The local streamline topology classification method of Chong et al. (1990) is adapted and extended to describe the geometry of infinitesimal vortex lines. Direct numerical simulation (DNS) data of forced isotropic turbulence reveals that joint probability density function (PDF) of the second ($q_ω$) and third ($r_ω$) normalized invariants of the vorticity gradient tensor asymptotes to a self-similar bell form beyond $Re_λ> 200$. The same PDF shape is also seen at the late stages of breakdown of Taylor-Green vortex suggesting the universality of the bell-shaped pdf form in turbulent flows. Additionally, vortex reconnection from different initial configurations is examined. The local topology and geometry of the reconnection bridge is shown to be identical in all cases with elliptic vortex lines on one side and hyperbolic filaments on the other. Overall, topological characterization of vorticity field provides a useful analytical basis for examining vorticity dynamics in turbulence and other fluid flows.
Aviation turbulence is modelled as an interaction between an aircraft and a vortex tube. The vortex tube can have an arbitrary orientation/offset with respect to the aircraft. We compare modelling the aircraft (i) as a point and (ii) having a finite area (wing and fuselage). We consider both vertical and horizontal acceleration experienced by the aircraft. The baseline vortex tube has an area which is of the order of the aircraft area.
The streamwise and transverse velocities are measured simultaneously in grid turbulence at Reynolds numbers 100-300. We extract typical intermittency patterns, which are consistent with velocity profiles of Burgers and Lamb-Oseen vortices. The radii of these vortex tubes are estimated to be several of the Kolmogorov length.
This report proposes a new method to study vortex tubes in one-dimensional velocity data of experimental turbulence. Vortex tubes are detected as local maxima on the scale-space plot of wavelet transforms of the velocity data. Then it is possible to extract a typical velocity pattern. The result for data obtained in a wind tunnel is consistent with those of three-dimensional direct numerical simulations.
The elementary structures of turbulence, i.e., vortex tubes, are studied using velocity data obtained in laboratory experiments for boundary layers and duct flows at microscale Reynolds numbers 332-1934. While past experimental studies focused on intense vortex tubes, the present study focuses on all vortex tubes with various intensities. We obtain the mean velocity profile. The radius scales with the Kolmogorov length. The circulation velocity scales with the Kolmogorov velocity, in contrast to the case of intense vortex tubes alone where the circulation velocity scales with the rms velocity …
Modelling the vortex structures and then translating them into the corresponding velocity fields are two essential aspects for the vortex-based modelling works in wall-bounded turbulence. This work develops a datadriven method, which allows an effective reconstruction for the velocity field based on a given vortex field. The vortex field is defined as a vector field by combining the swirl strength and the real eigenvector of the velocity gradient tensor. The distinctive properties for the vortex field are investigated, with the relationship between the vortex magnitude and orientation revealed…
An inverse turbulent cascade in a restricted two-dimensional periodic domain leads to the creation of condensate -- a pair of coherent system-size vortices. We perform extensive numerical simulations of this system and carry on detailed theoretical analysis based upon momentum and energy exchanges between the turbulence fluctuations and the mean coherent condensate (vortices). The theory predicts the vortex profile and amplitude which perfectly agree with the numerical data.
We examine the velocity profile of coherent vortices appearing as a consequence of the inverse cascade of two-dimensional turbulence in a finite box in the case of static pumping. We demonstrate that in the passive regime the flat velocity profile is realized, as in the case of pumping short correlated in time. However, in the static case the energy flux to large scales is dependent on the system parameters. We demonstrate that it is proportional to $f^{4/3}$ where $f$ is the characteristic force exciting turbulence.
Near absolute zero, superfluid liquid helium displays quantum properties at macroscopic length scales. One property, superfluidity, means flow with zero viscosity. Another property, the existence of a complex wavefunction, constrains the rotation to thin, discrete vortex lines carrying one quantum of circulation each. Therefore, if liquid helium is stirred, it becomes turbulent, but in a peculiar way: it is a state of turbulence consisting of a tangle of quantised vortex lines in a fluid without viscosity. Surprisingly, this disordered state, which I called quantum turbulence years ago, shares…
The streamwise and transverse velocities are measured simultaneously in isotropic grid turbulence at relatively high Reynolds numbers, Re(lambda) = 110-330. Using a conditional averaging technique, we extract typical intermittency patterns, which are consistent with velocity profiles of a model for a vortex tube, i.e., Burgers vortex. The radii of the vortex tubes are several of the Kolmogorov length regardless of the Reynolds number. Using the distribution of an interval between successive enhancements of a small-scale velocity increment, we study the spatial distribution of vortex tubes. The…
Compressing complex flows into a tangle of vortex filaments is the basic implication of the classical notion of the vortex representation. Various vortex identification criteria have been proposed to extract the vortex filaments from available velocity fields, which is an essential procedure in the practice of the vortex representation. This work focuses on the effectiveness of those identification criteria in the compressed representation of wall-bounded turbulence. Five local identification criteria regarding the vortex strength and three criteria for the vortex axis are considered. To facil…
The most elementary structures of turbulence, i.e., vortex tubes, are studied using velocity data obtained in a laboratory experiment for boundary layers with microscale Reynolds numbers 295-1258. We conduct conditional averaging for enhancements of a small-scale velocity increment and obtain the typical velocity profile for vortex tubes. Their radii are of the order of the Kolmogorov length. Their circulation velocities are of the order of the root-mean-square velocity fluctuation. We also obtain the distribution of the interval between successive enhancements of the velocity increment as the…
We report the results of three-dimensional direct numerical simulations for incompressible viscous fluid in a circular pipe flow with a sudden expansion. At the inlet, a parabolic velocity profile is applied together with a finite amplitude perturbation in the form of a vortex with its axis parallel to the axis of the pipe. At sufficiently high Reynolds numbers the recirculation region breaks into a turbulent patch that changes position axially depending on the strength of the perturbation. This vortex perturbation is believed to produce a less abrupt transition than in previous studies with a…
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…
Streamlines, vortex lines and magnetic flux tubes in turbulent fluids and plasmas display a great amount of coiling, twisting and linking, raising the question as to whether their topological complexity (continually created and destroyed by reconnections) can be quantified. In superfluid helium, the discrete (quantized) nature of vorticity can be exploited to associate to each vortex loop a knot invariant called the Alexander polynomial whose degree characterizes the topology of that vortex loop. By numerically simulating the dynamics of a tangle of quantum vortex lines, we find that this quan…
This paper addresses reconnection of vortex tubes, with particular focus on the topology of the vortex lines (field lines of the vorticity). This analysis of vortex line topology reveals previously undiscovered features of the reconnection process, such as the generation of many small flux rings, formed when reconnection occurs in multiple locations in the vortex sheet between the tubes. Consideration of three-dimensional reconnection principles leads to a robust measurement of the reconnection rate, even once instabilities break the symmetry. It also allows us to identify internal reconnectio…
Vortex has been considered as the building block and muscle of turbulence for long time. A new physical quantity called Liutex (previously named Rortex) has been defined as the rigid rotation part of fluid motion. From DNS and experiment, forest of hairpin vortices has been found in flow transition and low Reynolds turbulence, but more and more one leg vortices appear in fully developed turbulence. This letter shows hairpin vortex is unstable. Hairpin vortex will weaken or lose one leg by the shear and vortex interaction. This conclusion is made by Liutex definition and mathematical analysis w…
In two dimensional turbulence, vortex thinning process is one of the attractive mechanism to explain inverse energy cascade in terms of vortex dynamics. By direct numerical simulation to the two-dimensional Navier-Stokes equations with small-scale forcing and large-scale damping, Xiao-Wan-Chen-Eyink (2009) found an evidence that inverse energy cascade may proceed with the vortex thinning mechanism. The aim of this paper is to analyze the vortex-thinning mechanism mathematically (using the incompressible Euler equations), and give a mathematical evidence that large-scale vorticity gains energy …
Disentangling the evolution of a coherent mean-flow and turbulent fluctuations, interacting through the non-linearity of the Navier-Stokes equations, is a central issue in fluid mechanics. It affects a wide range of flows, such as planetary atmospheres, plasmas or wall-bounded flows, and hampers turbulence models. We consider the special case of a two-dimensional flow in a periodic box, for which the mean-flow, a pair of box-size vortices called \emph{condensate}, emerges from turbulence through an inverse cascade process. As was recently shown, a perturbative closure describes correctly the c…
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.
The abrupt expansion of the slender vortex core with changes in flow topology is commonly known as vortex breakdown. We present new experimental observations of an alteration of the helical vortex core in wall bounded turbulent flow with abrupt growth in core size, but without change in flow topology. The helical symmetry as such is preserved, though the characteristic parameters of helical symmetry of the vortex core transfer from a smooth linear variation to a different trend under the influence of a non-uniform pressure gradient, causing an increase in helical pitch without changing its sig…
In experimental study of very high Reynolds number turbulence, we found evidences that there are distinguished vortex structures in the intermediate range, that is, between the Kolmogorov and Taylor microscales, where they are indeed expected to be present. These structures are responsible for the intermittency, and, in the same time, they contribute into asymmetry of turbulent statistics, the latter following from the Kolmogorov law.
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…
Using direct numerical simulations of rotating Rayleigh-Bénard convection, we explore the transitions between turbulent states from a 3D flow state towards a quasi-2D condensate known as the large-scale vortex (LSV). We vary the Rayleigh number $Ra$ as control parameter and study the system response (strength of the LSV) in terms of order parameters assessing the energetic content in the flow and the upscale energy flux. By sensitively probing the boundaries of the region of existence of the LSV in parameter space, we find discontinuous transitions and we identify the presence of a hysteresis …
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 develop a computational model of quantum turbulence decay employing a kinematic prescription for the normal fluid. We find that after an initial transient, the length of the vortex tangle L decreases and for large times obeys a scaling law with a -0.45 exponent. The average magnitude (along the quantized vortices) of the superfluid and line-vortex velocity are close and differ significantly from the average magnitude of the normal fluid velocity.
This report addresses an important question discussed by the quantum turbulence community during the last decade: do quantized vortices form, in zero-temperature superfluids, coherent structures similar to vortex tubes in ordinary, viscous turbulence? So far the evidence provided by numerical simulations is that bundles of quantized vortices appear in finite-temperature superfluids, but from the interaction with existing coherent structures in the turbulent (viscous) normal fluid, rather than due to the intrinsic superfuid dynamics. In this report we show that, in very intense quantum turbulen…
We study formation of quasi two-dimensional (thin pancakes) vortex structures in three-dimensional flows, and quasi one-dimensional structures in two-dimensional hydrodynamics. These structures are formed at high Reynolds numbers, when their evolution is described at the leading order by the Euler equations for an ideal incompressible fluid. We show numerically and analytically that the compression of these structures and, as a consequence, the increase in their amplitudes is related to the compressibility of the frozen-in-fluid fields: the field of continuously distributed vortex lines in the…
Unstable shear layers in environmental and industrial flows roll up into a series of vortices, which often form complex nonlinear merging patterns like pairs and triplets. These patterns crucially determine the subsequent turbulence, mixing and scalar transport. We show that the late-time, highly nonlinear merging patterns are predictable from the linearized initial state. The initial asymmetry between consecutive wavelengths of the vertical velocity field provides an effective measure of the strength and pattern of vortex merging. The predictions of this measure are substantiated using direct…
The understanding of turbulent flows is one of the biggest current challenges in physics, as no first-principles theory exists to explain their observed spatio-temporal intermittency. Turbulent flows may be regarded as an intricate collection of mutually-interacting vortices. This picture becomes accurate in quantum turbulence, which is built on tangles of discrete vortex filaments. Here, we study the statistics of velocity circulation in quantum and classical turbulence. We show that, in quantum flows, Kolmogorov turbulence emerges from the correlation of vortex orientations, while deviations…
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…
The density fluctuations of quantum vortex lines are measured in a turbulent flow of superfluid He, at temperatures corresponding to superfluid fraction of 16%, 47% and 81%. The probe is a micro-fabricated second sound resonator that allows for local and small-scale measurements in the core of the flow, at a 10-mesh-size behind a grid. Remarkably, all the vortex power spectra collapse on a single master curve, independently from the superfluid fraction and the mean velocity. By contrast with previous measurements, we report an peculiar shape of the power spectra. The vortex density probability…
A rational theory is proposed to describe the large-scale motion in turbulence. The fluid element with inner orientational structures is proposed to be the building block of fluid dynamics. The variance of the orientational structures then constitutes new fields suitable to describe the vortex state in turbulence. When the fluid element is assumed to be an open subsystem, the differentiable manifold description of turbulence ought to be set up, and the complete fluid dynamics can be deduced from a variational calculus on the constructed Lagrangian dissipation energy density. The derived dynami…
We report recent 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 some results concerning acceleration statistics and the statistics of trapping by vortex filaments.
Turbulence in superfluid helium~II is a tangle of quantized vortex lines which interact via the classical Biot-Savart law. We show that vortex tangles with the same vortex line density will have different energy spectra, depending on the normal fluid which feeds energy into the superfluid component, and identify the spectral signature of two forms of superfluid turbulence: Kolmogorov tangles and Vinen tangles. By decomposing the superfluid velocity field into local and nonlocal contributions, we find that in Vinen tangles the motion of vortex lines depends mainly on the local curvature, wherea…
We compute the frequency spectrum of turbulent superfluid vortex density fluctuations and obtain the same Kolmogorov scaling which has been observed in a recent experiment in Helium-4. We show that the scaling can be interpreted in terms of the spectrum of reconnecting material lines. The calculation is performed using a vortex tree algorithm which considerably speeds up the evaluation of Biot-Savart integrals.
Quantum turbulence can be generated in superfluid helium either thermally (by applying a heat flux, as in thermal counterflow) or mechanically (by stirring the liquid). By modelling the superfluid vortex lines as reconnecting space curves with fixed circulation, and the driving normal fluid as a uniform flow (for thermal counterflow) and a synthetic turbulent flow (for mechanically-driven turbulence), we determine the difference between thermally-driven and mechanically-driven quantum turbulence. We find that in mechanically-driven turbulence the energy is concentrated at the large scales, the…
We report the numerical evidence of a new state of active turbulence in confined domains. By means of extensive numerical simulations of the Toner-Tu-Swift-Hohenberg model for dense bacterial suspensions in circular geometry, we discover the formation a stable, ordered state in which the angular momentum symmetry is broken. This is achieved by self-organization of a turbulent-like flow into a single, giant vortex of the size of the domain. The giant vortex is surrounded by an annular region close to the boundary, characterized by small-scale, radial vorticity streaks. The average radial veloci…
We discuss a recent experiment in which the spectrum of the vortex line density fluctuations has been measured in superfluid turbulence. The observed frequency dependence of the spectrum, $f^{-5/3}$, disagrees with classical vorticity spectra if, following the literature, the vortex line density is interpreted as a measure of the vorticity or enstrophy. We argue that the disagrement is solved if the vortex line density field is decomposed into a polarised field (which carries most of the energy) and an isotropic field (which is responsible for the spectrum).
Incompressible, homogeneous and isotropic turbulence is studied by solving the Navier-Stokes equations on a reduced set of Fourier modes, belonging to a fractal set of dimension $D$. By tuning the fractal dimension parameter, we study the dynamical effects of Fourier decimation on the vortex stretching mechanism and on the statistics of the velocity and the velocity gradient tensor. In particular, we show that as we move from $D=3$ to $D \sim 2.8$, the statistics gradually turns into a purely Gaussian one. This result suggests that even a mild fractal mode reduction strongly depletes the stret…
In a two-dimensional (2D) classical fluid, a large-scale flow structure emerges out of turbulence, which is known as the inverse energy cascade where energy flows from small to large length scales. An interesting question is whether this phenomenon can occur in a superfluid, which is inviscid and irrotational by nature. Atomic Bose-Einstein condensates (BECs) of highly oblate geometry provide an experimental venue for studying 2D superfluid turbulence, but their full investigation has been hindered due to a lack of the circulation sign information of individual quantum vortices in a turbulent …
Maintenance of a columnar vortex by inertial waves in rotational turbulence
Landau Institute for Theoretical Physics, Russian Academy of Sciences（朗道理论物理研究所，俄罗斯科学院）；National Research University Higher School of Economics（国立高等经济大学）；Osipyan Institute of Solid State Physics, Russian Academy of Sciences（奥西普扬固体物理研究所，俄罗斯科学院）
The title of this paper echoes the title of a paragraph in the famous book by Frisch on classical turbulence. In the relevant chapter, the author discusses the role of the statistical dynamics of vortex filaments in the fascinating problem of turbulence and the possibility of a breakthrough in constructing an advanced theory. This aspect arose due to the large amount of evidence, both experimental and numerical, that the vorticity field in turbulent flows has a pronounced filamentary structure. In fact, there is unquestionably a strong relationship between the dynamics of chaotic vortex filame…
Using high Reynolds number experimental data, we search for most dissipative, most intense structures. These structures possess a scaling predicted by log-Poisson model for the dissipation field $ε_r$. The probability distribution function for the exponents $α$, $ε_r\sim e^{αa}$, has been constructed, and compared with Poisson distribution. These new experimental data suggest that the most intense structures have co-dimension less than 2. The log-Poisson statistics is compared with log-binomial which follows from the random $β$-model.
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 …
Turbulence in a superfluid in the zero temperature limit consists of a dynamic tangle of quantized vortex filaments. Different types of turbulence are possible depending on the level of correlations in the orientation of vortex lines. We provide an overview of turbulence in superfluid $^4$He with a particular focus on recent experiments probing the decay of turbulence in the zero temperature regime below 0.5 K. We describe extensive measurements of the vortex line density during the free decay of different types of turbulence: ultraquantum and quasiclassical turbulence in both stationary and r…
We study the statistical properties of coherent, small-scales, filamentary-like structures in Turbulence. In order to follow in time such complex spatial structures, we integrate Lagrangian and Eulerian measurements by seeding the flow with light particles. We show that light particles preferentially concentrate in small filamentary regions of high persistent vorticity (vortex filaments). We measure the fractal dimension of the attracting set and the probability that two particles do not separate for long time lapses. We fortify the signal-to-noise ratio by exploiting multi-particles correlati…
The idea that chaotic set of quantum vortices can mimic classical turbulence, or at least reproduce many main features, is currently actively being developed. Appreciating significance of the challenging problem of the classical turbulence it can be expressed that the idea to study it in terms of quantized line is indeed very important and may be regarded as a breakthrough. For this reason, this theory should be carefully scrutinized. One of the basic arguments supporting this point of view is the fact that vortex tangle decays at zero temperature, when the apparent mechanism of dissipation (m…
