Energy and enstrophy transfer in decaying two-dimensional turbulence

We present experimental data on the direct enstrophy cascade in decaying two-dimensional turbulence. Velocity and vorticity fields are obtained using particle tracking velocimetry. From those fields we directly compute the enstrophy and energy flux by using a filtering technique inspired by large-eddy simulations. This allows considerable insight into the physical processes of turbulence when compared with structure-function or spectral analysis. The direct cascade of enstrophy is weakly forward, with almost as much backscatter as down-scale enstrophy transfer, whereas the inverse energy cascade is strongly upscale with a modest amount of backscatter.     

Intermittency in the enstrophy cascade of two-dimensional fully developed turbulence: Universal features

Intermittency (externally induced) in the two-dimensional (2D) enstrophy cascade is shown to be able to maintain a finite enstrophy along with a vorticity conservation anomaly. Intermittency mechanisms of three-dimensional (3D) energy cascade and 2D enstrophy cascade in fully developed turbulence (FDT) seem to have some universal features. The parabolic-profile approximation (PPA) for the singularity spectrum f(α) in multi-fractal model is used and extended to the appropriate microscale regimes to exhibit these features. The PPA is also shown to afford, unlike the generic multi-fractal model, an analytical calculation of probability distribution functions (PDF) of flow-variable gradients in these FDT cases and to describe intermittency corrections that complement those provided by the homogeneous-fractal model.     

Lagrangian chaos and the effect of drag on the enstrophy cascade in two-dimensional turbulence

We investigate the effect of drag force on the enstrophy cascade of two-dimensional Navier-Stokes turbulence. We find a power law decrease of the energy wave number (k) spectrum that is faster than the classical (no-drag) prediction of k(-3). It is shown that the enstrophy cascade with drag can be analyzed by making use of a previous theory for finite lifetime passive scalars advected by a Lagrangian chaotic fluid flow. Using this we relate the power law exponent of the energy wave number spectrum to the distribution of finite time Lyapunov exponents and the drag coefficient.     

Physical mechanism of the two-dimensional enstrophy cascade

In two-dimensional turbulence, irreversible forward transfer of enstrophy requires anticorrelation of the turbulent vorticity transport vector and the inertial-range vorticity gradient. We investigate the basic mechanism by numerical simulation of the forced Navier-Stokes equation. In particular, we obtain the probability distributions of the local enstrophy flux and of the alignment angle between vorticity gradient and transport vector. These are surprisingly symmetric and cannot be explained by a local eddy-viscosity approximation. The vorticity transport tends to be directed along streamlines of the flow and only weakly aligned down the fluctuating vorticity gradient. All these features are well explained by a local nonlinear model. The physical origin of the cascade lies in steepening of inertial-range vorticity gradients due to compression of vorticity level sets by the large-scale strain field.     

Slow decay of the finite Reynolds number effect of turbulence

The third-order structure function is used to study the finite Reynolds number (FRN) effect of turbulence, which refers to the deviation of turbulence statistics observed at finite Reynolds numbers from predictions of the Kolmogorov theories. It is found that the FRN effect decreases as CR(-mu)(lambda), when R(lambda) is high, and mu < or = 6/5. Here R(lambda) is the Taylor-microscale Reynolds number and C is a constant independent of R(lambda). From the exact spectral equations, the decay exponent mu and the constant C are determined for typical fully developed turbulent flows (freely decaying isotropic turbulence and shear flow turbulence), so that the quantitative prediction of the FRN effect is feasible.     

Similarity decay of enstrophy in an electron fluid

A similarity decay law is proposed for enstrophy of a one-signed-vorticity fluid in a circular free-slip domain. It excludes the metastable equilibrium enstrophy which cannot drive turbulence, and approaches Batchelor's t(-2) law for strong turbulence. Measurements of the decay of a turbulent electron fluid agree well with the predictions of the decay law for a variety of initial conditions.     

Probability density functions of the enstrophy flux in two dimensional grid turbulence

Probability density functions of the enstrophy flux in two dimensional grid turbulence are found to be strongly non-Gaussian and can be mimicked by stretched exponential functions. Evidence of this behavior is found in experiments using turbulent soap films and numerical simulations. The enstrophy flux itself is found to be constant for a range of scales corresponding to the enstrophy cascade.     

Decaying two-dimensional turbulence in a circular container

We present direct numerical simulations of two-dimensional decaying turbulence at initial Reynolds number 5 x 10(4) in a circular container with no-slip boundary conditions. Starting with random initial conditions the flow rapidly exhibits self-organization into coherent vortices. We study their formation and the role of the viscous boundary layer on the production and decay of integral quantities. The no-slip wall produces vortices which are injected into the bulk flow and tend to compensate the enstrophy dissipation. The self-organization of the flow is reflected by the transition of the initially Gaussian vorticity probability density function (PDF) towards a distribution with exponential tails. Because of the presence of coherent vortices the pressure PDF become strongly skewed with exponential tails for negative values.     

Structure functions in two-dimensional turbulence

The “variable range decomposition” mean field type approximation is applied to the enstrophy and energy balance equations in 2D homogeneous, isotropic turbulence. Enstrophy is seen to be transferred to smaller scales, energy to larger scales. The approximate enstrophy balance equation is supplemented by an exact relation between the velocity structure functionD(r) and the vorticity structure functionD _ω(r) to form a closed set of equations that is used to calculateD andD _ω from scale zero up to the input scale.D _ω is found to depend only on viscosity and the enstrophy dissipation ε_ω and tends to the constant ≈15ε _ω ^2/3 in the enstrophy inertial range.D(r) in addition to the well-knownr ²-law has a second power law term ∝r ^4/3, which is important in the intermediate range between the viscous range and the enstrophy inertial range. All numerical constants are calculated.     

The decay of turbulence in the burgers model

Statistical characteristics of the turbulence in the Burgers model are obtained     

On the decay law of quasi-two-dimensional turbulence

Laboratory measurements of decaying quasi-two-dimensional turbulence in thin fluid layers with various depths have been performed. It has been shown that decay at large Reynolds numbers corresponds to a non-linear bottom friction with the coefficient satisfying the law λ ∼ (ν/h ²)^1/2|K|^1/4 following from theoretical estimates, where K is the Okubo-Weiss function depending on the enstrophy and degree of ellipticity of vortices. It has been shown that the structure of the flow changes in the decay process.     

Crossover from symplectic to orthogonal class in a two-dimensional honeycomb lattice

We have calculated the weak-localization correction to the conductivity for disordered electrons in a two-dimensional honeycomb lattice and shown that it can be either positive or negative depending on the interaction range of impurity potentials. From symmetry considerations, the symplectic class turns out to be realized at nonzero temperatures and crossover to the orthogonal class is predicted with decreasing temperature.