Rotating turbulence is commonly known for being dominated by geostrophic vortices that are invariant along the rotation axis and undergo inverse cascade. Yet, it has recently been shown to sustain fully three-dimensional states with a downscale energy cascade. In this letter, we investigate the statistical properties of three-dimensional rotating turbulence by the means of direct numerical simulations in a triply periodic box where geostrophic vortices are specifically damped. The resulting turbulent flow is an inertial wave turbulence that verifies the Zakharov-Kolmogorov spectrum derived analytically by Galtier (Phys. Rev. E, 68, 2003), thus offering numerical proof of the relevance of wave turbulence theory for three-dimensional, anisotropic waves. Lastly, we show that the same forcing leads to either geostrophic or wave turbulence depending on the initial condition. Our results thus bring further evidence for bi-stability in rotating turbulent flows.
The Weak Turbulence Theory has been applied to waves in thin elastic plates obeying the Föppl-Von Kármán dynamical equations. Subsequent experiments have shown a strong discrepancy between the theoretical predictions and the measurements. Both the dynamical equations and the Weak Turbulence Theory treatment require some restrictive hypotheses. Here a direct numerical simulation of the Föppl-Von Kármán equations is performed and reproduces qualitatively and quantitatively the experimental results when the experimentally measured damping rate of waves $γ_\mathbf{k}= a + bk^2$ is used. This confi…
We suggest a new derivation of a kinetic equation of Kolmogorov-Zakharov (KZ) type for the spectrum of the weakly nonlinear Schrödinger equation with stochastic forcing. The kynetic equation is obtained as a result of a double limiting procedure. Firstly, we consider the equation on a finite box with periodic boundary conditions and send the size of the nonlinearity and of the forcing to zero, while the time is correspondingly rescaled; then, the size of the box is sent to infinity (with a suitable rescaling of the solution). We report here the results of the first limiting procedure, analyzed…
We use two related non-stationarity functions as measures of the degree of scale-by-scale non-equilibrium in homogeneous isotropic turbulence. The values of these functions indicate significant non-equilibrium at the upper end of the inertial range. Wind tunnel data confirm Lundgren's (2002, 2003) prediction that the two-point separation $r$ where the second and third order structure functions are closest to their Kolmogorov scalings is proportional to the Taylor length scale $λ$, and that both structure functions increasingly distance themselves from their Kolmogorov equilibrium form as $r$ i…
Turbulent flows laden with inertial particles present multiple open questions and are a subject of great interest in current research. Due to their higher density compared to the carrier fluid, inertial particles tend to form high concentration regions, i.e. clusters, and low concentration regions, i.e. voids, due to the interaction with the turbulence. In this work, we present an experimental investigation of the clustering phenomenon of heavy sub-Kolmogorov particles in homogeneous isotropic turbulent flows. Three control parameters have been varied over significant ranges: $Re_λ \in [170 - …
We examine the behavior of the Kolmogorov constants C_2, C_k, and C_{k1}, which are, respectively, the prefactors of the second order longitudinal structure function, the three dimensional and one-dimensional longitudinal energy spectrum in the inertial range. We show that their ratios, C_2/C_{k1} and C_k/C_{k1}, exhibit clear dependence on the micro-scale Reynolds number R_λ, implying that they cannot all be independent of R_λ. In particular, it is found that (C_{k1}/C_2-0.25) = 1.95R_λ^{-0.68}. The study further reveals that the widely-used relation C_2 = 4.02 C_{k1} holds only asymptoticall…
Using time-resolved measurements of the orientation of rigid inertial fibers in a turbulence-tank, we investigate the autocorrelation of their tumbling rate. The correlation time ($τ_d$) is well predicted by Kolmogorov inertial-range scaling based on the fiber length (L) when the fiber inertia can be neglected. For inertial fibers, we propose a simple model considering fiber inertia (measured by a tumbling Stokes number) and a viscous torque which accurately predicts both the correlation time and the variance of the tumbling rate.
Kolmogorov's similarity turbulence theory in a Lagrangian frame is assessed with new direct numerical simulations (DNS) of isotropic turbulence with and without hyperviscosity, which attain higher Reynolds numbers than previously available. It is demonstrated that hyperviscous simulations can be used to accurately predict second order Lagrangian velocity structure function (LVSF-2) in the inertial range. The results give strong support for Kolmogorov's Lagrangian similarity assumption and allow to compute the universal constant of the LVSF-2, which gives $C_0=7.5 \pm 0.2$, with a new level of …
Laboratoire de Physique des Plasmas (LPP), CNRS, École polytechnique, Institut Polytechnique de Paris, Université Paris-Saclay, Sorbonne Université, Observatoire de Paris（等离子体物理实验室，法国国家科学研究中心，巴黎综合理工学院，巴黎理工学院，巴黎萨克雷大学，索邦大学，巴黎天文台）
This paper develops a simple model of the inertial range of turbulent flow, based on a cascade of vortical filaments. A binary branching structure is proposed, involving the splitting of filaments at each step into pairs of daughter filaments with differing properties, in effect two distinct simultaneous cascades. Neither of these cascades has the Richardson-Kolmogorov exponent of 1/3. This bimodal structure is also different from bifractal models as vorticity volume is conserved. If cascades are assumed to be initiated continuously and throughout space we obtain a model of the inertial range …
Using the theory of wave turbulence for rapidly rotating incompressible fluids derived by Galtier (2003), we find the locality conditions that the solutions of the kinetic equation must satisfy. We show that the exact anisotropic Kolmogorov-Zakharov (KZ) spectrum satisfies these conditions, which justifies the existence of this constant (positive) energy flux solution. Although a direct cascade is predicted in the transverse ($\perp$) and parallel ($\parallel$) directions to the rotation axis, we show numerically that in the latter case some triadic interactions can have a negative contributio…
Direct numerical simulations of the forced Navier-Stokes equations were performed, in which each shell-averaged quantity evolved from a value appropriate to an initial Gaussian state, to fluctuate about a mean value. Once the transient had passed, mean values (and their associated statistics) were obtained by sampling the evolved time-series at intervals of the order of an eddy-turnover time. This was repeated for a range of Taylor-Reynolds numbers from 10.6 to 335.2. With increasing Reynolds number, our results for energy spectra, transfer spectra and inertial flux supported the Kolmogorov-Ob…
We consider an extension of the kinetic equation developed by Newell & Zakharov (A.C. Newell and V.E. Zakharov. The role of the generalized Phillips' spectrum in wave turbulence. Phys.Lett.A, 372:4230-4233, 2008). The new equation takes into account not only the resonant four-wave interactions but also the dissipation associated with the wave breaking. A dissipation function that depends on the spectral energy flux is introduced into the equation. This function is determined up to a functional parameter, which optimal choice should be made based on comparison with the experiment. A kinetic equ…
Qian Jian, a Chinese theoretical physicist and fluid dynamicist, devoted the second part of his scientific life to the physical understanding of small-scale turbulence to the exclusion of all else. Qian developed his own statistical theory of small-scale turbulence, based on the Liouville (1853) equation and a perturbation variational approach to non-equilibrium statistical mechanics, which is compatible with the Kolmogorov-Oboukhov energy spectrum. His statistical theory of small-scale turbulence, which appears mathematically and physically valid, successfully led to his contributions to (i) …
We ask whether the scaling exponents or the Kolmogorov constants depend on the anisotropy of the velocity fluctuations in a turbulent flow with no shear. According to our experiment, the answer is no for the Eulerian second-order transverse velocity structure function. The experiment consisted of 32 loudspeaker-driven jets pointed toward the centre of a spherical chamber. We generated anisotropy by controlling the strengths of the jets. We found that the form of the anisotropy of the velocity fluctuations was the same as that in the strength of the jets. We then varied the anisotropy, as measu…
We study the clustering of inertial particles using a periodic kinematic simulation. Particles clustering is observed for different pairs of Stokes number and Froude number and different spectral power laws ($1.4 \leqslant p \leqslant 2.1$). The main focus is to identify and then quantify the effect of $p$ on the clustering attractor - by attractor we mean the set of points in the physical space where the particles settle when time tends to infinity. It is observed that spectral power laws can have a dramatic effect on the attractor shape. In particular, we observed a new attractor type which …
We study experimentally the spatial distribution, settling, and interaction of sub-Kolmogorov inertial particles with homogeneous turbulence. Utilizing a zero-mean-flow air turbulence chamber, we drop size-selected solid particles and study their dynamics with particle imaging and tracking velocimetry at multiple resolutions. The carrier flow is simultaneously measured by particle image velocimetry of suspended tracers, allowing the characterization of the interplay between both the dispersed and continuous phases. The turbulence Reynolds number based on the Taylor microscale ranges from $Re_λ…
The fate of small particles in turbulent flows depends strongly on the surrounding fluid's velocity gradient properties such as rotation and strain-rates. For non-inertial (fluid) particles, the Restricted Euler model provides a simple, low-dimensional dynamical system representation of Lagrangian evolution of velocity gradients in fluid turbulence, at least for short times. Here we derive a new restricted Euler dynamical system for the velocity gradient evolution of inertial particles such as solid particles in a gas or droplets and bubbles in turbulent liquid flows. The model is derived in t…
The process of star formation in interstellar molecular clouds is believed to be controlled by driven supersonic magnetohydrodynamic turbulence. We suggest that in the inertial range such turbulence obeys the Kolmogorov law, while in the dissipative range it behaves as Burgers turbulence developing shock singularities. On the base of the She-Leveque analytical model we then predict the velocity power spectrum in the inertial range to be E_k ~ k^{-1.74}. This result reproduces the observational Larson law, <u^2_l>~ l^{0.74...0.76}, [Larson, MNRAS 194 (1981) 809] and agrees well with recent nume…
A wave turbulence theory is developed for inertial electron magnetohydrodynamics (IEMHD) in the presence of a relatively strong and uniform external magnetic field $\boldsymbol{B_0} = B_0 \hat{\boldsymbol{e}}_\|$. This regime is relevant for scales smaller than the electron inertial length $d_e$. We derive the kinetic equations that describe the three-wave interactions between inertial whistler or kinetic Alfvén waves. We show that for both invariants, energy and momentum, the transfer is anisotropic (axisymmetric) with a direct cascade mainly in the direction perpendicular ($\perp$) to $\bold…
This paper provides a detailed study of scale-by-scale budgets in turbulent Rayleigh-Bénard convection and aims at testing the applicability of Kolmogorov (1941) and Bolgiano (1959) theories for this flow. Particular emphasis is laid on anisotropic and inhomogeneous effects: the SO(3) decomposition of structure functions (Arad et al 1999) and a method of description of inhomogeneities proposed by Danaila et al (2001) are used to derive inhomogeneous and anisotropic generalizations of Kolmogorov and Yaglom equations applying to RB convection. The various terms in these equations are computed us…
Towards Gravitational Wave Turbulence within the Hadad-Zakharov metric
Laboratoire de Physique des Plasmas, Université Paris-Saclay, CNRS, École Polytechnique, Sorbonne Université, Observatoire de Paris（等离子体物理实验室，巴黎萨克雷大学，法国国家科学研究中心，巴黎综合理工学院，索邦大学，巴黎天文台）；Université Paris Saclay, CNRS/IN2P3, IJCLab（巴黎萨克雷大学，法国国家科学研究中心/欧洲核子研究组织粒子与核物理研究所，IJCLab）；Université Paris Saclay, Laboratoire de Physique des deux Infinis IJCLab, CNRS/IN2P3（巴黎萨克雷大学，无穷物理实验室IJCLab，法国国家科学研究中心/欧洲核子研究组织粒子与核物理研究所）
The classical turbulence theory by Kolmogorov is reconsidered using Navier-Stokes' equation generalized to 6D physical-plus-eddy space. Strong pseudo-singularity is shown to reveal itself along the boundary `ridge' line separating the dissipation and inertial sub-ranges surrounding the origin of the eddy space. A speculation is made that this singularity is generated by two dipoles of opposite sign aligned on the common axis. It is supported by the observation that the universal power spectrum calculated rediscovers the Kolmogorov's -5/3 power law as independent of the dimensional approach.
We present a link between the theory of deep water waves and that of bubble surface perturbations. Theory correspondence is shown analytically for small wavelengths in the linear regime and investigated numerically in the nonlinear regime. To do so, we develop the second-order spatial perturbation equations for the Rayleigh-Plesset equation and solve them numerically. Our code is publicly available. Studying capillary waves on stable bubbles, we recreate the Kolmogorov-Zakharov spectrum predicted by weak turbulence theory, putting wave turbulence theory to use for bubbles. In this investigatio…
We study intermittency of circulation moments in turbulent superfluid helium by using experimental grid turbulence and numerical simulations of the Hall-Vinen-Bekarevich-Khalatnikov model. More precisely, we compute the velocity circulation $Γ_r$ in loops of size $r$ laying in the inertial range. For both, experimental and numerical data, the circulation variance shows a clear Kolmogorov scaling $\langle Γ_r^2 \rangle \sim r^{8/3}$ in the inertial range, independently of the temperature. Scaling exponents of high-order moments are comparable, within error bars, to previously reported anomalous…
We investigate the dynamics of the two-dimensional periodic Kolmogorov flow of a viscoelastic fluid, described by the Oldroyd-B model, by means of direct numerical simulations. Above a critical Weissenberg number the flow displays a transition from stationary to randomly fluctuating states, via periodic ones. The increasing complexity of the flow in both time and space at progressively higher values of elasticity accompanies the establishment of mixing features. The peculiar dynamical behavior observed in the simulations is found to be related to the appearance of filamental propagating patter…
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…
For velocity and magnetic fields, the turbulent pressure and, more generally, the squared fields such as the components of the turbulent stress tensor, play important roles in astrophysics. For both one and three dimensions, we derive the equations relating the energy spectra of the fields to the spectra of their squares. We solve the resulting integrals numerically and show that for turbulent energy spectra of Kolmogorov type, the spectral slope of the stress spectrum is also of Kolmogorov type. For shallower turbulence spectra, the slope of the stress spectrum quickly approaches that of whit…
We report on direct numerical simulation of quasi-one-dimensional bidirectional capillary-wave turbulence. Although nontrivial three-wave and four-wave resonant interactions are absent in this peculiar geometry, we show that an energy transfer between scales still occurs concentrated around the linear dispersion relation that is broadened by nonlinearity. The wave spectrum displays a clear wave number power-law scaling that is found to be in good agreement with the dimensionally prediction for capillary-wave turbulence involving four-wave interactions. The carried out high-order correlation an…
In this paper, with the aid of direct numerical simulations (DNS) of forced turbulence in a periodic domain, we mathematically reformulate the Kolmogorov-Richardson energy cascade in terms of vortex stretching. By using the description, we prove that if the Navier-Stokes flow satisfies a new regularity criterion in terms of the enstrophy production rate, then the flow does not blow up. Our DNS results seem to support this regularity criterion. Next, we mathematically construct the hierarchy of tubular vortices, which is statistically self-similar in the inertial range. Under the assumptions of…
Since Kolmogorov proposed his phenomenological theory of hydrodynamic turbulence in 1941, the description of mechanism leading to the energy cascade and anomalous scaling remains an open problem in fluid mechanics. Soon after, in 1949 Onsager noticed that the scaling properties in inertial range imply non-differentiability of the velocity field in the limit of vanishing viscosity. This observation suggests that the turbulence mechanism may be related to a finite-time singularity (blowup) of incompressible Euler equations. However, the existence of such blowup is still an open problem too. In t…
We present an extension to Kolmogorov's refined similarity hypothesis for universal fully developed turbulence. The extension is applied within Z. She and E. Leveque's multifractal model of inertial range scaling and its generalizations. Our modification rectifies an apparent gap between the implicit continuum of length scales in Obukhov's conception of a turbulent energy cascade, and scaling law models derived from Kolmogorov's refined similarity hypothesis that lack infinite divisibility. The development has relevance to universal fully developed turbulence, a state we describe explicitly in…
We revisit the issue of spectral slope of MHD turbulence in the inertial range and argue that numerics favor Goldreich-Sridhar -5/3 slope rather than -3/2 slope. We also did precision measurements of anisotropy of MHD turbulence and determined the anisotropy constant C_A=0.34 of Alfvenic turbulence. Together with previously measured Kolmogorov constant C_K=4.2, or 3.3 for purely Alfvenic case, it constitutes a full description of MHD cascade in terms of spectral quantities, which is of high practical value for astrophysics.
We study the statistics of the vertical motion of inertial particles in strongly stratified turbulence. We use Kinematic Simulation (KS) and Rapid Distortion Theory (RDT) to study the mean position and the root mean square (rms) of the position fluctuation in the vertical direction. We vary the strength of the stratification and the particle inertial characteristic time. The stratification is modelled using the Boussinesq equation and solved in the limit of RDT. The validity of the approximations used here requires that$\sqrt{{L}/{g}}<{2π}/{\mathcal{N}}<τ\_η$,where $τ\_η$ is the Kolmogorov tim…
A defining feature of 3D hydrodynamic turbulence is that the rate of energy dissipation is bounded away from zero as viscosity is decreased (Reynolds number increased). This phenomenon - anomalous dissipation - is sometimes called the `zeroth law of turbulence' as it underpins many celebrated theoretical predictions. Another robust feature observed in turbulence is that velocity structure functions $S_p(\ell) :=\langle |δ_\ell u|^p\rangle$ exhibit persistent power-law scaling in the inertial range, namely $S_p(\ell) \sim |\ell|^{ζ_p}$ for exponents $ζ_p>0$ over an ever-increasing (with Reynold…
Multiscale statistical analyses of inertial particle distributions are presented to investigate the statistical signature of clustering and void regions in particle-laden incompressible isotropic turbulence. Three-dimensional direct numerical simulations of homogeneous isotropic turbulence at high Reynolds number ($Re_λ\gtrsim 200$) with up to $10^9$ inertial particles are performed for Stokes numbers ranging from $0.05$ to $5.0$. Orthogonal wavelet analysis is then applied to the computed particle number density fields. Scale-dependent skewness and flatness values of the particle number densi…
We present an experimental study of the preferential concentration of sub-Kolmogorov inertial particles in active-grid-generated homogeneous and isotropic turbulence, characterized via Voronoï tessellations. We show that the detection and quantification of clusters and voids is influenced by the intensity of the laser and high values of particles volume fraction $ϕ_v$. Different biases on the statistics of Voronoï cells are analyzed to improve the reliability of the detection and the robustness in the characterization of clusters and voids. We do this by adapting Big-Data techniques that allow…
We provide a numerical validation of a recently proposed phenomenological theory to characterize the space-time statistical properties of a turbulent puff, both in terms of bulk properties, such as the mean velocity, temperature and size, and scaling laws for velocity and temperature differences both in the viscous and in the inertial range of scales. In particular, apart from the more classical shear-dominated puff turbulence, our main focus is on the recently discovered new regime where turbulent fluctuations are dominated by buoyancy. The theory is based on an adiabaticity hypothesis which …
The effects of mechanical generation of turbulent kinetic energy and buoyancy forces on the statistics of air temperature and velocity increments are experimentally investigated at the cross over from production to inertial range scales. The ratio of an approximated mechanical to buoyant production (or destruction) of turbulent kinetic energy can be used to form a dimensionless stability parameter $ζ$ that classifies the state of the atmosphere as common in many atmospheric surface layer studies. Here, we assess how $ζ$ affects the scale-wise evolution of the probability of extreme air tempera…
We present an experimental study on the settling velocity of dense sub-Kolmogorov particles in active-grid-generated turbulence in a wind tunnel. Using phase Doppler interferometry, we observe that the modifications of the settling velocity of inertial particles, under homogeneous isotropic turbulence and dilute conditions $ϕ_v\leq O(10)^{-5}$, is controlled by the Taylor-based Reynolds number $Re_λ$ of the carrier flow. On the contrary, we did not find a strong influence of the ratio between the fluid and gravity accelerations (i.e., $γ\sim(η/τ_η^2)/g$) on the particle settling behavior. Rema…
Using 3D Vorono\text{ï} analysis, we explore the local dynamics of small, settling, inertial particles in isotropic turbulence using Direct Numerical Simulations (DNS). We independently vary the Taylor Reynolds number $R_λ\in[90,398]$, Froude number $Fr\equiv a_η/g\in[0.052,\infty]$ (where $a_η$ is the Kolmogorov acceleration, and $g$ is the acceleration due to gravity), and Kolmogorov scale Stokes number $St\equivτ_p/τ_η\in[0,3]$. In agreement with previous results using global measures of particle clustering, such as the Radial Distribution Function (RDF), we find that for small Vorono\text{…
The results of experimental and theoretical studies of the parametric decay instability of capillary waves on the surface of superfluid helium He-II are reported. It is demonstrated that in a system of turbulent capillary waves low-frequency waves are generated along with the direct Kolmogorov-Zakharov cascade of capillary turbulence. The effects of low-frequency damping and the discreteness of the wave spectrum are discussed.
The effects of turbulent dynamic range on scalar mixing in stably stratified turbulence are investigated by an adaptation of the theoretical passive scalar modelling arguments of Beguier et al. (1978) and demonstrated statistically using direct numerical simulations of statistically stationary homogeneous stratified and sheared turbulence (SHSST). By analysis of inertial and inertial-convective subrange scaling, we show that the relationship between active scalar and turbulence time scales is predicted by the ratio of the Kolmogorov and Oboukhov-Corrsin constants provided there is sufficient s…
In this paper, we investigate the statistical features of the fully developed, forced, rapidly rotating, {turbulent} system using numerical simulations, and model {the} energy {spectrum} that {fits} well with the numerical data. Among the wavenumbers ($k$) larger than the Kolmogorov dissipation wavenumber, the energy is distributed such that the suitably non-dimensionized energy spectrum is ${\bar E}({\bar k})\approx \exp(-0.05{\bar k})$, where overbar denotes appropriate non-dimensionalization. {For the wavenumbers smaller than that of forcing, the energy in a horizontal plane is much more th…
We propose an experimental study on the gravitational settling velocity of dense, sub-Kolmogorov inertial particles under different background turbulent flows. We report Phase Doppler Particle Analyzer measurements in a low-speed wind tunnel uniformly seeded with micrometer scale water droplets. Turbulence is generated with three different grids (two consisting on different active-grid protocols while the third is a regular static grid), allowing us to cover a very wide range of turbulence conditions in terms of Taylor-scale based Reynolds numbers ($Re_λ\in [30-520]$), Rouse numbers ($Ro \in […
In the study of weakly turbulent wave systems possessing incomplete self-similarity it is possible to use dimensional arguments to derive the scaling exponents of the Kolmogorov-Zakharov spectra, provided the order of the resonant wave interactions responsible for nonlinear energy transfer is known. Furthermore one can easily derive conditions for the breakdown of the weak turbulence approximation. It is found that for incompletely self-similar systems dominated by three wave interactions, the weak turbulence approximation usually cannot break down at small scales. It follows that such systems…
