We report experiments on gravity-capillary wave turbulence on the surface of a fluid. The wave amplitudes are measured simultaneously in time and space using an optical method. The full space-time power spectrum shows that the wave energy is localized on several branches in the wave-vector-frequency space. The number of branches depend on the power injected within the waves. The measurement of the nonlinear dispersion relation is found to be well described by a law suggesting that the energy transfer mechanisms involved in wave turbulence are not only restricted to purely resonant interaction between nonlinear waves. The power-law scaling of the spatial spectrum and the probability distribution of the wave amplitudes at a given wave number are also measured and compared to the theoretical predictions.
We report new measurements of single particle dispersion in turbulent two-dimensional (2D) flows. Laboratory experiments in electromagnetically driven and Faraday wave driven turbulence reveal a transition from weakly dispersing superdiffusive regime to strongly dispersing Brownian diffusion as the flow energy is increased in a broad range. The transition to fully developed 2D turbulence is characterized by the topological changes in the fluid particle trajectories and the development of self-similar diffusion. The degree of 2D turbulence development can be quantified by a parameter describing…
We apply a simple method to provide explicit expressions for different scaling exponents in intermittent fully developed turbulence, that before were only given through a Legendre transform. This includes predictability exponents for infinitesimal and non infinitesimal perturbations, Lagrangian velocity exponents, and dispersion exponents. We obtain also new results concerning inverse statistics corresponding to exit-time moments.
The statistical geometry of dispersing Lagrangian clusters of four particles (tetrahedra) is studied by means of high-resolution direct numerical simulations of three-dimensional homogeneous isotropic turbulence. We give the first evidence of a self-similar regime of shape dynamics characterized by almost two-dimensional, strongly elongated geometries. The analysis of four-point velocity-difference statistics and orientation shows that inertial-range eddies typically generate a straining field with a strong extensional component aligned with the elongation direction and weak extensional/compre…
We present models for single-particle dispersion in vertical and horizontal directions of stably stratified flows. The model in the vertical direction is based on the observed Lagrangian spectrum of the vertical velocity, while the model in the horizontal direction is a combination of a continuous-time eddy-constrained random walk process with a contribution to transport from horizontal winds. Transport at times larger than the Lagrangian turnover time is not universal and dependent on these winds. The models yield results in good agreement with direct numerical simulations of stratified turbu…
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 comp…
We study the effect of different forcing functions and of the local gradient Richardson number Rig on the vertical mixing of Lagrangian tracers in stably stratified turbulence under the Boussinesq approximation, and present a wave and continuous-time random walk model for single- and two- particle vertical dispersion. The model consists of a random superposition of linear waves with their amplitude based on the observed Lagrangian spectrum of vertical velocity, and a random walk process to capture overturning that depends on the statistics of Rig among other Eulerian quantities. The model is i…
Single-particle dispersion in the Bolgiano--Obukhov regime of two-dimensional turbulent convection is investigated. Unlike dispersion in a flow displaying the classical K41 phenomenology, here, the leading contribution to the Lagrangian velocity fluctuations is given by the largest eddies. This implies a linear behavior in time for a typical velocity fluctuation in the time interval $t$. The contribution to the Lagrangian velocity fluctuations of local eddies (i.e. with a characteristic time of order $t$), whose space/time scalings are ruled by the Bolgiano--Obukhov theory, is thus not detecta…
The concept of Nonlinear dispersion relation (NDR) is used in various fields of Physics (nonlinear optics, hydrodynamics, hydroelasticity, mechanics, quantum optics, plasma physics,...) to characterize fundamental phenomena induced by nonlinearity such as wave frequency shift or turbulence. Nonlinear random waves described by the onedimensional nonlinear Schrodinger equation (1DNLSE) exhibit a remarkable form of turbulence called "integrable turbulence" where solitons play a key role. Surprisingly, little attention has been paid to the NDR of such universal wave systems up to a very recent the…
Lagrangian studies of the local temperature mixing and heat transport in turbulent Rayleigh-Benard convection are presented, based on three-dimensional direct numerical simulations. Contrary to vertical pair distances, the temporal growth of lateral pair distances agrees with the Richardson law, but yields a smaller Richardson constant due to correlated pair motion in plumes. Our results thus imply that Richardson dispersion is also found in anisotropic turbulence. We find that extremely large vertical accelerations appear less frequently than lateral ones and are not connected with rising or …
In this paper we present a new model for modeling the diffusion and relative dispersion of particles in homogeneous isotropic turbulence. We use an Heisenberg-like Hamiltonian to incorporate spatial correlations between fluid particles, which are modeled by stochastic processes correlated in time. We are able to reproduce the ballistic regime in the mean squared displacement of single particles and the transition to a normal diffusion regime for long times. For the dispersion of particle pairs we find a $t^{2}$-dependence of the mean squared separation at short times and a $t$-dependence for l…
In this paper we investigate both theoretically and numerically the forward in time (FIT) and backward in time (BIT) dispersion of fluid and inertial particle pairs in isotropic turbulence. Fluid particles are known to separate faster BIT than FIT in three-dimensional turbulence, and we find that inertial particles do the same. However, we find that the irreversibility in the inertial particle dispersion is in general much stronger than that for fluid particles. For example, the ratio of the BIT to FIT mean-square separation can be up to an order of magnitude larger for inertial particles than…
Turbulent mixing of liquids and gasses is ubiquitous in nature. It is the basis of all industrial fluid mixing processes, and it determines the spread of pollutants or bioagents in the atmosphere and oceans. Biological organisms even use it to survive in marine ecosystems. A fundamental component of turbulent mixing is the separation of two nearby fluid elements, i.e., pair dispersion. Despite almost eighty years of intense scientific inquiry, no clear understanding of this fundamental aspect of turbulence has emerged. One critical unresolved question is the extent to which the initial separat…
In this paper we consider a recent theoretical prediction (Bragg \emph{et al.}, Phys. Fluids \textbf{28}, 013305 (2016)) that for inertial particles in 2D turbulence, the nature of the irreversibility of the particle-pair dispersion inverts when the particle inertia exceeds a certain value. In particular, when the particle Stokes number, ${\rm St}$, is below a certain value, the forward-in-time (FIT) dispersion should be faster than the backward-in-time (BIT) dispersion, but for ${\rm St}$ above this value, this should invert so that BIT becomes faster than FIT dispersion. This non-trivial beh…
Bubbles play an important role in the transport of chemicals and nutrients in many natural and industrial flows. Their dispersion is crucial to understand the mixing processes in these flows. Here we report on the dispersion of millimetric air bubbles in a homogeneous and isotropic turbulent flow with Reynolds number $\text{Re}_λ$ from $110$ to $310$. We find that the mean squared displacement (MSD) of the bubbles far exceeds that of fluid tracers in turbulence. The MSD shows two regimes. At short times, it grows ballistically ($\propto τ^2$), while at larger times, it approaches the diffusive…
Lagrangian dispersion in experimental stratified turbulence
Laboratoire des Ecoulements Geophysiques et Industriels (LEGI), Universite Grenoble Alpes, CNRS, Grenoble-INP（地球物理和工业流动实验室）；Imperial College London（伦敦帝国理工学院）；Laboratoire de Mécanique et ses Interfaces, ENSTA Paris, Institut Polytechnique de Paris（力学及其界面实验室）
In this paper, the results of an international collaborative test case relative to the production of a Direct Numerical Simulation and Lagrangian Particle Tracking database for turbulent particle dispersion in channel flow at low Reynolds number are presented. The objective of this test case is to establish a homogeneous source of data relevant to the general problem of particle dispersion in wall-bounded turbulence. Different numerical approaches and computational codes have been used to simulate the particle-laden flow and calculations have been carried on long enough to achieve a statistica…
In this work, single particle dispersion was analyzed for a bacterial turbulence by retrieving the virtual Lagrangian trajectory via numerical integration of the Lagrangian equation. High-order displacement functions were calculated for cases with and without mean velocity effect. Two-regime power-law behavior for short and long time evolutions were identified experimentally, which were separated by the Lagrangian integral time. For the case with the mean velocity effect, the experimental Hurst numbers were determined to be $0.94$ and $0.97$ for short and long times evolutions, respectively. F…
Turbulence in the upper ocean in the submesoscale range (scales smaller than the deformation radius) plays an important role for the heat exchange with the atmosphere and for oceanic biogeochemistry. Its dynamics should strongly depend on the seasonal cycle and the associated mixed-layer instabilities. The latter are particularly relevant in winter and are responsible for the formation of energetic small scales that extend over the whole depth of the mixed layer. The knowledge of the transport properties of oceanic flows at depth, which is essential to understand the coupling between surface a…
Large-scale magnetic fields thread through the electrically conducting matter of the interplanetary and interstellar medium, stellar interiors, and other astrophysical plasmas, producing anisotropic flows with regions of high-Reynolds-number turbulence. It is common to encounter turbulent flows structured by a magnetic field with a strength approximately equal to the root-mean-square magnetic fluctuations. In this work, direct numerical simulations of anisotropic magnetohydrodynamic (MHD) turbulence influenced by such a magnetic field are conducted for a series of cases that have identical res…
We study wave turbulence in shallow water flows in numerical simulations using two different approximations: the shallow water model, and the Boussinesq model with weak dispersion. The equations for both models were solved using periodic grids with up to $2048^2$ points. In all simulations, the Froude number varies between $0.015$ and $0.05$, while the Reynolds number and level of dispersion are varied in a broader range to span different regimes. In all cases, most of the energy in the system remains in the waves, even after integrating the system for very long times. For shallow flows, non-l…
We present an introductory overview of several challenging problems in the statistical characterisation of turbulence. We provide examples from fluid turbulence in three and two dimensions, from the turbulent advection of passive scalars, turbulence in the one-dimensional Burgers equation, and fluid turbulence in the presence of polymer additives.
Synthetic models of Eulerian turbulence like so called "Kinematic Simulations" (KS) have been criticized by Thomson & Devenish (TD) (2005), who argued that sweeping decorrelation effects suppress pair dispersion in such models. We derive analytical results for Eulerian turbulence modeled by Gaussian random fields for the case with zero mean velocity. Our starting point is an exact integrodifferential equation for the particle pair separation distribution. When memory times of particle locations are short, a Markovian approximation leads to a Richardson-type diffusion model. We obtain a diffusi…
Relative dispersion and eddy diffusivity in laboratory experiments of $β$-plane turbulence
University of St Andrews（圣安德鲁斯大学）；University of Glasgow（格拉斯哥大学）；Aix Marseille Univ, CNRS, Centrale Med, IRPHE（艾克斯-马赛大学、法国国家科学研究中心、中央马赛学院、IRPHE）；Department of Geosciences, University of Oslo（奥斯陆大学地球科学系）
We investigate experimentally turbulence of surface gravity waves in the Coriolis facility in Grenoble by using both high sensitivity local probes and a time and space resolved stereoscopic reconstruction of the water surface. We show that the water deformation is made of the superposition of weakly nonlinear waves following the linear dispersion relation and of bound waves resulting from non resonant triadic interaction. Although the theory predicts a 4-wave resonant coupling supporting the presence of an inverse cascade of wave action, we do not observe such inverse cascade. We investigate 4…
The classical statistics of turbulence are shown to be not specific to turbulence and can be derived from a solution for recurring unsteady state viscous flow. Care must be exercised in using them to make deductions about turbulence structures and mechanisms. The conditionally averaged statistics, particularly involving the velocities of the ejections in the burst phase, are more distinctive of turbulence. Key words: Turbulence statistics, classical, conditional average, probability density function, energy spectrum
Stratified turbulence shows scale- and direction-dependent anisotropy and the coexistence of weak turbulence of internal gravity waves and strong turbulence of eddies. Straightforward application of standard analyses developed in isotropic turbulence sometimes masks important aspects of the anisotropic turbulence. To capture detailed structures of the energy distribution in the wave-number space, it is indispensable to examine the energy distribution with non-integrated spectra by fixing the codimensional wave-number component or in the two-dimensional domain spanned by both the horizontal and…
We report an experimental investigation of the effect of finite depth on the statistical properties of wave turbulence at the surface of water in the gravity-capillary range. We tune the wave dispersion and the level of nonlinearity by modifying the depth of water and the forcing respectively. We use space-time resolved profilometry to reconstruct the deformed surface of water. When decreasing the water depth, we observe a drastic transition between weak turbulence at the weakest forcing and a solitonic regime at stronger forcing. We characterize the transition between both states by studying …
The statistics of the deformation and breakup of neutrally buoyant sub-Kolmogorov ellipsoidal drops is investigated via Lagrangian simulations of homogeneous isotropic turbulence. The mean lifetime of a drop is also studied as a function of the initial drop size and the capillary number. A vector model of drop previously introduced by Olbricht, Rallison and Leal [J. Non-Newtonian Fluid Mech. $\mathbf{10}$, 291 (1982)] is used to predict the behaviour of the above quantities analytically.
We carry out a direct numerical simulation (DNS) study that reveals the effects of polymers on statistically steady, forced, homogeneous, isotropic fluid turbulence. We find clear manifestations of dissipation-reduction phenomena: On the addition of polymers to the turbulent fluid, we obtain a reduction in the energy dissipation rate, a significant modification of the fluid energy spectrum, especially in the deep-dissipation range, a suppression of small-scale intermittency, and a decrease in small-scale vorticity filaments. We also compare our results with recent experiments and earlier DNS s…
The statistical properties of fluid particles transported by a fully developed turbulent flow are investigated by means of high resolution direct numerical simulations. Single trajectory statistics is investigated in a time range spanning more than three decades, from less than a tenth of the Kolmogorov timescale up to one large-eddy turnover time. Acceleration and velocity statistics show a neat quantitative agreement with recent experimental results. Trapping effects in vortex filaments give rise to enhanced small-scale intermittency on Lagrangian observables.
A statistically stationary turbulence with a mean shear gradient is realized in a flow driven by suitable body forces. The flow domain is periodic in downstream and spanwise directions and bounded by stress free surfaces in the normal direction. Except for small layers near the surfaces the flow is homogeneous. The fluctuations in turbulent energy are less violent than in the simulations using remeshing, but the anisotropy on small scales as measured by the skewness of derivatives is similar and decays weakly with increasing Reynolds number.
In Taylor's theory, the longitudinal dispersion in turbulent pipe flows approaches, on long timescales, a diffusive behavior with a constant diffusivity $K_L$, that depends \emph{empirically} on the Reynolds number $Re$. We show that the dependence on $Re$ can be determined from the turbulent energy spectrum. By using the intimate connection between the friction factor and longitudinal dispersion in wall-bounded turbulence, we predict different asymptotic scaling laws of $K_L(Re)$ depending on the different turbulent cascades in two-dimensional turbulence. We also explore numerically the $K_L(…
We provide evidence that the small-scale statistics of the acceleration of a test particle in high-Reynolds number Lagrangian turbulence is correctly described by Tsallis statistics with entropic index q=3/2. We present theoretical arguments why Tsallis statistics can naturally arise in Lagrangian turbulence and why at the smallest scales q=3/2 is relevant. A generalized Heisenberg-Yaglom formula is derived from the nonextensive model.
A calculational approach in fluid turbulence is presented. Use is made of the attracting nature of the fluid-dynamic dynamical system. An approach is offered that effectively propagates the statistics in time. Loss of sensitivity to an initial probability density functional and generation of stationary statistical effects is speculated.
New methods of flow visualization near absolute zero have opened the way to directly compare quantum turbulence (in superfluid helium) to classical turbulence (in ordinary fluids such as air or water) and explore analogies and differences. We present results of numerical simulations in which we examine the statistics of the superfluid acceleration in thermal counterflow. We find that, unlike the velocity, the acceleration obeys scaling laws similar to classical turbulence, in agreement with a recent quantum turbulence experiment of La Mantia et al.
We study the dispersion of light particles floating on a flat shear-free surface of an open channel in which the flow is turbulent. This configuration mimics the motion of buoyant matter (e.g. phytoplankton, pollutants or nutrients) in water bodies when surface waves and ripples are smooth or absent. We perform direct numerical simulation of turbulence coupled with Lagrangian particle tracking, considering different values of the shear Reynolds number (Reτ = 171 and 509) and of the Stokes number (0.06 < St < 1 in viscous units). Results show that particle buoyancy induces clusters that evolve …
We present high-resolution direct numerical simulations of turbulent three-dimensional Rayleigh-Benard convection with a focus on the Lagrangian properties of the flow. The volume is a Cartesian slab with an aspect ratio of four bounded by free-slip planes at the top and bottom and with periodic side walls. The turbulence is inhomogeneous with respect to the vertical direction. This manifests in different lateral and vertical two-particle dispersion and in a dependence of the dispersion on the initial tracer position for short and intermediate times. Similar to homogeneous isotropic turbulence…
Cellular suspensions such as dense bacterial flows exhibit a turbulence-like phase under certain conditions. We study this phenomenon of "active turbulence" statistically by using numerical tools. Following Wensink et al. [Proc. Natl. Acad. Sci. U.S.A. 109, 14308 (2012)], we model active turbulence by means of a generalized Navier-Stokes equation. Two-point velocity statistics of active turbulence, both in the Eulerian and the Lagrangian frame, is explored. We characterize the scale-dependent features of two-point statistics in this system. Furthermore, we extend this statistical study with me…
We present a description and validation of a finite volume solver aimed at solving the problems of microscale urban flows where vegetation is present. The solver is based on the five equation system of Reynolds-averaged Navier-Stokes equations for atmospheric boundary layer flows, which are complemented by the k-epsilon turbulence model. The vegetation is modelled as a porous zone, and the effects of the vegetation are included in the momentum and turbulence equations. A detailed dry deposition model is incorporated in the pollutant transport equation, allowing the investigation of the filteri…
Due to large scale flow inhomogeneities and the effects of temperature, turbulence small-scale structure in thermal convection is still an active field of investigation, especially considering sophisticated Lagrangian statistics. Here we experimentally study Lagrangian pair dispersion (one of the canonical problems of Lagrangian turbulence) in a Rayleigh-Bénard convection cell. A sufficiently high temperature difference is imposed on a horizontal layer of fluid to observe a turbulent flow. We perform Lagrangian tracking of sub-millimetric particles on a large measurement volume including part …
We study the statistical properties of helicity in direct numerical simulations of fully developed homogeneous and isotropic turbulence and in a class of turbulence shell models. We consider correlation functions based on combinations of vorticity and velocity increments that are not invariant under mirror symmetry. We also study the scaling properties of high-order structure functions based on the moments of the velocity increments projected on a subset of modes with either positive or negative helicity (chirality). We show that mirror symmetry is recovered at small-scales, i.e., chiral terms…
A calculational approach in fluid turbulence is presented. Use is made of the attracting nature of the fluid-dynamic dynamical system. An approximate approach is offerred that effectively propagates the statistics in time. Loss of sensitivity to an initial probability density functional and generation of steady-state statistical effects is suggested.
We present results from a numerical study of particle dispersion in the weakly nonlinear regime of Rayleigh-Bénard convection of a fluid with Prandtl number around unity, where bi-stability between ideal straight convection rolls and weak turbulence in the form of Spiral Defect Chaos exists. While Lagrangian pair statistics has become a common tool for studying fully developed turbulent flows at high Reynolds numbers, we show that key characteristics of mass transport can also be found in convection systems that show no or weak turbulence. Specifically, for short times, we find Richardson-like…
We solve an inverse problem for fluid particle pair-statistics: we show that a time sequence of probability density functions (PDF's) of separations can be exactly reproduced by solving the diffusion equation with a suitable time-dependent diffusivity. The diffusivity tensor is given by a time-integral of a conditional Lagrangian velocity structure-function, weighted by a ratio of PDF's. Physical hypotheses for hydrodynamic turbulence (sweeping, short memory, mean-field) yield simpler integral formulas, including one of Kraichnan and Lundgren. We evaluate the latter using a spacetime database …
The angle between subsequent particle displacement increments is evaluated as a function of the timelag in isotropic turbulence. It is shown that the evolution of this angle contains two well-defined power-laws, reflecting the multi-scale dynamics of high-Reynolds number turbulence. The proba-bility density function of the directional change is shown to be self-similar and well approximated by an analytically derived model assuming Gaussianity and independence of the velocity and the Lagrangian acceleration.
In this paper, we use our recently developed theory for the backward-in-time (BIT) relative dispersion of inertial particles in turbulence (Bragg \emph{et al.}, Phys. Fluids 28, 013305, 2016) to develop the theoretical model by Pan \& Padoan (J. Fluid Mech. 661 73, 2010) for inertial particle relative velocities in isotropic turbulence. We focus on the most difficult regime to model, the dissipation range, and find that the modified Pan \& Padoan model (that uses the BIT dispersion theory) can lead to significantly improved predictions for the relative velocities, when compared with Direct Num…
The structure and the dynamics of homogeneous turbulence are modified by the presence of body forces such that the Coriolis or the buoyancy forces, which may render a wide range of turbulence scales anisotropic. The corresponding statistical characterization of such effects is done in physical space using structure functions, as well as in spectral space with spectra of two-point correlations, providing two complementary viewpoints. In this framework, second-order and third-order structure functions are put in parallel with spectra of two-point second- and third-order velocity correlation func…
In ferrofluids the suspended ferromagnetic particles agglomerate due to the interaction between the particle magnetic moment and the external magnetic field, which in turn, influences the turbulence and relaxation time. The relaxation time becomes large when we are considering turbulence drag force instead of viscous drag force in Brownian motion. We investigate that the total energy conservation in ferrofluids taking into interaction between the external magnetic field and the ferrofluid.
We propose a unified framework to study the turbulent transport problem from the perspective of nonequilibrium statistical mechanics. By combining Krarichnan's turbulence thermalization assumption and Ruelle's recent work on nonequilibrium statistical mechanics settings for fluids, we show that the equation for viscous fluid can be viewed as the non-canonical Hamiltonian system perturbed by different thermostats. This allows an analogy between the viscous fluid and the nonequilibrium heat conduction model where the Fourier modes can be regarded as the ''particles''. With this framework, we ref…
We report on simulations of two-dimensional turbulence in the inverse energy cascade regime. Focusing on the statistics of Lagrangian tracer particles, scaling behavior of the probability density functions of velocity fluctuations is investigated. The results are compared to the three-dimensional case. In particular an analysis in terms of compensated cumulants reveals the transition from a strong non-Gaussian behavior with large tails to Gaussianity. The reported computation of correlation functions for the acceleration components sheds light on the underlying dynamics of the tracer particles…
Local regions of anomalous particle dispersion, and intermittent events that occur in turbulent flows can greatly influence the global statistical description of the flow. These local behaviors can be identified and analyzed by comparing the growth of neighboring convex hulls of Lagrangian tracer particles. Although in our simulations of homogeneous turbulence the convex hulls generally grow in size, after the Lagrangian particles that define the convex hulls begin to disperse, our analysis reveals short periods when the convex hulls of the Lagrangian particles shrink, evidence that particles …
