It is shown that the use of a high power α of the Laplacian in the dissipative term of hydrodynamical equations leads asymptotically to truncated inviscid conservative dynamics with a finite range of spatial Fourier modes. Those at large wavenumbers thermalize, whereas modes at small wavenumbers obey ordinary viscous dynamics [C. Cichowlas et al. Phys. Rev. Lett. 95, 264502 (2005)]. The energy bottleneck observed for finite α may be interpreted as incomplete thermalization. Artifacts arising from models with α > 1 are discussed. PACS numbers: 47.27 Gs, 05.20.Jj A single Maxwell daemon embedded in a turbulent flow would hardly notice that the fluid is not exactly in thermal equilibrium because incompressible turbulence, even at very high Reynolds numbers, constitutes a tiny perturbation on thermal molecular motion. Dissipation in real fluids is just the transfer of macroscopically organized (hydrodynamic) energy to molecular thermal energy. Artificial microscopic systems can act just like the real one as far as the emergence of hydrodynamics is concerned; for instance, in lattice gases the "molecules" are discrete Boolean entities [1] and thermalization is easily observed at high wavenumbers [2]. Another example has been found recently by Cichowlas et al. [3] wherein the Euler equations of ideal non-dissipative flow are (Galerkin) truncated by keeping only a finite -but largenumber of spatial Fourier harmonics. The modes with the highest wavenumbers k then rapidly thermalize through a mechanism discovered by T.D. Lee [4] and studied further by R.H. Kraichnan [5], leading in three dimensions (3D) to an equipartition energy spectrum ∝ k 2 . The thermalized modes act as a fictitious microworld on modes with smaller wavenumbers in such a way that the usual dissipative NavierStokes dynamics is recovered at large scales [25].All the known systems presenting thermalization are conservative. As we shall show themalization may be present in dissipative hydrodynamic systems when the dissipation rate increases so fast with the wavenumber that it mimics ideal hydrodynamics with a Galerkin truncation. This is best understood by considering hydrodynamics with hyperviscosity: the usual momentum diffusion operator (a Laplacian) is replaced by the αth power of the Laplacian, where α > 1 is the dissipativity. Hyperviscosity is frequently used in turbulence modeling to avoid wasting numerical resolution by reducing the range of scales over which dissipation is effective [6].The unforced hyperviscous 1D Burgers and multidimensional incompressible Navier-Stokes (NS) equations are:The equations must be supplemented with suitable initial and boundary conditions. We employ 2π-periodic boundary conditions in space, so that we can use Fourier decompositions such as v(x) = kv k e i k·x . Note that minus the Laplacian is a positive operator, with Fourier transform k 2 , which can be raised to an arbitrary power α. The coefficient µ is taken positive to make the hyperviscous operator dissipative. The Galerkin wavenumber k G > 0 is chos...

Heavy particles suspended in a turbulent flow settle faster than in a still fluid. This effect stems from a preferential sampling of the regions where the fluid flows downward and is quantified here as a function of the level of turbulence, of particle inertia, and of the ratio between gravity and turbulent accelerations. By using analytical methods and detailed, state-of-the-art numerical simulations, settling is shown to induce an effective horizontal two-dimensional dynamics that increases clustering and reduce relative velocities between particles. These two competing effects can either increase or decrease the geometrical collision rates between same-size particles and are crucial for realistic modeling of coalescing particles.Many industrial, atmospheric, and astrophysical phenomena ranging from the microphysics of cloud formation, to planet formation in a dusty circumstellar disk of gas, involves the modeling of the interactions between small solid particles suspended in a turbulent carrier flow. Two main effects are typically at play: a viscous drag that particles experience with the agitated fluid and an external force, such as gravity, that acts because of their density contrast with the fluid. While drag is predominant for small particles, gravity takes over the dynamics of large particles and most studies treat these two asymptotics independently. However it is usually at this critical transition that standard modeling fails, as is evident when estimating for instance the rate at which rain is triggered in warm clouds [1,2]. Most models are unable to circumvent a bottleneck in the droplet growth for diameters around 20-40µm. A key improvement might be to combine turbulent and gravitational effects.In this Letter we understand the intriguing interplay between turbulence, gravity, and particle sizes. This question is of fundamental importance in fluid dynamics, in particular, and in non-equilibrium statistical physics, in general, as it is central to modeling coalescences in natural or laboratory droplet suspensions. The most noticeable effect of turbulence on the settling of heavy particles is the increase of their terminal velocity induced by a preferential sweeping along the downward fluid flow [3][4][5]. This phenomenon is mostly understood on qualitative grounds and has been quantified only in model flows [6]. Furthermore very little is known on the effect of gravitational settling on two-particle statistics. Fundamental theoretical and numerical studies of the clustering of particle pairs [7,8] and of the enhancement of collisions due to inertia [9, 10] usually neglect gravity. We present here, by combining state-of-the-art direct numerical simulations with theoretical results based on our asymptotic analysis, a systematic study of the dynamical and statistical properties of particles as a function of (i) the level of turbulence of the carrier flow (Reynolds number), (ii) the inertia of the particles (Stokes number), and (iii) the ratio between the turbulent accelerations and gravity (Froude numb...

It is shown that the solutions of inviscid hydrodynamical equations with suppression of all spatial Fourier modes having wave numbers in excess of a threshold K(G) exhibit unexpected features. The study is carried out for both the one-dimensional Burgers equation and the two-dimensional incompressible Euler equation. For large K(G) and smooth initial conditions, the first symptom of truncation, a localized short-wavelength oscillation which we call a "tyger," is caused by a resonant interaction between fluid particle motion and truncation waves generated by small-scale features (shocks, layers with strong vorticity gradients, etc.). These tygers appear when complex-space singularities come within one Galerkin wavelength λ(G)=2π/K(G) from the real domain and typically arise far away from preexisting small-scale structures at locations whose velocities match that of such structures. Tygers are weak and strongly localized at first-in the Burgers case at the time of appearance of the first shock their amplitudes and widths are proportional to K(G)(-2/3) and K(G)(-1/3), respectively-but grow and eventually invade the whole flow. They are thus the first manifestations of the thermalization predicted by T. D. Lee [Q. J. Appl. Math. 10, 69 (1952)]. The sudden dissipative anomaly-the presence of a finite dissipation in the limit of vanishing viscosity after a finite time t(⋆)-which is well known for the Burgers equation and sometimes conjectured for the three-dimensional Euler equation, has as counterpart, in the truncated case, the ability of tygers to store a finite amount of energy in the limit K(G)→∞. This leads to Reynolds stresses acting on scales larger than the Galerkin wavelength and thus prevents the flow from converging to the inviscid-limit solution. There are indications that it may eventually be possible to purge the tygers and thereby to recover the correct inviscid-limit behavior.

We find that the effects of a localized perturbation in a chaotic classical many-body system-the classical Heisenberg chain at infinite temperature-spread ballistically with a finite speed even when the local spin dynamics is diffusive. We study two complementary aspects of this butterfly effect: the rapid growth of the perturbation, and its simultaneous ballistic (light-cone) spread, as characterized by the Lyapunov exponents and the butterfly speed, respectively. We connect this to recent studies of the out-of-time-ordered commutators (OTOC), which have been proposed as an indicator of chaos in a quantum system. We provide a straightforward identification of the OTOC with a natural correlator in our system and demonstrate that many of its interesting qualitative features are present in the classical system. Finally, by analyzing the scaling forms, we relate the growth, spread, and propagation of the perturbation with the growth of one-dimensional interfaces described by the Kardar-Parisi-Zhang equation.