Large time behavior and asymptotic stability of the two-dimensional Euler and linearized Euler equations

We study the asymptotic behavior and the asymptotic stability of the two-dimensional Euler equations and of the two-dimensional linearized Euler equations close to parallel flows. We focus on spectrally stable jet profiles $U(y)$ with stationary streamlines $y_{0}$ such that $U'(y_{0})=0$, a case that has not been studied previously. We describe a new dynamical phenomenon: the depletion of the vorticity at the stationary streamlines. An unexpected consequence, is that the velocity decays for large times with power laws, similarly to what happens in the case of the Orr mechanism for base flows without stationary streamlines. The asymptotic behaviors of velocity and the asymptotic profiles of vorticity are theoretically predicted and compared with direct numerical simulations. We argue on the asymptotic stability of these flow velocities even in the absence of any dissipative mechanisms.Comment: To be published in Physica D, nonlinear phenomena (accepted January 2010

Hysteresis phenomenon in turbulent convection

Coherent large-scale circulations of turbulent thermal convection in air have been studied experimentally in a rectangular box heated from below and cooled from above using Particle Image Velocimetry. The hysteresis phenomenon in turbulent convection was found by varying the temperature difference between the bottom and the top walls of the chamber (the Rayleigh number was changed within the range of $10^7 - 10^8$). The hysteresis loop comprises the one-cell and two-cells flow patterns while the aspect ratio is kept constant ($A=2 - 2.23$). We found that the change of the sign of the degree of the anisotropy of turbulence was accompanied by the change of the flow pattern. The developed theory of coherent structures in turbulent convection (Elperin et al. 2002; 2005) is in agreement with the experimental observations. The observed coherent structures are superimposed on a small-scale turbulent convection. The redistribution of the turbulent heat flux plays a crucial role in the formation of coherent large-scale circulations in turbulent convection.Comment: 10 pages, 9 figures, REVTEX4, Experiments in Fluids, 2006, in pres

Enviromental patterns and intermittent cascades

Real environmental flows are non-homogeneous, of fundamental interest is to determine and quantify turbulent diffusion from the available conditions of the flow, because the role of buoyancy and rotation modify the flow topology with often the dominant scale occurring when these two forces are in equilibrium. In geophysical flows both in the Atmosphere and the Ocean, the main forcing occurs at the Rossby deformation Radius with both direct and inverse energy cascades [1,2]. The role of the spectra of steady and decaying turbulence is important as well as its scale to scale conditions, so that a large range of scales has to be taken into account. When mixing and dispersion processes are studied, the behaviour of reactants or pollutants is seen to depend of both the intermittency of the vorticity and energy spectra. If irreversible molecular mixing has to be accounted, the range of scales spans from hundreds of Kilometres to the Bachelor or Kolmogorov sub millimeter scales. It is important to evaluate mixing and compare with oscillating grid experiments, Redondo [3], across a density interface measuring entrainment and grid decaying non steady mixing. These experiments are evaluated and compared with results of a Kinematic simulation model, Castilla [4]. The local vorticity is evaluated confirming the trapping of tracers in the strong vertical regions in 2D flows, but showing also that hyperdiffusion may also occur. Intermittency was evaluated using numerical evaluation of higher order moments in different types of 2D and 3D turbulence.Peer ReviewedPostprint (published version

Marangoni driven turbulence in high energy surface melting processes

Experimental observations of high-energy surface melting processes, such as laser welding, have revealed unsteady, often violent, motion of the free surface of the melt pool. Surprisingly, no similar observations have been reported in numerical simulation studies of such flows. Moreover, the published simulation results fail to predict the post-solidification pool shape without adapting non-physical values for input parameters, suggesting the neglect of significant physics in the models employed. The experimentally observed violent flow surface instabilities, scaling analyses for the occurrence of turbulence in Marangoni driven flows, and the fact that in simulations transport coefficients generally have to be increased by an order of magnitude to match experimentally observed pool shapes, suggest the common assumption of laminar flow in the pool may not hold, and that the flow is actually turbulent. Here, we use direct numerical simulations (DNS) to investigate the role of turbulence in laser melting of a steel alloy with surface active elements. Our results reveal the presence of two competing vortices driven by thermocapillary forces towards a local surface tension maximum. The jet away from this location at the free surface, separating the two vortices, is found to be unstable and highly oscillatory, indeed leading to turbulence-like flow in the pool. The resulting additional heat transport, however, is insufficient to account for the observed differences in pool shapes between experiment and simulations

Beyond scaling and locality in turbulence

An analytic perturbation theory is suggested in order to find finite-size corrections to the scaling power laws. In the frame of this theory it is shown that the first order finite-size correction to the scaling power laws has following form $S(r) \cong cr^{\alpha_0}[\ln(r/\eta)]^{\alpha_1}$, where $\eta$ is a finite-size scale (in particular for turbulence, it can be the Kolmogorov dissipation scale). Using data of laboratory experiments and numerical simulations it is shown shown that a degenerate case with $\alpha_0 =0$ can describe turbulence statistics in the near-dissipation range $r > \eta$, where the ordinary (power-law) scaling does not apply. For moderate Reynolds numbers the degenerate scaling range covers almost the entire range of scales of velocity structure functions (the log-corrections apply to finite Reynolds number). Interplay between local and non-local regimes has been considered as a possible hydrodynamic mechanism providing the basis for the degenerate scaling of structure functions and extended self-similarity. These results have been also expanded on passive scalar mixing in turbulence. Overlapping phenomenon between local and non-local regimes and a relation between position of maximum of the generalized energy input rate and the actual crossover scale between these regimes are briefly discussed.Comment: extended versio

Quantized vortices in superfluid helium and atomic Bose-Einstein condensates

This article reviews recent developments in the physics of quantized vortices in superfluid helium and atomic Bose-Einstein condensates. Quantized vortices appear in low-temperature quantum condensed systems as the direct product of Bose-Einstein condensation. Quantized vortices were first discovered in superfluid 4He in the 1950s, and have since been studied with a primary focus on the quantum hydrodynamics of this system. Since the discovery of superfluid 3He in 1972, quantized vortices characteristic of the anisotropic superfluid have been studied theoretically and observed experimentally using rotating cryostats. The realization of atomic Bose-Einstein condensation in 1995 has opened new possibilities, because it became possible to control and directly visualize condensates and quantized vortices. Historically, many ideas developed in superfluid 4He and 3He have been imported to the field of cold atoms and utilized effectively. Here, we review and summarize our current understanding of quantized vortices, bridging superfluid helium and atomic Bose-Einstein condensates. This review article begins with a basic introduction, which is followed by discussion of modern topics such as quantum turbulence and vortices in unusual cold atom condensates.Comment: 99 pages, 20 figures, Review articl

Towards an experimental von Karman dynamo: numerical studies for an optimized design

Numerical studies of a kinematic dynamo based on von Karman type flows between two counterrotating disks in a finite cylinder are reported. The flow has been optimized using a water model experiment, varying the driving impellers configuration. A solution leading to dynamo action for the mean flow has been found. This solution may be achieved in VKS2, the new sodium experiment to be performed in Cadarache, France. The optimization process is described and discussed, then the effects of adding a stationary conducting layer around the flow on the threshold, on the shape of the neutral mode and on the magnetic energy balance are studied. Finally, the possible processes involved into kinematic dynamo action in a von Karman flow are reviewed and discussed. Among the possible processes we highlight the joint effect of the boundary-layer radial velocity shear and of the Ohmic dissipation localized at the flow/outer-shell boundary

Theoretical and Computational Analyses on Transition and Turbulence in Purely Oscillating Pipe Flow

The hydrodynamic instability of purely oscillating pipe flows is investigated in terms of the quasi-steady formulation assuming the temporal changes of the laminar base flow relative to disturbances are slow. A simple model equation is introduced to compare the exact solution of the current approach with those of the major theories dedicated for unsteady flows. The results of the analysis show that the quasi-steady assumption and the multiple scales method are more efficient and accurate than the formal Floquet theory in predicting the transient instabilities within a period in addition to the long-term growth or decay of disturbances. The most significant contribution of the present quasi-steady analysis is the neutral stability curves, from which the critical Reynolds numbers are obtained for a wide range of oscillation frequencies. The stability criterion pertains to the cycle-averaged growth rates obtained from the eigenvalues of the parametric stability problem. Moreover, the approximate accuracy of the quasi-steadiness is assessed by proposing a new mathematical relation, confirming the validity of the method for the stability analysis. The theoretical findings are consistent with some experimental results although some others show quantitative discrepancies, which can be attributed mostly to the deviations in the second spatial derivative of base flow. In the computational analyses of this research, direct numerical simulations (DNS) based on the spectral element method are performed to verify the theoretical predictions and to accurately examine the transition to turbulence. The onset of transition in smooth pipe, identified as a disturbed laminar flow after imposing small random perturbations as initial conditions, qualitatively agrees with that estimated by the quasi-steady theory. The later transition stage at which the turbulence and relaminarization phenomena first emerge is detected from the high-amplitude velocity fluctuations. The turbulence intensity increases with the Stokes number proportional to oscillation frequency. Furthermore, surface roughness constructed utilizing the overset-grid technique is also considered as the triggering mechanism to induce transition and turbulence with small wavy imperfections distributed along the inner wall of a pipe. The influence of the surface roughness on flow stability is evaluated and the critical Reynolds number is close to that of the smooth pipe unless the roughness height is large. The friction coefficients at a few flow conditions for both smooth and rough pipes are determined according to the maximum values of the wall shear stress and found to be compatible with those shown by experiments in the literature