A direct numerical simulation at resolution 2403 is used to obtain a statistically stationary three-dimensional homogeneous and isotropic turbulent field at a Reynolds number around 1000 (Rλ ≈ 150). The energy spectrum displays an inertial subrange. The velocity derivative distribution, known to be strongly non-Gaussian, is found to be close to, but not, exponential. The nth-order moments of this distribution, as well as the velocity structure functions, do not scale with n as predicted by intermittency models. Visualization of the flow confirms the previous finding that the strongest vorticity is organized in very elongated thin tubes. The width of these tubes is of the order of a few dissipation scales, while their length can reach the integral scale of the flow.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112091001957

The spatial structure and statistical properties of homogeneous turbulence

/pdf/lagrangian-coherent-structures-and-mixing-in-two-dimensional-4o5udmdfy8.pdf

Lagrangian coherent structures and mixing in two-dimensional turbulence

In physical systems, a reduction in dimensionality often leads to exciting new phenomena. Here we discuss the novel effects arising from the consideration of fluid turbulence confined to two spatial dimensions. The additional conservation constraint on squared vorticity relative to three-dimensional (3D) turbulence leads to the dual-cascade scenario of Kraichnan and Batchelor with an inverse energy cascade to larger scales and a direct enstrophy cascade to smaller scales. Specific theoretical predictions of spectra, structure functions, probability distributions, and mechanisms are presented, and major experimental and numerical comparisons are reviewed. The introduction of 3D perturbations does not destroy the main features of the cascade picture, implying that 2D turbulence phenomenology establishes the general picture of turbulent fluid flows when one spatial direction is heavily constrained by geometry or by applied body forces. Such flows are common in geophysical and planetary contexts, are beautiful to observe, and reflect the impact of dimensionality on fluid turbulence.

http://personalpages.to.infn.it/~boffetta/Papers/be_arfm12.pdf

Two-Dimensional Turbulence

We consider the canonical Gibbs measure associated to aN-vortex system in a bounded domain Λ, at inverse temperature$\widetilde\beta $ and prove that, in the limitN→∞,$\widetilde\beta $/N→β, αN→1, where β∈(−8π, + ∞) (here α denotes the vorticity intensity of each vortex), the one particle distribution function ϱN = ϱNx,x∈Λ converges to a superposition of solutions ϱα of the following Mean Field Equation:

$$\left\{ {\begin{array}{*{20}c} {\varrho _{\beta (x) = } \frac{{e^{ - \beta \psi } }}{{\mathop \smallint \limits_\Lambda e^{ - \beta \psi } }}; - \Delta \psi = \varrho _\beta in\Lambda } \\ {\psi |_{\partial \Lambda } = 0.} \\ \end{array} } \right.$$

/pdf/a-special-class-of-stationary-flows-for-two-dimensional-yoy3grwgu3.pdf

A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description

The lattice Boltzmann equation: a new tool for computational fluid-dynamics

Forced two-dimensional turbulence is simulated by high-resolution (5122) numerical integrations of the Navier-Stokes equations. Small-scale behaviour differs significantly from the one previously observed at lower resolutions. Coherent structures are observed at the injection scale with spontaneous formation of dipolar and tripolar structures here analysed for the first time.

High-Resolution Numerical Experiments for Forced Two-Dimensional Turbulence

The authors present a detailed study of the emergence and the long-time behaviour of coherent vortices in two-dimensional decaying turbulence. By high-resolution numerical experiments they find that the coherent structures are self-similar, i.e. their energy, enstrophy and size satisfy scaling laws. Moreover, the knowledge of the statistical distribution of the size of these vortices is sufficient to compute the energy spectrum of the 2D turbulent flow and to explain the significant deviations from the Kraichnan-Batchelor theory. At long times the motion of the fluid is dominated by the vortex dynamics, which is strikingly similar to the Hamiltonian motion of few point vortices, as is confirmed by a comparison between the numerical simulations of the two systems.

Self-similar coherent structures in two-dimensional decaying turbulence

The results of a very‐high‐resolution numerical integration of a two‐dimensional turbulent flow is presented. The very long integration time allows a discussion of both the initial and the ultimate stage of the decay. It is established that the k−3 enstrophy inertial range is only obtained as a transient state of the system. For very long time the emergence of coherent vortices destroys the scale invariance and produces a rather steep spectrum. In addition, the long term behavior strongly depends on the initial conditions. For steep enough initial spectrum, a large‐scale energy containing a range in which vortices are directly linked to the initial conditions separates from a small‐scale range in which many coherent structures originate from inviscid instabilities. The results are compared with previous studies and the origin of self‐similarity breaking in turbulent decay is discussed.

The generation of vortices in high‐resolution, two‐dimensional decaying turbulence and the influence of initial conditions on the breaking of self‐similarity

By high-resolution numerical integration of two-dimensional Navier-Stokes equations we show that the turbulent flow at high Reynolds number is dominated by a simple and weakly unstable Hamiltonian system of pointlike vortices. The large instabilities, typical of the turbulent flow, are found uniquely outside vortices, in the wide dissipative region which results to be only a small perturbation of the vortex system. Moreover, the statistical distribution of vortex sizes determines the slope of the energy spectrum, which is steeper than that predicted by phenomenological theories.

On the Statistical Properties of Two-Dimensional Decaying Turbulence

The authors study the mechanisms of enstrophy transfer in 2D turbulence. By theoretical discussion of the fragmentation process at very small scales, the authors show that enstrophy transfer must be space filling, due to the regularity of the flow at any time. Detailed analysis of a numerical decaying experiment confirms this picture. Yet 'intermittency' is acting at large scale due to the presence of coherent structures, namely vortices, which inhibit local enstrophy transfer.

https://iopscience.iop.org/article/10.1088/0305-4470/19/18/023/pdf

P. Santangelo

Papers

High-Resolution Numerical Experiments for Forced Two-Dimensional Turbulence

Self-similar coherent structures in two-dimensional decaying turbulence

The generation of vortices in high‐resolution, two‐dimensional decaying turbulence and the influence of initial conditions on the breaking of self‐similarity

On the Statistical Properties of Two-Dimensional Decaying Turbulence

Intermittency and coherent structures in two-dimensional turbulence