scholarly journals Fokker-Planck Kinetic Description of Small-scale Fluid Turbulence for Classical Incompressible Fluids§

2008 ◽  
Author(s):  
M. Tessarotto ◽  
M. Ellero ◽  
D. Sarmah ◽  
P. Nicolini ◽  
Takashi Abe
Keyword(s):  
Incompressible Fluids ◽  
Small Scale ◽  
Kinetic Description ◽  
Fluid Turbulence ◽  
Fokker Planck

Non-Gaussian probability distributions of solar wind fluctuations

1994 ◽  
Vol 12 (12) ◽  
pp. 1127-1138 ◽  
Author(s):  
E. Marsch ◽  
C. Y. Tu
Keyword(s):  
Solar Wind ◽  
Probability Distributions ◽  
Time Lag ◽  
Small Scale ◽  
Time Lags ◽  
Mhd Turbulence ◽  
Theoretical Concepts ◽  
Fluid Turbulence ◽  
Long Time ◽  
Non Gaussian

Abstract. The probability distributions of field differences ∆x(τ)=x(t+τ)-x(t), where the variable x(t) may denote any solar wind scalar field or vector field component at time t, have been calculated from time series of Helios data obtained in 1976 at heliocentric distances near 0.3 AU. It is found that for comparatively long time lag τ, ranging from a few hours to 1 day, the differences are normally distributed according to a Gaussian. For shorter time lags, of less than ten minutes, significant changes in shape are observed. The distributions are often spikier and narrower than the equivalent Gaussian distribution with the same standard deviation, and they are enhanced for large, reduced for intermediate and enhanced for very small values of ∆x. This result is in accordance with fluid observations and numerical simulations. Hence statistical properties are dominated at small scale τ by large fluctuation amplitudes that are sparsely distributed, which is direct evidence for spatial intermittency of the fluctuations. This is in agreement with results from earlier analyses of the structure functions of ∆x. The non-Gaussian features are differently developed for the various types of fluctuations. The relevance of these observations to the interpretation and understanding of the nature of solar wind magnetohydrodynamic (MHD) turbulence is pointed out, and contact is made with existing theoretical concepts of intermittency in fluid turbulence.

scholarly journals On universal features of the turbulent cascade in terms of non-equilibrium thermodynamics

2018 ◽  
Vol 848 ◽  
pp. 117-153 ◽  
Author(s):  
Nico Reinke ◽  
André Fuchs ◽  
Daniel Nickelsen ◽  
Joachim Peinke
Keyword(s):  
Probability Density ◽  
Entropy Production ◽  
Turbulent Flows ◽  
Planck Equation ◽  
Small Scale ◽  
Equilibrium Thermodynamics ◽  
Fokker Planck Equation ◽  
Velocity Increments ◽  
Turbulent Cascade ◽  
Fokker Planck

Features of the turbulent cascade are investigated for various datasets from three different turbulent flows, namely free jets as well as wake flows of a regular grid and a cylinder. The analysis is focused on the question as to whether fully developed turbulent flows show universal small-scale features. Two approaches are used to answer this question. First, two-point statistics, namely structure functions of longitudinal velocity increments, and, second, joint multiscale statistics of these velocity increments are analysed. The joint multiscale characterisation encompasses the whole cascade in one joint probability density function. On the basis of the datasets, evidence of the Markov property for the turbulent cascade is shown, which corresponds to a three-point closure that reduces the joint multiscale statistics to simple conditional probability density functions (cPDFs). The cPDFs are described by the Fokker–Planck equation in scale and its Kramers–Moyal coefficients (KMCs). The KMCs are obtained by a self-consistent optimisation procedure from the measured data and result in a Fokker–Planck equation for each dataset. Knowledge of these stochastic cascade equations enables one to make use of the concepts of non-equilibrium thermodynamics and thus to determine the entropy production along individual cascade trajectories. In addition to this new concept, it is shown that the local entropy production is nearly perfectly balanced for all datasets by the integral fluctuation theorem (IFT). Thus, the validity of the IFT can be taken as a new law of the turbulent cascade and at the same time independently confirms that the physics of the turbulent cascade is a memoryless Markov process in scale. The IFT is taken as a new tool to prove the optimal functional form of the Fokker–Planck equations and subsequently to investigate the question of universality of small-scale turbulence in the datasets. The results of our analysis show that the turbulent cascade contains universal and non-universal features. We identify small-scale intermittency as a universality breaking feature. We conclude that specific turbulent flows have their own particular multiscale cascades, in other words, their own stochastic fingerprints.

scholarly journals Small-scale universality in fluid turbulence

2014 ◽  
Vol 111 (30) ◽  
pp. 10961-10965 ◽  
Author(s):  
J. Schumacher ◽  
J. D. Scheel ◽  
D. Krasnov ◽  
D. A. Donzis ◽  
V. Yakhot ◽  
...  
Keyword(s):  
Small Scale ◽  
Fluid Turbulence

Cross-independence closure for statistical mechanics of fluid turbulence

2011 ◽  
Vol 670 ◽  
pp. 365-403 ◽  
Author(s):  
TOMOMASA TATSUMI
Keyword(s):  
Statistical Mechanics ◽  
Velocity Distribution ◽  
Quantum Fluids ◽  
Small Scale ◽  
Mean Velocity ◽  
Fluid Turbulence ◽  
The Mean ◽  
The Cross ◽  
Point Velocity ◽  
Statistics Of Turbulence

The infinite set of the Lundgren-Monin equations for the multi-point velocity distributions of fluid turbulence is closed by making use of the cross-independence closure hypothesis proposed by Tatsumi (Geometry and Statistics of Turbulence, 2001, p. 3), and the minimum deterministic set of equations is obtained as the equations for the one-point velocity distribution f, the two-point velocity distribution f(2) and the two-point local velocity distribution f(2)*. In practice, the two-point distributions f(2) and f(2)* are more conveniently expressed in terms of the velocity-sum and -difference distributions g+, g− and g+*, g−*, respectively.As an outstanding result, the energy dissipation rate is expressed in terms of the distribution g− which is mainly contributed from small-scale turbulent fluctuations, making clear analogy with the ‘fluctuation-dissipation theorem’ in non-equilibrium statistical mechanics.It is to be remarked that the integral moments of the equations for the distributions f and f(2) give the equations for the mean flow and the mean velocity procducts of various orders, which are identical with the corresponding equations derived directly from the Navier--Stokes equation. This results clearly shows the exact consistency of the cross-independence closure and gives an overall solution for the classical closure problem concerning the mean velocity products since they are derived from the known distributions.Although the present work is confined to the two-point statistics of turbulence, the analysis can be extended to the higher-order statistics and even to turbulence in other fluids such as magneto and quantum fluids.

scholarly journals Incompressible Fluids Simulation by Relaxing the Density-Invariant Condition in a Marine Simulator

2019 ◽  
Vol 2019 ◽  
pp. 1-11 ◽  
Author(s):  
Xingfeng Duan ◽  
Hongxiang Ren ◽  
Haijiang Li
Keyword(s):  
Large Time ◽  
Relaxation Method ◽  
Average Density ◽  
Incompressible Fluids ◽  
Small Scale ◽  
Incompressible Sph ◽  
Fluid Particles ◽  
Sph Algorithm ◽  
The Stability ◽  
And Control

To achieve small-scale ocean scene simulation in a marine simulator, we present an incompressible SPH algorithm by relaxing the density-invariant condition for incompressible fluids. As there are larger density errors of the fluid particles near or on the boundary, more iteration numbers are required. Taking boundary handling to modify the density of the fluid particles, the relaxation method is used to optimize the iterative method to solve the density-invariant condition, which can reduce the iteration numbers and average density deviation and improve the accuracy. Our proposed approach can achieve incompressible SPH and solve the problem of the particle deficiency. While ensuring the stability, the approach can allow large time steps and control the density deviation below 0.01% and improve the efficiency by reducing the iteration numbers and optimizing the calculating procedure and the initial value selection.

scholarly journals Fractional Brownian motion, the Matérn process, and stochastic modeling of turbulent dispersion

2017 ◽  
Vol 24 (3) ◽  
pp. 481-514 ◽  
Author(s):  
Jonathan M. Lilly ◽  
Adam M. Sykulski ◽  
Jeffrey J. Early ◽  
Sofia C. Olhede
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Broad Class ◽  
Low Frequency ◽  
Turbulent Dispersion ◽  
Small Scale ◽  
Fluid Turbulence ◽  
Constant Rate ◽  
Initial Location ◽  
Zero Frequency

Abstract. Stochastic processes exhibiting power-law slopes in the frequency domain are frequently well modeled by fractional Brownian motion (fBm), with the spectral slope at high frequencies being associated with the degree of small-scale roughness or fractal dimension. However, a broad class of real-world signals have a high-frequency slope, like fBm, but a plateau in the vicinity of zero frequency. This low-frequency plateau, it is shown, implies that the temporal integral of the process exhibits diffusive behavior, dispersing from its initial location at a constant rate. Such processes are not well modeled by fBm, which has a singularity at zero frequency corresponding to an unbounded rate of dispersion. A more appropriate stochastic model is a much lesser-known random process called the Matérn process, which is shown herein to be a damped version of fractional Brownian motion. This article first provides a thorough introduction to fractional Brownian motion, then examines the details of the Matérn process and its relationship to fBm. An algorithm for the simulation of the Matérn process in O(NlogN) operations is given. Unlike fBm, the Matérn process is found to provide an excellent match to modeling velocities from particle trajectories in an application to two-dimensional fluid turbulence.

scholarly journals Fractional Brownian motion, the Matérn process, and stochastic modeling of turbulent dispersion

2017 ◽  
Author(s):  
Jonathan M. Lilly ◽  
Adam M. Sykulski ◽  
Jeffrey J. Early ◽  
Sofia C. Olhede
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Broad Class ◽  
Low Frequency ◽  
Turbulent Dispersion ◽  
Small Scale ◽  
Fluid Turbulence ◽  
Constant Rate ◽  
Initial Location ◽  
Zero Frequency

Abstract. Stochastic process exhibiting power-law slopes in the frequency domain are frequently well modeled by fractional Brownian motion (fBm). In particular, the spectral slope at high frequencies is associated with the degree of small-scale roughness or fractal dimension. However, a broad class of real-world signals have a high-frequency slope, like fBm, but a plateau in the vicinity of zero frequency. This low-frequency plateau, it is shown, implies that the temporal integral of the process exhibits diffusive behavior, dispersing from its initial location at a constant rate. Such processes are not well modeled by fBm, which has a singularity at zero frequency corresponding to an unbounded rate of dispersion. A more appropriate stochastic model is a much lesser-known random process called the Matérn process, which is shown herein to be a damped version of fractional Brownian motion. This article first provides a thorough introduction to fractional Brownian motion, then examines the details of the Matérn process and its relationship to fBm. An algorithm for the simulation of the Matérn process in O(N log N) operations is given. Unlike fBm, the Matérn process is found to provide an excellent match to modeling velocities from particle trajectories in an application to two-dimensional fluid turbulence.

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