Statistical mechanics of 2D turbulence with a prior vorticity distribution
P. H. Chavanis
备注：
In Proceedings of the Workshop on ``Interdisciplinary Aspects of Turbulence'' at Ringberg Castle, Tegernsee, Germany (Max-Planck Institut fur Astrophysik, 2005)
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.
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…
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…
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 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 …
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 …
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 $Re…
Building upon the intrinsic properties of Navier-Stokes dynamics, namely the prevalence of intense vortical structures and the interrelationship between vorticity and strain rate, we propose a simple framework to quantify the extreme events and the smallest scales of turbulence. We demonstrate that our approach is in excellent agreement with the best available data from direct numerical simulations of isotropic turbulence, with Taylor-scale Reynolds number up to 1300. We additionally highlight a shortcoming of prevailing intermittency models due to their disconnection from observed correlation…
In this paper we study the enstrophy transers in helical turbulence using direct numerical simulation. We observe that the helicity injection does not have significant effects on the inertial-range energy and helicity spectra ($\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 …
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 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 …
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…
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…
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.
Small scale characteristics of turbulence such as velocity gradients and vorticity fluctuate rapidly in magnitude and oscillate in sign. Much work exists on the characterization of magnitude variations, but far less on sign oscillations. While averages performed on large scales tend to zero because of the oscillatory character, those performed on increasingly smaller scales will vary with the averaging scale in some characteristic way. This characteristic variation at high Reynolds numbers is captured by the so-called cancellation exponent, which measures how local averages tend to cancel out …
