We derive a formula for the entropy of two dimensional incompressible inviscid flow, by determining the volume of the space of vorticity distributions with fixed values for the moments Q_k= \int_w(x)^k d^2 x. This space is approximated by a sequence of spaces of finite volume, by using a regularization of the system that is geometrically natural and connected with the theory of random matrices. In taking the limit we get a simple formula for the entropy of a vortex field. We predict vorticity distributions of maximum entropy with given mean vorticity and enstrophy; also we predict the cylindrically symmetric vortex field with maximum entropy. This could be an approximate description of a hurricane.
We propose an exact analytical formula for the anomalous scaling exponents of inertial range structure functions in incompressible fluid turbulence. The formula is a gravitational Knizhnik-Polyakov-Zamolodchikov (KPZ)-type relation, and is valid in any number of space dimensions. It incorporates intermittency by gravitationally dressing the Kolmogorov linear scaling via a coupling to a random geometry. The formula has one real parameter $γ$ that depends on the number of space dimensions. The scaling exponents satisfy the convexity inequality, and the supersonic bound constraint. They agree wit…
We propose a new Eulerian turbulence theory to obtain a closed set of equations for homogeneous, isotropic turbulent velocity field correlations and propagator functions by incorporating constraints of random Galilean invariance. This incorporation generates a few different equations for propagator and the present theory suggests a way to utilize them into the closure solutions of two-time and single-time velocity correlations' equations so as to properly account for random sweeping phenomena. The present theory yields exact solutions when applied to simple model problem of random oscillator.
We present pseudospectral direct-numerical-simulation (DNS) studies of the three-dimensional magnetohydrodynamic (MHD) equations (3DRFMHD) with a stochastic force that has zero mean and a variance $\sim k^{-3}$, where $k$ is the wavenumber, because 3DRFMHD is used in field-theoretic studies of the scaling of energy spectra in MHD turbulence. We obtain velocity and magnetic-field spectra and structure functions and, from these, the multiscaling exponent ratios $ζ_p/ζ_3$, by using the extended self similarity (ESS) procedure. These exponent ratios lie within error bars of their counterparts for …
Turbulence is a challenging feature common to a wide range of complex phenomena. Random fibre lasers are a special class of lasers in which the feedback arises from multiple scattering in a one-dimensional disordered cavity-less medium. Here, we report on statistical signatures of turbulence in the distribution of intensity fluctuations in a continuous-wave-pumped erbium-based random fibre laser, with random Bragg grating scatterers. The distribution of intensity fluctuations in an extensive data set exhibits three qualitatively distinct behaviours: a Gaussian regime below threshold, a mixture…
The Random Phase and Amplitude Formalism (RPA) has significantly extended the scope of weak turbulence studies. Because RPA does not assume any proximity to the Gaussianity in the wavenumber space, it can predict, for example, how the fluctuation of the complex amplitude of each wave mode grows through nonlinear interactions with other modes, and how it approaches the Gaussianity. Thus, RPA has a great potential capability, but its validity has been assessed neither numerically nor experimentally. We compare the theoretical predictions given by RPA with the results of direct numerical simulati…
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…
Whether turbulence intermittencies shall be described by a log-Poisson, a log-stable pdf or other distributions is still debated nowadays. In this paper, a bridge between polymer physics, self-avoiding walk and random vortex stretching is established which may help in getting a new insight on this topics. Actually a very simple relationship between stability index of the stable law and the well known Flory exponent stemming from polymer physics is established. Moreover the scaling of turbulence intermittencies with Reynolds number is also obtained and the overall picture is very close to Tenne…
This work considers the distribution of inertial particles in turbulence using the point-particle approximation. We demonstrate that the random point process formed by the positions of particles in space is a Poisson point process with log-normal random intensity ("log Gaussian Cox process" or LGCP). The probability of having a finite number of particles in a small volume is given in terms of the characteristic function of a log-normal distribution. Corrections due to discreteness of the number of particles to the previously derived statistics of particle concentration in the continuum limit a…
We investigate the statistical properties, based on numerical simulations and analytical calculations, of a recently proposed stochastic model for the velocity field of an incompressible, homogeneous, isotropic and fully developed turbulent flow. A key step in the construction of this model is the introduction of some aspects of the vorticity stretching mechanism that governs the dynamics of fluid particles along their trajectory. An additional further phenomenological step aimed at including the long range correlated nature of turbulence makes this model depending on a single free parameter $…
In this work we address the open problem of high Reynolds number limit in hydrodynamic turbulence, which we modify by considering a vanishing random (instead of deterministic) viscosity. In this formulation, a small-scale noise propagates to large scales in an inverse cascade, which can be described using qualitative arguments of the Kolmogorov-Obukhov theory. We conjecture that the limit of the resulting probability distribution exists as $\mathrm{Re} \to \infty$, and the limiting flow at finite time remains stochastic even if forcing, initial and boundary conditions are deterministic. This c…
A nested polyhedra model has been developed for magnetohydrodynamic (MHD) turbulence. Driving only the velocity field at large scales with random, divergence free forcing results in a clear, stationary $k^{-5/3}$ spectrum for both kinetic and magnetic energies. Since the model naturaly effaces disparate scale interactions, does not have a guide field and avoids injecting any sign of helicity by random forcing, the resulting three dimensional k-spectrum is statistically isotropic. The strengths and weaknesses of the model are demonstrated by considering large or small magnetic Prandtl numbers. …
Max Planck Institute for Dynamics and Self-Organization（马克斯·普朗克动力学与自组织研究所）；Georg August University of Göttingen（哥廷根乔治·奥古斯特大学）；Cornell University（康奈尔大学）
The vortices that appear repeatedly and suggest turbulent dynamics are crucial to the understanding of sheared turbulence. These vortices produce order out of chaos, benefiting the turbulence modelling that focuses only on statistically stable quantities. In three dimensions, the hairpin vortices [J. Zhou, R. Adrian, S. Balachandar, T. Kendall, J. Fluid Mech., 387, 353 (1999); R. Adrian, Phys. Fluids, 19, 041301 (2007)] play such a fundamental role in the transport of momentum and energy for wall bounded sheared turbulence. In two dimensions where the atmospheric flow has a great play, the fun…
Structure formation in turbulence is effectively an instability of "plasma" formed by fluctuations serving as particles. These "particles" are quantumlike; namely, their wavelengths are non-negligible compared to the sizes of background coherent structures. The corresponding "kinetic equation" describes the Wigner matrix of the turbulent field, and the coherent structures serve as collective fields. This formalism is usually applied to manifestly quantumlike or scalar waves. Here, we extend it to compressible Navier--Stokes turbulence, where the fluctuation Hamiltonian is a five-dimensional ma…
The appearance of vortex filaments, the power-law dependence of velocity and vorticity correlators and their multiscaling behavior are derived from the Navier-Stokes equation. This is possible due to interpretation of the Navier-Stokes equation as an equation with multiplicative noise, and remarkable properties of random matrix products.
When a planar shock wave interacts with a random pattern of pre-shock density non-uniformities, it generates an anisotropic turbulent velocity/vorticity field. This turbulence plays an important role at the early stages of the mixing process in the compressed fluid. This situation emerges naturally in shock interaction with weakly inhomogeneous deuterium-wicked foam targets in Inertial Confinement Fusion (ICF) and with density clumps/clouds in astrophysics. We present an exact small-amplitude linear theory describing such interaction. It is based on the exact theory of time and space evolution…
We report on the experimental observation of a transition from a dispersive wave turbulence regime to a nondispersive regime involving shock waves on the surface of a fluid. We use a magnetic fluid in a canal subjected to an external horizontal magnetic field to tune the dispersivity of the system. For a low magnetic field, gravity-capillary wave turbulence is observed, whereas for a high enough field, random steep coherent structures arise which are found to be shock waves. These shock waves create singularities in the second-order difference of the surface elevation, leading to an $ω^{-4}$ f…
The transport dynamo mechanism, which describes the magnetic field generation by diffusion flow is reviewed. In this mechanism, the cross-field transport caused by the random motion of fluid breaks the frozen-flux approximation, and the resulting cross-field diffusion that can generate the magnetic field. Turbulence can play an important role in inducing such random motion. Compared to the conventional dynamo mechanism, this transport mechanism has several special features that the field generation can occur on a very slow time scale because the mechanism is mediated by diffusion and that this…
Bacterial cellulose is an important class of biomaterials which can be grown in well-controlled laboratory and industrial conditions. The cellulose structure is affected by several biological, chemical and environmental factors, including hydrodynamic flows in bacterial suspensions. In this work, we explore the possibility of using well controlled turbulent flow to control the bacterial cellulose production. Turbulent flows leading to random motion of fluid elements may affect the structure of the extracellular polymeric matrix produced by bacteria. Here we show that two-dimensional turbulence…
We consider the one-dimensional Burgers equation randomly stirred at large scales by a Gaussian short-time correlated force. Using the method of dissipative anomalies, we obtain velocity and velocity-difference probability density functions and confirm the results with high-resolution numerical simulations.
Turbulence problem is often considered as "the last unsolved problem of classical physics". It is due to strong interaction between velocity and/or velocity gradient fluctuations, a high Reynolds number flow is a fascinating mixture of purely random, close to Gaussian, fields and coherent structures where substantial fraction of kinetic energy is dissipated into heat. To evaluate intensity of fluctuations, one usually studies different moments of velocity increments and/or dissipation rate, characterized by scaling exponents $ζ_{n}$ and $d_{n}$, respectively. In high Reynolds number flows, the…
We have found that in two-dimensional Kolmogorov flow a single spatially-localized turbulence (SLT) exists stably and travels with a constant speed on average switching the moving direction randomly and intermittently for moderate values of control parameters: Reynolds number and the flow rate. We define the coarse-grained position and velocity of an SLT and separate the motion of the SLT from its internal turbulent dynamics by introducing a co-moving frame. The switching process of an SLT represented by the coarse-grained velocity seems to be a random telegraph signal. Focusing on the asymmet…
With the ever-increasing use of Reynolds-Averaged Navier--Stokes (RANS) simulations in mission-critical applications, the quantification of model-form uncertainty in RANS models has attracted attention in the turbulence modeling community. Recently, a physics-based, nonparametric approach for quantifying model-form uncertainty in RANS simulations has been proposed, where Reynolds stresses are projected to physically meaningful dimensions and perturbations are introduced only in the physically realizable limits. However, a challenge associated with this approach is to assess the amount of infor…
Using the FLUIDICS (Fluid Dynamics in Space) experiment in the International Space Station, turbulence of capillary waves at the air-water interface is experimentally investigated in weightlessness. Capillary waves are excited in a spherical container partially filled with water and undergoing sinusoidal or random oscillations. The fluctuations of the interface, recorded with two capacitive probes are analyzed by means of the frequency power spectrum of wave elevation. For high enough forcing amplitudes, we report power-law spectra with exponents close to the prediction of weak wave turbulence…
A novel machine learning algorithm is presented, serving as a data-driven turbulence modeling tool for Reynolds Averaged Navier-Stokes (RANS) simulations. This machine learning algorithm, called the Tensor Basis Random Forest (TBRF), is used to predict the Reynolds-stress anisotropy tensor, while guaranteeing Galilean invariance by making use of a tensor basis. By modifying a random forest algorithm to accept such a tensor basis, a robust, easy to implement, and easy to train algorithm is created. The algorithm is trained on several flow cases using DNS/LES data, and used to predict the Reynol…
Experiments and numerical simulations show that quantum turbulence exists in two distinct limiting regimes: Kolmogorov turbulence (which shares with classical turbulence the important property of a cascade of kinetic energy from large eddies to small eddies) and Vinen turbulence (which is more similar to a random flow). In this work, we define a mesoscale helicity for the superfluid, which, tested in numerical experiments, distinguishes the two turbulent regimes, quantifying the amount of nonlocal vortex interactions and the orientation of the vortex lines.
Following recent evidence that the vortices in decaying two-dimensional turbulence can be classified into small--mobile, and large--quasi-stationary, this paper examines the evidence that the latter might be considered a `crystal' whose formation embodies the inverse cascade of energy towards larger scales. Several diagnostics of order are applied to the ostensibly disordered large vortices. It is shown that their geometric arrangement is substantially more regular than random, that they move more slowly than could be expected from simple mean-field arguments, and that their energy is signific…
We analyse the formation and the dynamics of quantum turbulence in a two-dimensional Bose-Einstein condensate with a Josephson junction barrier modelled using the Gross-Pitaevskii equation. We show that a sufficiently high initial superfluid density imbalance leads to randomisation of the dynamics and generation of turbulence, namely, the formation of a quasi-1D dispersive shock consisting of a train of grey solitons that eventually breakup into chains of distinct quantised vortices of alternating vorticity followed by random turbulent flow. The Josephson junction barrier allows us to create t…
This paper serves as a review of our recent new DNS study on physics of late boundary layer transition. This includes mechanism of the large coherent vortex structure formation, small length scale generation and flow randomization. The widely spread concept vortex breakdown to turbulence,which was considered as the last stage of flow transition, is not observed and is found theoretically incorrect. The classical theory on boundary layer transition is challenged and we proposed a new theory with five steps, i.e. receptivity, linear instability, large vortex formation, small length scale generat…
The dynamics of the multi-dimensional randomly forced Burgers equation is studied in the limit of vanishing viscosity. It is shown both theoretically and numerically that the shocks have a universal global structure which is determined by the topology of the configuration space. This structure is shown to be particularly rigid for the case of periodic boundary conditions.
Measurements of the energy spectrum and of the vortex-density fluctuation spectrum in superfluid turbulence seem to contradict each other. Using a numerical model, we show that at each instance of time the total vortex line density can be decomposed into two parts: one formed by metastable bundles of coherent vortices, and one in which the vortices are randomly oriented. We show that the former is responsible for the observed Kolmogorov energy spectrum, and the latter for the spectrum of the vortex line density fluctuations.
We consider a general model of Hamiltonian wave systems with triple resonances, in the standard kinetic limit of a continuum of weakly interacting dispersive waves with random phases. In this asymptotic limit we show that the correct dynamical equation for multimode amplitude distributions is not the well-known equation of Peierls but is instead a reduced equation with only a subset of the terms in that equation. The equations that we derive are the direct analogue of the Boltzmann hierarchy obtained from the BBGKY hierarchy in the low-density limit for gases. We show that the asymptotic multi…
We present a new approach to determine numerically the statistical behavior of small-scale structures in hydrodynamic turbulence. Starting from the functional integral representation of the random-force-driven Burgers equation we show that Monte Carlo simulations allow us to determine the anomalous scaling of high-order moments of velocity differences. Given the general applicability of Monte Carlo methods, this opens up the possibility to address also other systems relevant to turbulence within this framework.
