Real-space Manifestations of Bottlenecks in Turbulence Spectra

Bottleneck crossover between classical and quantum superfluid turbulence

We consider superfluid turbulence near absolute zero of temperature generated by classical means, e.g. towed grid or rotation but not by counterflow. We argue that such turbulence consists of a {\em polarized} tangle of mutually interacting vortex filaments with quantized vorticity. For this system we predict and describe a bottleneck accumulation of the energy spectrum at the classical-quantum crossover scale $\ell$. Demanding the same energy flux through scales, the value of the energy at the crossover scale should exceed the Kolmogorov-41 spectrum by a large factor $\ln^{10/3} (\ell/a_0)$ (…

Bottleneck effect in three-dimensional turbulence simulations

At numerical resolutions around $512^3$ and above, three-dimensional energy spectra from turbulence simulations begin to show noticeably shallower spectra than $k^{-5/3}$ near the dissipation wavenumber (`bottleneck effect'). This effect is shown to be significantly weaker in one-dimensional spectra such as those obtained in wind tunnel turbulence. The difference can be understood in terms of the transformation between one-dimensional and three-dimensional energy spectra under the assumption that the turbulent velocity field is isotropic. Transversal and longitudinal energy spectra are similar…

Voigt正则化湍流中的动力学减速、瓶颈及多标度行为

Dynamical slowdown, bottlenecks, and multiscaling in Voigt-regularised turbulence

该研究采用Voigt正则化SABRA模型与NSV方程DNS，发现湍流存在三个谱区，揭示瓶颈形成与尺度相关减速及初始热化有关，证实小α下Voigt模型是NS方程的优秀近似。

Bottleneck effects in turbulence: Scaling phenomena in r- versus p-space

We (analytically) calculate the energy spectrum corresponding to various experimental and numerical turbulence data analyzed by Benzi et al.. We find two bottleneck phenomena: While the local scaling exponent $ζ_r(r)$ of the structure function decreases monotonically, the local scaling exponent $ζ_p(p)$ of the corresponding spectrum has a minimum of $ζ_p(p_{min})\approx 0.45$ at $p_{min}\approx (10 η)^{-1}$ and a maximum of $ζ_p(p_{max})\approx 0.77$ at $p_{max}\approx 8 L^{-1}$. A physical argument starting from the constant energy flux in p--space reveals the general mechanism underlying the…

Inertial range scaling in numerical turbulence with hyperviscosity

Numerical turbulence with hyperviscosity is studied and compared with direct simulations using ordinary viscosity and data from wind tunnel experiments. It is shown that the inertial range scaling is similar in all three cases. Furthermore, the bottleneck effect is approximately equally broad (about one order of magnitude) in these cases and only its height is increased in the hyperviscous case--presumably as a consequence of the steeper decent of the spectrum in the hyperviscous subrange. The mean normalized dissipation rate is found to be in agreement with both wind tunnel experiments and di…

Intermittency, cascades and thin sets in three-dimensional Navier-Stokes turbulence

Visual manifestations of intermittency in computations of three dimensional Navier-Stokes fluid turbulence appear as the low-dimensional or `thin' filamentary sets on which vorticity and strain accumulate as energy cascades down to small scales. In order to study this phenomenon, the first task of this paper is to investigate how weak solutions of the Navier-Stokes equations can be associated with a cascade and, as a consequence, with an infinite sequence of inverse length scales. It turns out that this sequence converges to a finite limit. The second task is to show how these results scale wi…

Cascade time-scales for energy and helicity in homogeneous isotropic turbulence

We extend the Kolmogorov phenomenology for the scaling of energy spectra in high-Reynolds number turbulence, to explicitly include the effect of helicity. There exists a time-scale $τ_H$ for helicity transfer in homogeneous, isotropic turbulence with helicity. We arrive at this timescale using the phenomenological arguments used by Kraichnan to derive the timescale $τ_E$ for energy transfer (J. Fluid Mech. {\bf 47}, 525--535 (1971)). We show that in general $τ_H$ may not be neglected compared to $τ_E$, even for rather low relative helicity. We then deduce an inertial range joint cascade of ene…

Universality in fully developed turbulence

We extend the numerical simulations of She et al. [Phys.\ Rev.\ Lett.\ 70, 3251 (1993)] of highly turbulent flow with $15 \le$ Taylor-Reynolds number $Re_λ\le 200$ up to $Re_λ\approx 45000$, employing a reduced wave vector set method (introduced earlier) to approximately solve the Navier-Stokes equation. First, also for these extremely high Reynolds numbers $Re_λ$, the energy spectra as well as the higher moments -- when scaled by the spectral intensity at the wave number $k_p$ of peak dissipation -- can be described by {\it one universal} function of $k/k_p$ for all $Re_λ$. Second, the ISR sc…

Anisotropy and scaling corrections in turbulence

Two parametrizations for second order velocity moments, the Batchelor parametrization for the r-space structure function and a common parametrization for the energy spectrum, $E(p)\propto p^{-5/3}\exp(-p/p_d)$, are examined and compared. In particular, we investigate corrections to the local scaling exponents induced by finite size effects. The behavior of local r-- and p--space exponents differs dramatically. The Batchelor type parametrization leads to energy pileups in p-space at the ends of the ISR. These bottleneck effects result in an extended r-space scaling range, comparable to experime…

真正的蝴蝶效应：从流行文化到数学与物理

The real butterfly effect: from the pop culture to mathematics and physics

本文借助有限尺寸李雅普诺夫指数，通过萨布拉壳模型、克莱奇南模型等，研究蝴蝶效应在湍流中的多尺度物理，弥合不同尺度现象差距，阐明高雷诺数流动预测界限的物理机制。