We present a detailed numerical study of anisotropic statistical fluctuations in stationary, homogeneous turbulent flows. We address both problems of intermittency in anisotropic sectors, and the relative importance of isotropic and anisotropic fluctuations at different scales on a direct numerical simulation of a three-dimensional random Kolmogorov flow. We review a simple argument to predict the dimensional scaling for all velocity moments, in all anisotropic sectors. We extend a previous analysis made on the same data set (Phys. Rev. Lett. 86 (2001) 4831) presenting (i) the statistical behavior of spectra and co-spectra; (ii) high-order longitudinal structure functions; (iii) anisotropic fluctuations of the full tensorial two-points velocity correlations. Among the many issues discussed, we stress the problem of the return-to-isotropy, the universality of anisotropic fluctuations and the foliation mechanism. A new a priori test on sub-grid quantities used in Large-Eddy Simulations is also presented.

We study the transition from the collisionless to the hydrodynamic regime in a two-component spin-polarized mixture of 40K atoms by exciting its dipolar oscillation modes inside harmonic traps. The time evolution of the mixture is described by the Vlasov–Landau equations and numerically solved with a fully three-dimensional concurrent code. We observe a master/slave behaviour of the oscillation frequencies depending on the dipolar mode that is excited. Regardless of the initial conditions, the transition to hydrodynamics is found to shift to lower values of the collision rate as the temperature decreases.

The achievement of Bose-Einstein condensation in ultra-cold vapours of alkali atoms has given enormous impulse to the study of dilute atomic gases in condensed quantum states inside magnetic traps and optical lattices. High-purity and easy optical access make them ideal candidates to investigate fundamental issues on interacting quantum systems. This review presents some theoretical issues which have been addressed in this area and the numerical techniques which have been developed and used to describe them, from mean-field models to classical and quantum simulations for equilibrium and dynamical properties. After an introductory overview on dilute quantum gases, both in the homogeneus state and under harmonic or periodic confinement, the article is organized in three main sections. The first concerns Bose-condensed gases at zero temperature, with main regard to the properties of the ground state in different confinements and to collective excitations and transport in the condensate. Bose-Einstein-condensed gases at finite temperature are addressed in the next section, the main emphasis being on equilibrium properties and phase transitions and on dynamical and transport properties associated with the presence of the thermal cloud. Finally, the last section is focused on theoretical and computational issues that have emerged from the efforts to drive gases of fermionic atoms and boson-fermion mixtures deep into the quantum degeneracy regime, with the aim of realizing novel superfluids from fermion pairing. The attention given in this article to methods beyond standard mean-field approaches should make it a useful reference point for future advances in these areas.

We present a numerical study of the micro-dynamical roots of dissipation in two colliding mesoscopic clouds of point-like fermions as a function of the scattering length and of temperature approaching full quantum degeneracy. This study, which is motivated by current experiments on ultracold gaseous mixtures of fermionic atoms inside magnetic traps, combines the solution of the coupled Vlasov-Landau equations for the Wigner distribution functions with a locally adaptive importance-sampling technique for handling collisional interactions. The results illustrate the consequences of genuinely quantum collisional phenomena, and in particular the role of Pauli blocking in the transition to hydrodynamic behaviour. We also compare the computed quantum collision rate as a function of temperature in the weak-coupling case with theoretical results assuming that equilibrium distributions determine the quantum collision integral.

We review some ideas about the physics of small-scale turbulent statistics, focusing on the scaling behavior of anisotropic fluctuations. We present results from direct numerical simulations of three-dimensional homogeneous, anisotropically forced, turbulent systems: the Rayleigh–Bénard system, the random-Kolmogorov-flow, and a third flow with constant anisotropic energy spectrum at low wave numbers. A comparison of the anisotropic scaling properties displays good similarity among these very different flows. Our findings support the conclusion that scaling exponents of anisotropic fluctuations are universal, i.e., independent of the forcing mechanism sustaining turbulence.

We discuss a stochastic closure for the equation of motion satisfied by multiscale correlation functions in the framework of shell models of turbulence. We present a plausible closure scheme to calculate the anomalous scaling exponents of structure functions by using the exact constraints imposed by the equation of motion. We present an explicit calculation for fifth-order scaling exponent by varying the free parameter entering in the nonlinear term of the model. The same method applied to the case of shell models for Kraichnan passive scalar provides a connection between the concept of zero-modes and time-dependent cascade processes.

We present the results of a numerical investigation of three-dimensional decaying turbulence with statistically homogeneous and anisotropic initial conditions. We show that at large times, in the inertial range of scales: (i) isotropic velocity fluctuations decay self-similarly at an algebraic rate which can be obtained by dimensional arguments; (ii) the ratio of anisotropic to isotropic fluctuations of a given intensity falls off in time as a power law, with an exponent approximately independent of the strength of the fluctuation; (iii) the decay of anisotropic fluctuations is not self-similar, their statistics becoming more and more intermittent as time elapses. We also investigate the early stages of the decay. The different short-time behavior observed in two experiments differing by the phase organization of their initial conditions gives a new hunch on the degree of universality of small-scale turbulence statistics, i.e., its independence of the conditions at large scales.

We develop a numerical method to study the dynamics of a two-component atomic Fermi gas trapped inside a harmonic potential at temperature T well below the Fermi temperature TF. We examine the transition from the collisionless to the collisional regime down to T = 0.2 TF and find a good qualitative agreement with the experiments of B. DeMarco and D.S. Jin [Phys. Rev. Lett. 88, 040405 (2002)]. We demonstrate a twofold role of temperature on the collision rate and on the efficiency of collisions. In particular, we observe a hitherto unreported effect, namely, the transition to hydrodynamic behavior is shifted towards lower collision rates as temperature decreases.

The motion and the action of microbubbles in homogeneous and isotropic turbulence are investigated through (three-dimensional) direct numerical simulations of the Navier–Stokes equations and applying the Lagrangian approach to track the bubble trajectories. The forces acting on the bubbles are added mass, drag, lift, and gravity. The bubbles are found to accumulate in vortices, preferably on the side with downward velocity. This effect, mainly caused by the lift force, leads to a reduced average bubble rise velocity. Once the reaction of the bubbles on the carrier flow is embodied using a point-force approximation, an attenuation of the turbulence on large scales and an extra forcing on small scales is found.

The ultimate regime of thermal convection, the so called Kraichnan regime (R. H. Kraichnan, Phys. Fluids 5, 1374 (1962)), hitherto has been elusive. Here, numerical evidence for that regime is presented by performing simulations of the bulk of turbulence only, eliminating the thermal and kinetic boundary layers and replacing them by periodic boundary conditions.

We present a numerical study of anisotropic statistical fluctuations in homogeneous turbulent flows. We give an argument to predict the dimensional scaling exponents, (p+j)/3, for the projections of p-th order structure function in the j-th sector of the rotational group. We show that measured exponents are anomalous, showing a clear deviation from the dimensional prediction. Dimensional scaling is subleading and it is recovered only after a random reshuffling of all velocity phases, in the stationary ensemble. This supports the idea that anomalous scaling is the result of a genuine inertial evolution, independent of large-scale behavior.