The sharp maximal function approach to LpL^{p} estimates for operators structured on H\"{o}rmander's vector fields

We consider a nonvariational degenerate elliptic operator structured on a system of left invariant, 1-homogeneous, H\"ormander's vector fields on a Carnot group in $R^{n}$, where the matrix of coefficients is symmetric, uniformly positive on a bounded domain of $R^{n}$ and the coefficients are bounded, measurable and locally VMO in the domain. We give a new proof of the interior $L^{p}$ estimates on the second order derivatives with respect to the vector fields, first proved by Bramanti-Brandolini in [Rend. Sem. Mat. dell'Univ. e del Politec. di Torino, Vol. 58, 4 (2000), 389-433], extending to this context Krylov' technique, introduced in [Comm. in P.D.E.s, 32 (2007), 453-475], consisting in estimating the sharp maximal function of the second order derivatives.Comment: 26 page

Intermittency in Turbulence: Multiplicative random process in space and time

We present a simple stochastic algorithm for generating multiplicative processes with multiscaling both in space and in time. With this algorithm we are able to reproduce a synthetic signal with the same space and time correlation as the one coming from shell models for turbulence and the one coming from a turbulent velocity field in a quasi-Lagrangian reference frame.Comment: 23 pages, 12 figure

Extreme events in the dispersions of two neighboring particles under the influence of fluid turbulence

We present a numerical study of two-particle dispersion from point-sources in 3D incompressible Homogeneous and Isotropic turbulence, at Reynolds number Re \simeq 300. Tracer particles are emitted in bunches from localized sources smaller than the Kolmogorov scale. We report the first quantitative evidence, supported by an unprecedented statistics, of the deviations of relative dispersion from Richardson's picture. Deviations are due to extreme events of pairs separating much faster than average, and of pairs remaining close for long times. The two classes of events are the fingerprint of complete different physics, the former being dominated by inertial subrange and large-scale fluctuations, while the latter by the dissipation subrange. A comparison of relative separation in surrogate white-in-time velocity field, with correct viscous-, inertial- and integral-scale properties allows us to assess the importance of temporal correlations along tracer trajectories.Comment: 5 pages, 6 figure

Accurate lubrication corrections for spherical and non-spherical particles in discretized fluid simulations

Discretized fluid solvers coupled to a Newtonian dynamics method are a popular tool to study suspension flow. As any simulation technique with finite resolution, the lattice Boltzmann method, when coupled to discrete particles using the momentum exchange method, resolves the diverging lubrication interactions between surfaces near contact only insufficiently. For spheres, it is common practice to account for surface-normal lubrication forces by means of an explicit correction term. A method that additionally covers all further singular interactions for spheres is present in the literature as well as a link-based approach that allows for more general shapes but does not capture non-normal interactions correctly. In this paper, lattice-independent lubrication corrections for aspherical particles are outlined, taking into account all leading divergent interaction terms. An efficient implementation for arbitrary spheroids is presented and compared to purely normal and link-based models. Good consistency with Stokesian dynamics simulations of spheres is found. The non-normal interactions affect the viscosity of suspensions of spheres at volume fractions \Phi >= 0.3 but already at \Phi >= 0.2 for spheroids. Regarding shear-induced diffusion of spheres, a distinct effect is found at 0.1 <= \Phi <= 0.5 and even increasing the resolution of the radius to 8 lattice units is no substitute for an accurate modeling of non-normal interactions.Comment: 19 pages, 10 figure

Universality in passively advected hydrodynamic fields: the case of a passive vector with pressure

Universality of statistical properties of passive quantities advected by turbulent velocity fields at changing the passive forcing mechanism is discussed. In particular, we concentrate on the statistical properties of an hydrodynamic system with pressure. We present theoretical arguments and preliminary numerical results which show that the fluxes of passive vector field and of the velocity field have the same scaling behavior. By exploiting such a property, we propose a way to compute the anomalous exponents of three dimensional turbulent velocity fields. Our findings are in agreement within 5% with experimental values of the anomalous exponents.Comment: 15 pages, 6 figure

Statistics of small scale vortex filaments in turbulence

We study the statistical properties of coherent, small-scales, filamentary-like structures in Turbulence. In order to follow in time such complex spatial structures, we integrate Lagrangian and Eulerian measurements by seeding the flow with light particles. We show that light particles preferentially concentrate in small filamentary regions of high persistent vorticity (vortex filaments). We measure the fractal dimension of the attracting set and the probability that two particles do not separate for long time lapses. We fortify the signal-to-noise ratio by exploiting multi-particles correlations on the dynamics of bunches of particles. In doing that, we are able to give a first quantitative estimation of the vortex-filaments life-times, showing the presence of events as long as the integral correlation time. The same technique introduced here could be used in experiments as long as one is capable to track clouds of bubbles in turbulence for a relatively long period of time, at high Reynolds numbers; shading light on the dynamics of small-scale vorticity in realistic turbulent flows.Comment: 5 pages, 5 figure

Earthquake statistics inferred from plastic events in soft-glassy materials

We propose a new approach for generating synthetic earthquake catalogues based on the physics of soft glasses. The continuum approach produces yield-stress materials based on Lattice-Boltzmann simulations. We show that, if the material is stimulated below yield stress, plastic events occur, which have strong similarities with seismic events. Based on a suitable definition of displacement in the continuum, we show that the plastic events obey a Gutenberg-Richter law with exponents similar to those for real earthquakes. We further find that average acceleration, energy release, stress drop and recurrence times scale with the same exponent. The approach is fully self-consistent and all quantities can be calculated at all scales without the need of ad hoc friction or statistical laws. We therefore suggest that our approach may lead to new insight into understanding of the physics connecting the micro and macro scale of earthquakes.Comment: 13 pages, 7 figure

On the Heat Transfer in Rayleigh-Benard systems

In this paper we discuss some theoretical aspects concerning the scaling laws of the Nusselt number versus the Rayleigh number in a Rayleigh-Benard cell. We present a new set of numerical simulations and compare our findings against the predictions of existing models. We then propose a new theory which relies on the hypothesis of Bolgiano scaling. Our approach generalizes the one proposed by Kadanoff, Libchaber and coworkers and solves some of the inconsistencies raised in the recent literature.Comment: 10 pages, 5 figure