& Frisch (1980), by a shift of origin. It offers the advantage of the symmetric Fourier representation (1 2 ) . Early-time behaviour of the inviscid flowThe symmetries of the TG vortex are listed in appendix A. Here we emphasize only those that help to visualize the qualitative features of the flow and those that may be important in making this flow atypical of general three-dimensional flow. First, for all times, no fluid crosses any of the boundaries x , y or z = nn, where n is an integer.Therefore the flow can be visualized as flow in the box 0 < x , y , z < n with impermeable stress-free faces. I n the following discussion, the region 0 < x, y , z < n is termed the impermeable box, as it confines the flow, while the region 0 < x , y , z < 2n is termed the periodicity box, as it reflects the periodicity of the Fourier series (1.2). Also, because of the symmetries listed in appendix A, the flow at any point in space can be inferred from its values in the fundamental box 0 < x, y, z < in.Secondly, if near each face we write the velocity field in terms of components parallel or perpendicular to the face, i.e. u = u,, + vl, then ul and au,,/an vanish on the face. This implies that the vorticity on each face is normal to that face so it may be written w = @, where fi is the unit normal. Note that g must vanish on all edges of the box where faces meet. I n contrast, a general incompressible flow will have only isolated points of vanishing vorticity. (Both velocity and vorticity also vanish for all time a t the centre x = y = z = in.)Thirdly, the vanishing of ul and au,,/an on each face also implies that the tensor V v is partly diagonal. One principal axis of the strain rate or symmetric part of this tensor is then perpendicular to the face. Furthermore, the magnitude of the strain rate along this axis determines the fractional growth rate of the normal vorticity on the face, i.e. d % = --~r * v --lnlcl. an ' Idt (2.1) 14-2

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International audienc

International audienc

Cavitation in a liquid seeded with bubbles is used as a new visualization technique to single out the regions of very low pressure of a fully developed turbulent flow. By this means, the sudden appearance of high vorticity filaments is observed. These structures are very thin and short lived and display a high degree of temporal as well as spatial intermittency. They contribute to the flow organization: In particular their disintegration corresponds to the formation of large eddies.

A new transient regime in the relaxation towards absolute equilibrium of the conservative and time-reversible 3-D Euler equation with high-wavenumber spectral truncation is characterized. Large-scale dissipative effects, caused by the thermalized modes that spontaneously appear between a transition wavenumber and the maximum wavenumber, are calculated using fluctuation dissipation relations. The large-scale dynamics is found to be similar to that of high-Reynolds number Navier-Stokes equations and thus to obey (at least approximately) Kolmogorov scaling.PACS numbers: 47.27. Eq,05.20.Jj, 83.60.Df Turbulence has been observed in inviscid and conservative systems, in the context of (compressible) lowtemperature superfluid turbulence [1,2,3]. This behavior has also been reproduced using simple (incompressible) Biot-Savart vortex methods, which amount to Eulerian dynamics with ad hoc vortex reconnection [4]. The purpose of the present letter is to study the dynamics of spectrally truncated 3-D incompressible Euler flows. Our main result is that the inviscid and conservative Euler equation, with a high-wavenumber spectral truncation, has long-lasting transients which behave just as those of the dissipative (with generalized dissipation) Navier-Stokes equation. This is so because the thermalized modes between some transition wavenumber and the maximum wavenumber can act as a fictitious microworld providing an effective viscosity to the modes with wavenumbers below the transition wavenumber.We thus study general solutions to the finite system of ordinary differential equations for the complex variableŝwhereThis system is time-reversible and exactly conserves the kinetic energy E = k E(k, t), where the energy spectrum E(k, t) is defined by averagingv(k ′ , t) on spherical shells of width ∆k = 1,The discrete equations (1) are classically obtained [5] by performing a Galerkin truncation (v(k) = 0 for sup α |k α | ≤ k max ) on the Fourier transform v(x, t) = v(k, t)e ik·x of a spatially periodic velocity field obeying the (unit density) three-dimensional incompressible Euler equations,The short-time, spectrally-converged truncated Eulerian dynamics (1) has been studied [6,7] to obtain numerical evidence for or against blowup of the original (untruncated) Euler equations (3). We will study here the behavior of solutions of (1) when spectral convergence to solutions of (3) is lost. Long-time truncated Eulerian dynamics is relevant to the limitations of standard simulations of high Reynolds number (small viscosity) turbulence which are performed using Galerkin truncations of the Navier-Stokes equation [8].Equations (1) are solved numerically using standard [9] pseudo-spectral methods with resolution N . The solutions are dealiased by spectrally truncating the modes for which at least one wave-vector component exceeds N/3 (thus a 1600 3 run is truncated at k max = 534). This method allows the exact evaluation of the Galerkin convolution in (1) in only N 3 log N operations. Time marching is done with a second-order ...

The continuous limit of quantum walks (QWs) on the line is revisited through a recently developed method. In all cases but one, the limit coincides with the dynamics of a Dirac fermion coupled to an artificial electric and/or relativistic gravitational field. All results are carefully discussed and illustrated by numerical simulations.

The three-dimensional stability of two-dimensional vortical states of planar mixing layers is studied by direct numerical integration of the Navier-Stokes equations. Small-scale instabilities are shown to exist for spanwise scales at which classical linear modes are stable. These modes grow on convective timescales, extract their energy from the mean flow and exist at moderately low Reynolds numbers. Their growth rates are comparable with the most rapidly growing inviscid instability and with the growth rates of two-dimensional subharmonic (pairing) modes. At high amplitudes, they can evolve into pairs of counter-rotating, streamwise vortices, connecting the primary spanwise vortices, which are very similar to the structures observed in laboratory experiments. The three-dimensional modes do not appear to saturate in quasi-steady states as do the purely two-dimensional fundamental and subharmonic modes in the absence of pairing. The subsequent evolution of the flow depends on the relative amplitudes of the pairing modes. Persistent pairings can inhibit three-dimensional instability and, hence, keep the flow predominantly two-dimensional. Conversely, suppression of the pairing process can drive the three-dimensional modes to more chaotic, turbulent-like states. An analysis of high-resolution simulations of fully turbulent mixing layers confirms the existence of rib-like structures and that their coherence depends strongly on the presence of the two-dimensional pairing modes.