The appearance of sharp vorticity gradients in two-dimensional hydrodynamic turbulence and their influence on the turbulent spectra is considered. We have developed the analog of the vortex line representation as a transformation to the curvilinear system of coordinates moving together with the di-vorticity lines. Compressibility of this mapping can be considered as the main reason for the formation of the vorticity discontinuities at high Reynolds numbers. For two-dimensional turbulence in the case of strong anisotropy the vorticity discontinuities can generate spectra with the fall-off at large $k$ proportional to $k^{-3}$ resembling the Kraichnan spectrum for the enstrophy cascade. For turbulence with weak anisotropy the $k$ dependence of the spectrum due to discontinuities coincides with that of the Saffman spectrum: $k^{-4}$. We have compared the analytical predictions with direct numerical solutions of the two-dimensional Euler equation for decaying turbulence. We observe that the di-vorticity is reaching very high values and is distributed locally in space along piecewise straight lines. Thus, indicating strong anisotropy and accordingly we found a spectrum close to the $k^{-3}$-spectrum.
Viscous depletion of vorticity is an essential and well known property of turbulent flows, balancing, in the mean, the net vorticity production associated with the vortex stretching mechanism. In this letter we however demonstrate that viscous effects are not restricted to a mere destruction process, but play a more complex role in vorticity dynamics that is as important as vortex stretching. Based on results from particle tracking experiments (3D-PTV) and direct numerical simulation (DNS) of homogeneous and quasi isotropic turbulence, we show that the viscous term in the vorticity equation ca…
We investigate the conditional vorticity budget of fully developed three-dimensional homogeneous isotropic turbulence with respect to coherent and incoherent flow contributions. The Coherent Vorticity Extraction based on orthogonal wavelets allows to decompose the vorticity field into coherent and incoherent contributions, of which the latter are noise-like. The impact of the vortex structures observed in fully developed turbulence on statistical balance equations is quantified considering the conditional vorticity budget. The connection between the basic structures present in the flow and the…
We adapt the formalism of the statistical theory of 2D turbulence in the case where the Casimir constraints are replaced by the specification of a prior vorticity distribution. A new relaxation equation is obtained for the evolution of the coarse-grained vorticity. It can be used as a thermodynamical parametrization of forced 2D turbulence (determined by the prior), or as a numerical algorithm to construct arbitrary nonlinearly dynamically stable stationary solutions of the 2D Euler equation.
This note discusses basic properties of vorticity in groundwater flow theory. An evolution equation for the vorticity vector is derived to demonstrate that, when present, vorticity decreases very rapidly. In addition, it is shown how vorticity affects, though very little, the hydraulic head directional variation in the vicinity of a point.
An intrinsic feature of turbulent flows is an enhanced rate of mixing and kinetic energy dissipation due to the rapid generation of small-scale motions from large-scale excitation. The transfer of kinetic energy from large to small scales is commonly attributed to the stretching of vorticity by the strain-rate, but strain self-amplification also plays a role. Previous treatments of this connection are phenomenological or inexact, or cannot distinguish the contribution of vorticity stretching from that of strain self-amplification. In this paper, an exact relationship is derived which quantitat…
According to statistical turbulence theory, the ensemble averaged squared vorticity rho_E is expected to grow not faster than drho_E/dt ~ rho_E^{3/2}. Solving a variational problem for maximal bulk enstrophy (E) growth, velocity fields were found for which the growth rate is as large as dE/dt ~ E^3. Using numerical simulations with well resolved small scales and a quasi-Lagrangian advection to track fluid subvolumes with rapidly growing vorticity, we study spatially resolved statistics of vorticity growth. We find that the volume ensemble averaged growth bound is satisfied locally to a remarka…
We study the generation of 2D turbulence in Faraday waves by investigating the creation of spatially periodic vortices in this system. Measurements which couple a diffusing light imaging technique and particle tracking algorithms allow the simultaneous observation of the three-dimensional fluid motion and of the temporal changes in the wave field topography. Quasi-standing waves are found to coexist with a spatially extended fluid transport. More specifically, the destruction of regular patterns of oscillons coincides with the emergence of a complex fluid motion whose statistics are similar to…
Stretching in continuum mechanics is naturally described using the Cauchy-Green strain tensors. These tensors quantify the Lagrangian stretching experienced by a material element, and provide a powerful way to study processes in turbulent fluid flows that involve stretching such as vortex stretching and alignment of anisotropic particles. Analyzing data from a simulation of isotropic turbulence, we observe preferential alignment between anisotropic particles and vorticity. We show that this alignment arises because both of these quantities independently tend to align with the strongest Lagrang…
We adapt the formalism of the statistical theory of 2D turbulence in the case where the Casimir constraints are replaced by the specification of a prior vorticity distribution. A phenomenological relaxation equation is obtained for the evolution of the coarse-grained vorticity. This equation monotonically increases a generalized entropic functional (determined by the prior) while conserving circulation and energy. It can be used as a thermodynamical parametrization of forced 2D turbulence, or as a numerical algorithm to construct (i) arbitrary statistical equilibrium states in the sense of Ell…
Although the current Reply by Grebenev et al. (2021a) makes their original analysis in Grebenev et al. (2017) more transparent, the actual problem remains. Their claim to have analytically proven conformal invariance in 2D turbulence for a zero-vorticity characteristic equation is not true. We refuted this claim in Frewer & Khujadze (2021a,b), which we will briefly summarize here again with respect to the presented Reply. In particular our proof on the symmetry-breaking property of the integral normalization constraint is misrepresented and misconstrued, especially in their second Reply (Grebe…
We study sign changes and scaling laws in the Cartesian components of the velocity and vorticity of rotating turbulence, in the helicity, and in the components of vertically-averaged fields. Data for the analysis is provided by high-resolution direct numerical simulations of rotating turbulence with different forcing functions, with up to 1536^3 grid points, with Reynolds numbers between ~1100 and ~5100, and with moderate Rossby numbers between ~0.06 and ~8. When rotation is negligible, all Cartesian components of the velocity show similar scaling, in agreement with the expected isotropy of th…
Grid turbulence is investigated using cross-correlation digital Particle Image Velocimetry (PIV) over a range of Taylor Reynolds Number (Reλ) from 5 to 44. Instantaneous velocity is measured directly and vorticity and velocity gradients are obtained indirectly. Measurements are taken at various downstream locations from the generating grid. Probability distribution functions (PDFs) are calculated for the fluctuating component of the velocity, the spatial velocity gradients and vorticity. The PDF of the velocity fluctuations has a Gaussian distribution while velocity gradients and vorticity are…
The mechanisms promoting scalar dissipation through scalar gradient production are scrutinized in terms of vorticity alignment with respect to strain principal axes. For that purpose, a stochastic Lagrangian model for the velocity gradient tensor and the scalar gradient vector is used. The model results show that the major part of scalar dissipation occurs for stretched vorticity, namely when the vorticity vector aligns with the extensional and intermediate strain eigenvectors. More specifically, it appears that the mean scalar dissipation is well represented by the sample defined by alignment…
The tendency of turbulent flows to produce fine-scale motions from large-scale energy injection is often viewed as a scale-wise cascade of kinetic energy driven by vorticity stretching. This has been recently evaluated by an exact, spatially-local relationship [Johnson, P. L. 2020, Phys. Rev. Lett., \textbf{124}, 104501], which also highlights the contribution of strain self-amplification. In this paper, the role of these two mechanisms is explored in more detail. Vorticity stretching and strain amplification interactions between velocity gradients filtered at the same scale account for about …
In the context of two-dimensional (2D) turbulence, we apply the maximum entropy production principle (MEPP) by enforcing a local conservation of energy. This leads to an equation for the vorticity distribution that conserves all the Casimirs, the energy, and that increases monotonically the mixing entropy ($H$-theorem). Furthermore, the equation for the coarse-grained vorticity dissipates monotonically all the generalized enstrophies. These equations may provide a parametrization of 2D turbulence. They do not generally relax towards the maximum entropy state. The vorticity current vanishes for…
In this paper, the scaling property of the inverse energy cascade and forward enstrophy cascade of the vorticity filed $ω(x,y)$ in two-dimensional (2D) turbulence is analyzed. This is accomplished by applying a Hilbert-based technique, namely Hilbert-Huang Transform, to a vorticity field obtained from a $8192^2$ grid-points direct numerical simulation of the 2D turbulence with a forcing scale $k_f=100$ and an Ekman friction. The measured joint probability density function $p(C,k)$ of mode $C_i(x)$ of the vorticity $ω$ and instantaneous wavenumber $k(x)$ is separated by the forcing scale $k_f$ …
Viscous vortex layers subject to a more general uniform strain are considered. They include Townsend's steady solution for plane strain (corresponding to a parameter $a = 1$) in which all the strain in the plane of the layer goes toward vorticity stretching, as well as Migdal's recent steady asymmetric solution for axisymmetric strain ($a = 1/2$) in which half of the strain goes into vorticity stretching. In addition to considering asymmetric, symmetric and antisymmetric steady solutions $\forall a \ge 0$, it is shown that for $a < 1$, i.e., anything less than the Townsend case, the vorticity …
A framework is developed to describe the two-point statistics of potential vorticity in rotating and stratified turbulence as described by the Boussinesq equations. The Karman-Howarth equation for the dynamics of the two-point correlation function of potential vorticity reveals the possibility of inertial-range dynamics in certain regimes in the Rossby, Froude, Prandtl and Reynolds number parameters. For the case of large Rossby and Froude numbers, and for the case of quasi-geostrophic dynamics, a linear scaling law with 2/3 prefactor is derived for the third-order mixed correlation between po…
We report results from a high resolution numerical study of fluid particles transported by a fully developed turbulent flow. Single particle trajectories were followed for a time range spanning more than three decades, from less than a tenth of the Kolmogorov time-scale up to one large-eddy turnover time. We present results concerning acceleration statistics and the statistics of trapping by vortex filaments conditioned to the local values of vorticity and enstrophy. We distinguish two different behaviors between the joint statistics of vorticity and centripetal acceleration or vorticity and l…
We analyze the vector nulls of velocity, Lagrangian acceleration, and vorticity, coming from direct numerical simulations of forced homogeneous isotropic turbulence at $Re_λ\in [40-610]$. We show that the clustering of velocity nulls is much stronger than those of acceleration and vorticity nulls. These acceleration and vorticity nulls, however, are denser than the velocity nulls. We study the scaling of clusters of these null points with $Re_λ$ and with characteristic turbulence lengthscales. We also analyze datasets of point inertial particles with Stokes numbers $St = 0.5$, 3, and 6, at $Re…
Based on the mechanics of the Euler equation at short time, we show that a Recent Fluid Deformation (RFD) closure for the vorticity field, neglecting the early stage of advection of fluid particles, allows to build a 3D incompressible velocity field that shares many properties with empirical turbulence, such as the teardrop shape of the R-Q plane. Unfortunately, non gaussianity is weak (i.e. no intermittency) and vorticity gets preferentially aligned with the wrong eigenvector of the deformation. We then show that slightly modifying the former vectorial field in order to impose the long range …
We report results on the geometrical statistics of the vorticity vector obtained from experiments in electromagnetically forced rotating turbulence. A range of rotation rates $Ω$ is considered, from non-rotating to rapidly rotating turbulence with a maximum background rotation rate of $Ω=5$ rad/s (with Rossby number much smaller than unity). Typically, in our experiments ${\rm{Re}}_λ\approx 100$. The measurement volume is located in the centre of the fluid container above the bottom boundary layer, where the turbulent flow can be considered locally statistically isotropic and horizontally homo…
Motivated by results from recent particle tracking experiments in turbulence (Xu et al., Nat. Phys. 7, 709 (2011)), we study the Lagrangian time correlations of vorticity alignments with the three eigenvectors of the deformation-rate tensor. We use data from direct numerical simulations (DNS), and explore the predictions of a Lagrangian model for the velocity gradient tensor. We find that the initial increase of correlation of vorticity direction with the most extensive eigen-direction observed by Xu et al. is reproduced accurately using the Lagrangian model, as well as the evolution of correl…
Building upon the intrinsic properties of Navier-Stokes dynamics, namely the prevalence of intense vortical structures and the interrelationship between vorticity and strain rate, we propose a simple framework to quantify the extreme events and the smallest scales of turbulence. We demonstrate that our approach is in excellent agreement with the best available data from direct numerical simulations of isotropic turbulence, with Taylor-scale Reynolds number up to 1300. We additionally highlight a shortcoming of prevailing intermittency models due to their disconnection from observed correlation…
In this paper we study the enstrophy transers in helical turbulence using direct numerical simulation. We observe that the helicity injection does not have significant effects on the inertial-range energy and helicity spectra ($\sim k^{-5/3}$) and fluxes (constants). We also calculate the separate contributions to enstrophy transfers via velocity to vorticity and vorticity to vorticity channels. There are four different enstrophy fluxes associated with the former channel or vorticity stretching, and one flux associated with the latter channel or vorticity advection. In the inertial range, the …
The aim of this paper is to understand the tendency to organization of the turbulence in two-dimensional ideal fluids. We show that nonlinear processes as inverse cascade of the energy and vorticity concentration are essentially determined by trajectory trapping or eddying. The statistics of the trajectories of the vorticity elements is studied using a semianalytic method. The separation of the positive and negative vorticity is due to the attraction produced by a large scale vortex on the small scale vortices of the same sign. More precisely, a large scale velocity is shown to determine avera…
We describe a numerical investigation of a continuum model of an active nematic, concentrating on the regime of active turbulence. Results are presented for the effect of three parameters, activity, elastic constant and rotational diffusion constant, on the order parameter and flow fields. Defects and distortions in the director field act as sources of vorticity, and thus vorticity is strongly correlated to the director field. In particular the characteristic length of decay of vorticity and order parameter correlations is controlled by the defect density. By contrast the decay of velocity cor…
The statistics of rotational motion of small, inertialess triaxial ellipsoids are computed along Lagrangian trajectories extracted from direct numerical simulations of homogeneous isotropic turbulence. The particle angular velocity and its components along the three principal axes of the particle are considered, expanding on the results presented by \citet{ChevillardMeneveau13}. The variance of the particle angular velocity, referred to as the particle enstrophy, is found to increase for particles with elongated shapes. This trend is explained by considering the contributions of vorticity and …
We investigate the vorticity dynamics in a turbulent vortex using scattering of acoustic waves. Two ultrasonic beams are adjusted to probe simultaneously two spatial scales in a given volume of the flow, thus allowing a dual channel recording of the dynamics of coherent vorticity structures. Our results show that this allows to measure the average energy transfer time between different spatial length scales, and that such transfer goes faster at smaller scales.
We discuss a mean-field theory of generation of large-scale vorticity in a rotating density stratified developed turbulence with inhomogeneous kinetic helicity. We show that the large-scale nonuniform flow is produced due to ether a combined action of a density stratified rotating turbulence and uniform kinetic helicity or a combined effect of a rotating incompressible turbulence and inhomogeneous kinetic helicity. These effects result in the formation of a large-scale shear, and in turn its interaction with the small-scale turbulence causes an excitation of the large-scale instability (known …
The vorticity rotor field ${\bf B}=\mbox{rot}\,\mathbfω$ (di-vorticity) for freely decaying two-dimensional hydrodynamic turbulence due to a tendency to breaking is concentrated in the vicinity of the lines corresponding to the position of the vorticity quasi-shocks. The maximum value of the di-vorticity $B_{max}$ at the stage of quasi-shocks formation increases exponentially in time, while the thickness $\ell(t)$ of the maximum area in the transverse direction to the vector ${\bf B}$ decreases in time also exponentially. It is numerically shown that $B_{max} (t)$ depends on the thickness acco…
Fuel sprays produce high-velocity, jet-like flows that impart turbulence onto the ambient flow field. The spray-induced turbulence augments fuel-air mixing, which has a primary role in controlling pollutant formation and cyclic variability in engines. This paper presents tomographic particle image velocimetry (TPIV) measurements to analyse the 3D spray-induced turbulence during the intake stroke of a direct-injection engine. The spray produces a strong spray-induced jet in the far field, which travels through the cylinder and imparts turbulence onto the surrounding flow. Planar high-speed PIV …
We find an instability resulting in generation of large-scale vorticity in a fast rotating small-scale turbulence or turbulent convection with inhomogeneous fluid density along the rotational axis in anelastic approximation. The large-scale instability causes excitation of two modes: (i) the mode with dominant vertical vorticity and with the mean velocity being independent of the vertical coordinate; (ii) the mode with dominant horizontal vorticity and with the mean momentum being independent of the vertical coordinate. The mode with the dominant vertical vorticity can be excited in a fast rot…
We present simultaneous experimental measurements of the dynamics of anisotropic particles transported by a turbulent flow and the velocity gradient tensor of the flow surrounding them. We track both rod-shaped particles and small spherical flow tracers using stereoscopic particle tracking. By using scanned illumination, we are able to obtain a high enough seeding density of tracers to measure the full velocity gradient tensor near the rod. The alignment of rods with the vorticity and the eigenvectors of the strain rate show agreement with numerical simulations. A full description of the tumbl…
The modeling of surface pressure wave number spectra beneath a turbulent boundary layer is reviewed and reconsidered in terms of the vorticity in the flow. Using a solution based on the vorticity equation and Squires theorem, which was originally given by Chase(1991), it is shown that the complete solution for surface pressure spectrum can be specified using the non-linear turbulence-turbulence interaction terms as sources. It is then shown that the surface pressure can be directly related to the vorticity in the flow. The results are checked against a Direct Numerical Simulation of a channel …
We investigate the spatial distributions and production mechanisms of vorticity and turbulent kinetic energy around a finite NACA 0018 wing with square wingtip profile at $Re_c=10^4$ and $10^{\circ}$ angle of attack with the aid of Direct Numerical Simulation (DNS). The analysis focuses on the highly inhomogeneous region around the tip and the near wake; this region is highly convoluted, strongly three-dimensional, and far from being self-similar. The flow separates close to the leading edge creating a large, open recirculation zone around the central part of the wing. In the proximity of the …
We present a stochastic turbulence generator based on a vorticity formulation where the generated turbulent field implicitly fulfills the kinematic constraints of an incompressible flow. The generator allows direct access to the turbulent velocity and vorticity field. Enforcing additional constraints such as a divergence-free vorticity field and a specified differentiability of the flow field can also implemented directly within this formulation. The resulting turbulent field contain improved kinematic properties and may be imported into numerical simulations without an excessive loss of energ…
We present a statistical analysis of the two-point vorticity probability density of the vorticity field generated in the inverse cascade of stationary two-dimensional turbulence.
Vorticity in turbulent flows is often organized into complex geometries that influence the dynamics. We use a relatively novel approach to describe these geometries: that of obtaining segments of vortex lines embedded in the flow. This enables us to quantify the geometric features of these objects. Vortex lines differ widely in their behaviour but we find some unifying features. The decay from high levels of vorticity is shown to happen over a short fraction of the vortex length. The local curvature is inversely related with the vorticity magnitude. Strong parts of vortex lines bundle together…
We describe the structure and dynamics of turbulence by the scale-dependent perceived velocity gradient tensor as supported by following four tracers, i.e. fluid particles, that initially form a regular tetrahedron. We report results from experiments in a von Kármán swirling water flow and from numerical simulations of the incompressible Navier-Stokes equation. We analyze the statistics and the dynamics of the perceived rate of strain tensor and vorticity for initially regular tetrahedron of size $r_0$ from the dissipative to the integral scale. Just as for the true velocity gradient, at any i…
The paper defines and discusses the concept of hidden drifts in two-dimensional turbulence. These are ordered components of the trajectories that average to zero and do not produce direct transport. Their effects appear in the evolution of the turbulence as a special type of fluxes, which consist of average motion of positive and negative fluctuations in opposite directions. We show that these fluxes have important nonlinear effects in turbulent fluids and in confined plasmas. In the first case, they determine the increase of the large scale vorticity and velocity at the expense of the small s…
Direct numerical simulations of turbulent flow in a channel with one rigid and one viscoelastic wall are performed. An Eulerian-Eulerian model is adopted with a level-set approach to identify the fluid-compliant material interface. Focus is placed on the propagation of Rayleigh waves in the compliant material, whose speed depends on the shear modulus of elasticity and whose dominant wavelength depends on the thickness of the viscoelastic layer. These parameters are selected to ensure coupling between the compliant surface and turbulence. When the phase speed of Rayleigh waves is commensurate w…
An essential ingredient of turbulent flows is the vortex stretching mechanism, which emanates from the non-linear interaction of vorticity and strain-rate tensor and leads to formation of extreme events. We analyze the statistical correlations between vorticity and strain rate by using a massive database generated from very well resolved direct numerical simulations of forced isotropic turbulence in periodic domains. The grid resolution is up to $12288^3$, and the Taylor-scale Reynolds number is in the range $140-1300$. In order to understand the formation and structure of extreme vorticity fl…
A vorticity surge event that could be a paradigm for a wide class of bursting events in turbulence is studied to examine how the energy cascade is established and how this event could serve as a new test of LES turbulence models. This vorticity surge event is tied to the formation of the energy cascade in a direct numerical simulation by the traditional signatures of a turbulent energy cascade such as spectra approaching -5/3 and strongly Beltramized vortex tubes. A coherent mechanism is suggested by the nearly simultaneous development of a maximum of the peak vorticity $\|ω\|_\infty$, growth …
The current claim by Grebenev et al. [J. Phys. A: Math. Theor. 52, 335501 (2019)], namely that the inviscid and unclosed 2D Lundgren-Monin-Novikov (LMN) equations on a zero-vorticity Lagrangian path admit conformal invariance, is based on a flawed and misleading analysis published earlier by Grebenev et al. (2017). All false results and conclusions made before in the Eulerian picture were now extended by Grebenev et al. (2019) to the Lagrangian picture. Although we have already commented on these errors and consistently refuted their previous study (Frewer & Khujadze, 2018), we deem it necessa…
Predicting the long time or late time states of two-dimensional incompressible, high Reynolds number, slowly decaying turbulence has been one of the long-standing problems. Using ``point vortices'' as ``inviscid'' building blocks, which do not respect incompressibility, statistical mechanical models conserving only total energy and zero total circulation result in the well-known sinh-Poisson relation between vorticity and stream function. On the other hand, statistical mechanics of ``inviscid patch'' vortices, which respects incompressibility by conserving regions of zero and nonzero vorticity…
We investigate the dynamics of energy cascades in quantum turbulence by directly observing the vorticity distributions in numerical simulations of the Gross-Pitaevskii equation. By Fourier filtering each scale of the vorticity distribution, we find that antiparallel vortex tubes at a large scale generate small-scale vortex tubes orthogonal to those at the large scale, which is a manifestation of the energy cascade from large to small scales. We reveal the dynamics of quantized vortex lines in these processes.
Nonlocal generalization of the Chapman-Enskog method and hydrodynamic equations. Nonlocal vorticity transport in a turbulent medium
National Research Center “Kurchatov Institute,” Moscow, Russia（库尔恰托夫国家研究中心）；National Research Nuclear University MEPhI, Moscow, Russia（莫斯科国立核研究大学MEPhI）
