We present results on the connection between the vorticity equation and the shape of the single-point vorticity PDF. The statistical framework for these observations is cast in form of conditional averages. The numerical evaluation of these conditional averages provides insights into the intimate relation of dynamical effects like vortex stretching and vorticity diffusion and non-Gaussian vorticity statistics.
The velocity-vorticity correlation integral (Chkhetiani invariant) is an invariant of a viscous Karman-Howarth equation. In the tree-dimensional space this invariant can naturally determine a turbulent diffusivity (viscosity). It is shown, using results of direct numerical simulations, that distributed chaos dominated by this integral can provide an adequate description of the turbulent Rayleigh-Bénard convection, stably stratified turbulence and Rayleigh-Taylor mixing at Prandtl number $Pr \sim 1$.
This article presents direct numerical simulations of the growth of turbulent spots in the transitional regime of plane Couette flow. A quantitative description of the growth process and of the detail of the quadrupolar flow around the spot is given. Focus is put on formation of spanwise vorticity in the velocity streaks that resembles a secondary shear instability. The main features of the instability (coherence lengths, advection velocity) are studied in the context of the turbulent spot, below and above the threshold Reynolds number of appearance of the oblique turbulent bands of plane Coue…
A direct Biot-Savart integration is used to decompose the strain rate into its local and nonlocal constituents, allowing the vorticity alignment with the local and nonlocal strain rate eigenvectors to be investigated. These strain rate tensor constituents are evaluated in a turbulent flow using data from highly-resolved direct numerical simulations. While the vorticity aligns preferentially with the intermediate eigenvector of the \textit{combined} strain rate, as has been observed previously, the present results for the first time clearly show that the vorticity aligns with the most extension…
We study the physics of turbulent/non-turbulent interface of an isolated turbulent region in dilute polymer solutions and Newtonian fluid. The performance of a FENE-P model with a localized homogeneous forcing is verified using the specially designed experimental setup of a turbulent patch growing in water/dilute polymer solution, without mean shear and far from the walls. The results of the small scale dynamics of vorticity and strain help to reveal the key mechanism of polymer action in turbulent flows without mean shear. Modified degrees of alignment between vorticity, the polymer conformat…
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 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.
Using a Maximum Entropy Production Principle (MEPP), we derive a new type of relaxation equations for two-dimensional turbulent flows in the case where a prior vorticity distribution is prescribed instead of the Casimir constraints [Ellis, Haven, Turkington, Nonlin., 15, 239 (2002)]. The particular case of a Gaussian prior is specifically treated in connection to minimum enstrophy states and Fofonoff flows. These relaxation equations are compared with other relaxation equations proposed by Robert and Sommeria [Phys. Rev. Lett. 69, 2776 (1992)] and Chavanis [Physica D, 237, 1998 (2008)]. They c…
Local and nonlocal contributions to the total strain rate tensor at any point in a flow are formulated from an expansion of the vorticity field in a local spherical neighborhood of radius R centered on x. The resulting exact expression allows the nonlocal (background) strain rate tensor to be obtained from the total strain rate tensor. In turbulent flows, where the vorticity naturally concentrates into relatively compact structures, this allows the local alignment of vorticity with the most extensional principal axis of the background strain rate tensor to be evaluated. In the vicinity of any …
This article investigates the formation of spanwise vorticity in the velocity streaks of the oblique laminar- turbulent bands of plane Couette flow (PCF) by mean of Direct Numerical Simulations (DNS). The spanwise vorticity is created by a roll-up type development of the streamwise-wall normal shear layer of the velocity streaks. It is advected by the large scale flow along the bands. We propose a criterion on spanwise vorticity which detects these events in order to perform systematic measurements. Beside of the streamwise and spanwise correlation lengths of the rolls, their advection velocit…
We highlight the differing roles of vorticity and strain in the transport of coarse-grained scalars at length-scales larger than $\ell$ by smaller scale (subscale) turbulence. %subscale flux/stress which appear in the evolution of coarse-grained (resolved) scalars/momentum account for the effect of (subgrid) scales smaller than the coarse-graining length $\ell$. We use the first term in a multiscale gradient expansion due to Eyink \cite{Eyink06a}, which exhibits excellent correlation with the exact subscale physics when the partitioning length $\ell$ is any scale smaller than that of the spect…
We present a construction of isotropic boundary adapted wavelets, which are orthogonal and yield a multi-resolution analysis. We analyze direct numerical simulation data of turbulent channel flow computed at a friction Reynolds number of 395, and investigate the role of coherent vorticity. Thresholding of the vorticity wavelet coefficients allows to split the flow into two parts, coherent and incoherent vorticity. The coherent vorticity is reconstructed from their few intense wavelet coefficients. The statistics of the coherent part, i.e., energy and enstrophy spectra, are close to the statist…
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…
Time scales of turbulent strain activity, denoted as the strain persistence times of first and second order, are obtained from time-dependent expectation values and correlation functions of lagrangian rate-of-strain eigenvalues taken in particularly defined statistical ensembles. Taking into account direct numerical simulation data, our approach relies on heuristic closure hypotheses which allow us to establish a connection between the statistics of vorticity and strain. It turns out that softly divergent prefactors correct the usual "1/s" strain time-scale estimate of standard turbulence phen…
The nonlinear and nonlocal coupling of vorticity and strain-rate constitutes a major hindrance in understanding the self-amplification of velocity gradients in turbulent fluid flows. Utilizing highly-resolved direct numerical simulations of isotropic turbulence in periodic domains of up to $12288^3$ grid points, and Taylor-scale Reynolds number $R_λ$ in the range $140-1300$, we investigate this non-locality by decomposing the strain-rate tensor into local and non-local contributions obtained through Biot-Savart integration of vorticity in a sphere of radius $R$. We find that vorticity is predo…
High-amplitude events of the out-of-plane vorticity component $ω_z$ are analyzed by stereoscopic particle image velocimetry (PIV) in the bulk region of turbulent Rayleigh-Bénard convection in air. The Rayleigh numbers ${\rm Ra}$ vary from $1.7 \times 10^4$ to $5.1 \times 10^5$. The experimental investigation is connected with a comprehensive statistical analysis of long-term time series of $ω_z$ and individual velocity derivatives $\partial u_i/\partial x_j$. A statistical convergence for derivative moments up to an order of 6 is demonstrated. Our results are found to agree well with existing …
We study the distribution of spatial extrema of vorticity and the determinant of the strain rate tensor for a two-dimensional turbulent flow forced by a Kolmogorov forcing. The distribution of these quantities follow non-Gaussian behaviour and they do not fall into the Generalised Extreme value distributions. It is found that for the truncated Euler equations the spatial extrema of vorticity and strain rate tensor are well described by the Gumbel distribution. The spatial extrema for the vorticity is found to be at the core of the vortices while the velocity gradients are found near the edges …
The vorticity random field of turbulent flow is singled out as the main dynamical variable for the description of turbulence, and the evolution equation of the probability density function (PDF) of the vorticity field has been obtained. This PDF evolution equation is a mixed type partial differential equation (PDE) of second order which depends only on the conditional mean (first order) of the underlying turbulent flow, which is in contrast with Reynolds' mean flow equation which relies on a quadratic statistics. Therefore the new PDF PDE may provide new closure scheme based on the conditional…
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…
This article presents a modelling of the formation of spanwise vorticity in the turbulent streaks of the oblique bands and spots of transitional plane Couette flow. A functional model is designed to mimic the coherent flow in the streaks. The control parameters of the model are extracted from Direct Numerical Simulations (DNS) statistical data. A Reynolds stress is proposed to study the effect on the instability of this additional force maintaining the baseflow. Local (quasi-parallel) temporal stability analysis is performed on that model to investigate the linear development of the spanwise v…
A commonplace view of pressure-driven turbulence in pipes and channels is as "cascades" of streamwise momentum toward the viscous layer at the wall. We present in this paper an alternative picture of these flows as "inverse cascades" of spanwise vorticity, in the cross-stream direction but away from the viscous sublayer. We show that there is a constant spatial flux of spanwise vorticity, due to vorticity conservation, and that this flux is necessary to produce pressure-drop and energy dissipation. The vorticity transport is shown to be dominated by viscous diffusion at distances closer to the…
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）
We report an experimental investigation of various statistical properties of the spatial Fourier modes of the vorticity field in turbulent jets for a large range of Reynolds numbers ($530 \leq R_λ \leq 6100$). The continuous time evolution of a spatial Fourier mode of the vorticity distribution, characterized by a well defined wave-vector, is obtained from acoustic scattering measurements. The spatial enstrophy spectrum, as a function of the spatial wave-vector, is determined by scanning the incoming sound frequencies. Time-frequency analysis of the turbulent vorticity fluctuations is also per…
The expansion of the rotating fluid will change the vorticity and rotational speed of the expanding region. In turbulent thermal convection, this microscopic effect is preserved. Tracking the fluid micelles shows that the average vorticity varies with density, producing vorticity transport and angular momentum transport from the low-density area to the high-density area, forming a macroscopic vorticity difference and rotational speed difference. Taking the axial thermal convection model of the solar polar region, it can generate axial differential rotation, and the centrifugal force difference…
The results of an analysis of turbulent pipe flow based on a Karhunen-Lo`eve decomposition are presented. The turbulent flow is generated by a direct numerical simulation of the Navier-Stokes equations using a spectral element algorithm at a Reynolds number Re_τ=150. This simulation yields a set of basis functions that captures 90% of the energy after 2,453 modes. The eigenfunctions are categorised into two classes and six subclasses based on their wavenumber and coherent vorticity structure. Of the total energy, 81% is in the propagating class, characterised by constant phase speeds; the rema…
This paper examines the mechanisms of coherent structure interactions in spatially evolving turbulent free shear layers at different values of the velocity ratio parameter λ=$(U_1-U_2)/(U_1+U_2)$, where $U_1$ and $U_2 (\leq U_1)$ are the free stream velocities on either side of the layer. The study employs the point-vortex (or vortex-gas) model presented in part I (arXiv:1509.00603) which predicts spreading rates that are in the close neighborhood of results from most high Reynolds number experiments and 3D simulations. The present (2D) simulations show that the well-known steep-growth merger …
It has been broadly acknowledged that vortex detection algorithms, usually based on linear-algebraic properties of the velocity gradient tensor, can be plagued with severe shortcomings and may become, in practical terms, dependent on the choice of subjective threshold parameters in their implementations. In two-dimensions, a large class of standard vortex identification prescriptions turn out to be equivalent to the "swirling strength criterion" ($λ_{ci}$-criterion), which is critically revisited in this work. We classify the instances where the accuracy of the $λ_{ci}$-criterion is affected b…
Assimilation of nearly turbulent Rayleigh-Bénard flow through vorticity or local circulation measurements: a computational study
University of Virginia（弗吉尼亚大学）；University of Massachusetts, Amherst（马萨诸塞大学阿默斯特分校）；Indiana University（印第安纳大学）；Texas A&M University（德克萨斯农工大学）；Weizmann Institute of Science（魏茨曼科学研究所）
The discrete wavelet packet transform (DWPT) and discrete wavelet transform (DWT) are used to extract and study the dynamics of coherent structures in a turbulent rotating fluid. Three-dimensional (3D) turbulence is generated by strong pumping through tubes at the bottom of a rotating tank (48.4 cm high, 39.4 cm diameter). This flow evolves toward two-dimensional (2D) turbulence with increasing height in the tank. Particle Image Velocimetry (PIV) measurements on the quasi-2D flow reveal many long-lived coherent vortices with a wide range of sizes. The vorticity fields exhibit vortex birth, mer…
We put forward the idea of defining vortex boundaries in planar flows as closed material barriers to the diffusive transport of vorticity. Such diffusive vortex boundaries minimize the leakage of vorticity from the fluid mass they enclose when compared to other nearby material curves. Building on recent results on passive diffusion barriers, we develop an algorithm for the automated identification of such structures from general, two-dimensional unsteady flow data. As examples, we identify vortex boundaries as vorticity diffusion barriers in two flows: an explicitly known laminar flow and a nu…
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 …
Courant Institute of Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）；Initiative for the Theoretical Sciences, The Graduate Center, CUNY（CUNY研究生院理论科学倡议）；Weizmann Institute of Science（魏茨曼科学研究所）
Numerical simulations show that microscopic rod-like bodies suspended in a turbulent flow tend to align with the vorticity vector, rather than with the dominant eignevector of the strain-rate tensor. This paper investigates an analytically solvable limit of a model for alignment in a random velocity field with isotropic statistics. The vorticity varies very slowly and the isotropic random flow is equivalent to a pure strain with statistics which are axisymmetric about the direction of the vorticity. We analyse the alignment in a weakly fluctuating uniaxial strain field, as a function of the pr…
This investigation is based on a DNS of a steady self-preserving incompressible axisymmetric turbulent jet at a Reynolds number of 2400. The DNS data enable accurate maps of the outer irrotational flow field, and also the vorticity field in the turbulent core of the jet. It is found necessary to define two separate boundaries of the jet. The first is an inner boundary (turbulent/nonturbulent, T/NT), from where vorticity rises steeply towards to the core. The second is an outer rotational/irrotational boundary, beyond which the flow may be considered irrotational. The velocity field beyond the …
Statistical equilibria of two-dimensional turbulent flows for generic initial vorticity fields on a sphere, calculated on the basis of the original Miller-Robert-Sommeria theory
Research Institute for Applied Mechanics, Kyushu University（九州大学应用力学研究所）；Graduate School of Science, Kyoto University（京都大学理学研究科）
In this paper, we show that the spatio-temporal evolution of incompressible flows in a long circular pipe can be described by vorticity dynamics. The principal techniques to obtain solutions are similar to those used for flows in the whole space. As the consideration of the Navier-Stokes equations is given in a cylindrical co-ordinates system, two aspects of complication arise. One is the interaction of the velocity components in the radial and azimuthal directions, due to the fictitious centrifugal force in the equations of motion. The rate of the vorticity production at the pipe wall depends…
We demonstrate that numerical solutions of Burgers' equation can be obtained by a scale-totality algorithm for fluids of small viscosity (down to one billionth). Two sets of initial data, modelling simple shears and wall boundary layers, are chosen for our computations. Most of the solutions are carried out well into the fully turbulent regime over finely-resolved scales in space and in time. It is found that an abrupt spatio-temporal concentration in shear constitutes an essential part during the flow evolution. The vorticity surge has been instigated by the non-linearity complying with insta…
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…
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 employ detailed numerical simulations to probe the mechanism of flow reversals in two-dimensional turbulent convection. We show that the reversals occur via vortex reconnection of two attracting corner rolls having same sign of vorticity, thus leading to major restructuring of the flow. Large fluctuations in heat transport are observed during the reversal due to this flow reconfiguration. The flow configurations during the reversals have been analyzed quantitatively using large-scale modes. Using these tools, we also show why flow reversals occur for a restricted range of Rayleigh and Prand…
The turbulent/non-turbulent interface is analysed in a direct numerical simulation of a boundary layer in the range $Re_θ=2800-6600$, with emphasis on the behaviour of the relatively large-scale fractal intermittent region. This requires the introduction of a new definition of the distance between a point and a general surface, which is compared with the more usual vertical distance to the top of the layer. Interfaces are obtained by thresholding the enstrophy field and the magnitude of the rate-of-strain tensor, and it is concluded that, while the former are physically relevant features, the …
No evidence of vorticity production from irrotational turbulent gravitational collapse yet
Nordita, KTH Royal Institute of Technology and Stockholm University（诺迪塔，皇家理工学院和斯德哥尔摩大学）；The Oskar Klein Centre, Department of Astronomy, Stockholm University（奥斯卡·克莱因中心，天文学系，斯德哥尔摩大学）；McWilliams Center for Cosmology & Department of Physics, Carnegie Mellon University（麦克威廉斯宇宙学中心与物理系，卡内基梅隆大学）；School of Natural Sciences and Medicine, Ilia State University（自然科学与医学院，伊利亚州立大学）；Scuola Normale Superiore（比萨高等师范学院）；Argelander-Institut für Astronomie, Universität Bonn（阿尔根德天文学研究所，波恩大学）
We present results of the study of a turbulent air flow with a large scale circulation in Rayleigh-Benard rectangular convective cell with a heated bottom wall and a cooled top wall. The mean horizontal velocity of the main roll and the mean vorticity of eddy rings are almost aligned in a large part of the flow. The helicity of the mean flow is quite high, and is the source of turbulent helicity. Since helicity of the mean flow and turbulence is quite large, the flow in Rayleigh-Benard convective cell is well suited to study properties of helical turbulence.
Expressing the evolution equations for the filtered velocity gradient tensor (FVGT) in the strain-rate eigenframe provides an insightful way to disentangle and understand various processes such as strain self-amplification, vortex stretching and tilting, and to consider their properties at different scales in the flow. Using data from Direct Numerical Simulation (DNS) of the forced Navier-Stokes equation, we consider the relative importance of local and non-local terms in the FVGT eigenframe equations across the scales using statistical analysis. The analysis of the eigenframe rotation-rate, t…
We study the three-dimensional clustering of velocity stagnation points, of nulls of the vorticity and of the Lagrangian acceleration, and of inertial particles in turbulent flows at fixed Reynolds numbers, but under different large-scale flow geometries. To this end, we combine direct numerical simulations of homogeneous and isotropic turbulence and of the Taylor-Green flow, with particle tracking velocimetry in a von Kármán experiment. While flows have different topologies (as nulls cluster differently), particles behave similarly in all cases, indicating that Taylor-scale neutrally buoyant …
Boundary-layer transition is accompanied by a significant increase in skin friction whose origin is rigorously explained using the stochastic Lagrangian formulation of the Navier-Stokes equations. This formulation permits the exact analysis of vorticity dynamics in individual realizations of a viscous incompressible fluid flow. The Lagrangian reconstruction formula for vorticity is here extended for the first time to Neumann boundary conditions (Lighthill source). We can thus express the wall vorticity, and therefore the wall stress, as the expectation of a stochastic Cauchy invariant in backw…
We report the experimental observation of $1/f$ fluctuations in three different turbulent flow configurations: the large scale velocity driven by a two-dimensional turbulent flow, the magnetic field generated by a turbulent swirling flow of liquid sodium and the pressure fluctuations due to vorticity filaments in a swirling flow. For these three systems, $1/f$ noise is shown to result from the dynamics of coherent structures that display transitions between a small number of states. The interevent duration is distributed as a power-law. The exponent of this power-law and the nature of the dyna…
The transfer of kinetic energy from large to small scales is a hallmark of turbulent flows. Yet, a precise mechanistic description of this transfer, which is expected to occur via an energy cascade, is still missing. Several conceptually simple configurations with vortex tubes have been proposed as a testing ground to understand the energy cascade. Here, we focus on incompressible flows and compare the energy transfer occurring in a statistically steady homogeneous isotropic turbulent (HIT) flow with the generation of fine-scale motions in configurations involving vortex tubes. We start by fil…
A new flamelet model is developed for sub-grid modeling and coupled with the resolved flow for turbulent combustion. The model differs from current models in critical ways. (i) Non-premixed flames, premixed flames, or multi-branched flame structures are determined rather than prescribed. (ii) The effects of shear strain and vorticity are determined. (iii) The strain rates and vorticity applied at the sub-grid level are directly determined from the resolved-scale strain rates and vorticity without a contrived progress variable. (iv) The flamelet model is three-dimensional . (v) The effect of va…
