characterized by two well-defined phases: a laminar phase where the kinetic energy grows linearly for a (random) time t(w) followed by abrupt avalanche-like energy drops of sizes S due to strong intermittent fluctuations of energy dissipation. We study the probability distribution P[t(w)] and P[S] which both exhibit a quite well-defined scaling behaviour. Although t(w) and S are not statistically correlated, we suggest and numerically checked that their scaling properties are related based on a simple, but non-trivial, scaling argument. We propose that the same approach can be used for other systems showing avalanche-like behaviour such as amorphous solids and seismic events. This article is part of the theme issue 'Scaling the turbulence edifice (part 1)'.
查看更多>>摘要：The purpose of this article is to give a complete proof of a C-0,C-alpha regularity result for the pressure for weak solutions of the two-dimensional 'incompressible Euler equations' when the fluid velocity enjoys the same type of regularity in a compact simply connected domain with C-2 boundary. To accomplish our result, we realize that it is compulsory to introduce a new weak formulation for the boundary condition of the pressure, which is consistent with, and equivalent to, that of classical solutions. This article is part of the theme issue 'Scaling the turbulence edifice (part 1)'.
查看更多>>摘要：This note is devoted to broken and emerging scale invariance of turbulence. Pumping breaks the symmetry: the statistics of every mode explicitly depend on the distance from the pumping. And yet the ratios of mode amplitudes, called Kolmogorov multipliers, are known to approach scale-invariant statistics away from the pumping. This emergent scale invariance deserves an explanation and a detailed study. We put forward the hypothesis that the invariance of multipliers is due to an extreme non-locality of their interactions (similar to the appearance of mean-field properties in the thermodynamic limit for systems with long-range interaction). We analyse this phenomenon in a family of models that connects two very different classes of systems: resonantly interacting waves and wave-free incompressible flows. The connection is algebraic and turns into an identity for properly discretized models. We show that this family provides a unique opportunity for an analytic (perturbative) study of emerging scale invariance in a system with strong interactions. This article is part of the theme issue 'Scaling the turbulence edifice (part 1)'.
查看更多>>摘要：Variation of the statistical properties of an incompressible velocity, passive vector and passive scalar in isotropic turbulence was studied using direct numerical simulation. The structure functions of the gradients, and the moments of the dissipation rates, began to increase at about R-lambda similar to 2 from the Gaussian state and grew rapidly at R-lambda > 20 in the turbulent state. A contour map of the probability density functions (PDFs) indicated that PDF expansion of the gradients of the passive vector and passive scalar begins at around R-lambda approximate to 4, whereas that of the longitudinal velocity gradient PDF is more gradual. The left tails of the dissipation rate PDF were found to follow a power law with an exponent of 3/2 for the incompressible velocity and passive vector dissipation rates, and 1/2 for the scalar dissipation rate and the enstrophy; they remained constant for all Reynolds numbers, indicating the universality of the left tail. The analytical PDFs of the dissipation rates and enstrophy of the Gaussian state were obtained and found to be the Gamma distribution. It was shown that the number of terms contributing to the dissipation rates and the enstrophy determines the decay rates of the two PDFs for low to moderate amplitudes. This article is part of the theme issue 'Scaling the turbulence edifice (part 1)'.
NSTL
Royal Soc London