Scaling of Ln (Permeability) in Sediments and Velocity Distributions in Turbulence: The Possibility of an Analogy (original) (raw)
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Geophysical Monograph Series, 2000
It is increasingly apparent that sediment property distributions on sufficiently small scales are probably irregular. This has led to the development of stochastic theory in subsurface hydrology, including statistically heterogeneous concepts based mainly on the Gaussian and Lévy-stable probability density functions (PDFs), the mathematical basis for stochastic fractals. Gaussian and Levy-stable stochastic fractals have been applied both in the field of turbulence and subsurface hydrology. However, measurements have shown that the increment frequency distributions do not always follow Gaussian or Lévy-stable PDFs. Provided herein is an overview of the origin and development of a new non-stationary stochastic process, called fractional Laplace motion (flam) with stationary, correlated, increments called fractional Laplace noise (fLan). It is based on the Laplace PDF and known generalizations, and does not display self-similarity. Uncorrelated versions are equivalent to a Brownian motion subordinated to the gamma process. In analogy to the development of fractional Brownian motion (fBm) from Brownian motion, fLam is equivalent to fBm subordinated to a gamma process. The new stochastic fractal has increment PDFs that compare better with measurements, the moments of the PDF family remain bounded, and decay of the increment distribution tails vary from being slower than exponential through exponential and on to a Gaussian decay as the lag size increases. This leads to increasingly more intermittent fluctuations as the lag size decreases. It may be that the geometric central limit theorem, and possible generalizations, will play an important role in connecting the abstract mathematics to the physics underlying applications. 1 1. HISTORICAL OVERVIEW
Moments and probability density functions (PDF) of (absolute value) velocity increments |v(x + )−v(x)| in turbulence are linked by simple integral relations. It is shown that the steepest descent method can be applied to evaluate the integrals if the moments (the absolute value structure functions) obey multifractal scaling laws of the type |v(x + ) − v(x)| n = A n ζ n . A double asymptotic relation then relates the moments to the PDF. The dominant (exponential) terms of the asymptotic relation naturally yield the Legendre transform that is at the core of the Parisi-Frisch model of inertial-range intermittency. Using the asymptotic relation, the PDF can be reconstructed from the multifractal exponent spectrum ζ n and the statistics of large scale moments. On the basis of experimental results, it is shown that moments are quantitatively represented by multifractal scaling laws and large scale Gaussian (or quasi-Gaussian) statistics. The large scale at which the statistics are Gaussian (or quasi-Gaussian) is determined from inertial-range data alone and is of the order of the integral scale for Taylor-scale Reynolds numbers R λ in the range (300-2200). This representation of moments together with the double asymptotic relations is able to reconstruct quantitatively the experimental inertial-range PDF. Analytic expressions (She-Lévêque and Log-normal) of scaling exponents are both shown to lead to reconstructed PDF with systematic deviations from experiment. : S 0 1 6 7 -2 7 8 9 ( 9 8 ) 0 0 3 0 7 -8 94 J. M. Tchéou et al. / Physica D 129 (1999) and n = 3. This prediction was latter extrapolated to arbitrary n. However, the actual situation is known to be more complex than this K41 model. In 1962, Kolmogorov and Obukhov [4,5] introduced the log-normal (K62) model which is defined using a random multiplicative process that was presented as a model of the (random) fluctuating cascade of energy in turbulence. Although the K62 model is known to lead to inconsistency in incompressible turbulence in the limit of high Reynolds numbers , several other models using the concept of a random cascade were developed. An example is the 1964 'black and white' model of Novikov and Stewart [7]. A reformulation of the black and white model using inertial-range quantities, the so-called β-model, was proposed in 1978 by Frisch, Sulem and Nelkin . Independently, at the end of the 1960s, Mandelbrot applied the notion of non-integer Hausdorff dimension (fractal dimension) to the measure of the sets where energy is dissipated . A few years later, several experimental teams measured the exponents ζ n , for high n-vaues . These experiments showed that neither the log-normal nor the β models were adequate. In 1985, in order to interpret the experimental results, Frisch and Parisi introduced the multifractal model that represents the exponents ζ n as the Legendre transform of a function µ(h) = 3 − D(h), interpreted as the Hausdorff co-dimension of a family of sets. Let us recall that the scaling is termed 'unifractal' if the exponent ζ n is an affine function of n and 'multifractal' if it is a non-linear (convex) function of n. Our starting point will be the probabilistic reformulation of the multifractal model by Frisch , which gives for the velocity, a formulation analogous to that called 'Cramér renormalization' by Mandelbrot .
Multifractal statistics of Lagrangian velocity and acceleration in turbulence
2005
The statistical properties of velocity and acceleration fields along the trajectories of fluid particles transported by a fully developed turbulent flow are investigated by means of high resolution direct numerical simulations. We present results for Lagrangian velocity structure functions, the acceleration probability density function and the acceleration variance conditioned on the instantaneous velocity. These are compared with predictions of the multifractal formalism and its merits and limitations are discussed. Understanding the Lagrangian statistics of particles advected by a turbulent velocity field, u(x, t), is important both for its theoretical implications [1] and for applications, such as the development of phenomenological and stochastic models for turbulent mixing . Recently, several authors have attempted to describe Lagrangian statistics such as acceleration by constructing models based on equilibrium statistics (see e.g. , critically reviewed in [6]). In this letter we show how the multifractal formalism offers an alternative approach which is rooted in the phenomenology of turbulence. Here, we derive the Lagrangian statistics from the Eulerian statistics without introducing ad hoc hypotheses.
Applied and Computational Harmonic Analysis, 1999
We use a wavelet-based deconvolution method to extract some multiplicative cascading process from experimental turbulent velocity signals. We show that at the highest accessible Reynolds numbers, the experimental data do not allow us to discriminate between various phenomenological cascade models recently proposed to account for intermittency and their log-normal approximations. We further report evidence that velocity fluctuations are not scale invariant but possess more complex self-similarity properties that are likely to depend on the Reynolds number. We comment on the possible asymptotic validity of the multifractal description. © 1999 Academic Press One of the challenging questions in fully developed turbulence is the possible existence of universal scaling behavior as a result of strong nonlinear interactions [1, 2]. In this respect, a very important issue is the scaling properties of velocity structure functions, S p ͑l ͒ ϭ ͗␦v l p ͘ ϳ l ͑ p͒ , (1)
Multifractality in the statistics of the velocity gradients in turbulence
Physical Review Letters, 1991
Using the multifractal approach, we derive the probability distribution function (PDF) of the velocity gradients in fully developed turbulence. The PDF is given by a nontrivial superposition of stretched exponentials, corresponding to the various singularity exponents. The form of the distribution is explicitly dependent on the Reynolds number. The experimental data are in good agreement with the PDF predicted by the same random beta model used to fit the scaling of the velocity structure functions.
2009
Investigating the multi-point correlation (MPC) equations for the velocity and pressure fluctuations in the limit of homogeneous turbulence a new scaling symmetry has been discovered. Interesting enought this property is not shared with the Euler or Navier-Stokes equations from which the MPC equations have orginally emerged. This was first observed for parallel wall-bounded shear flows in [1]. Presently using this extended set of symmetry groups a much wider class of invariant solutions or turbulent scaling laws is derived for the decay of homogeneous-isotropic turbulence which is in stark contrast to the classical power law decay. In particular, we show that the experimentally observed specific scaling properties of fractal-generated turbulence i.e. a constant integral and Taylor length scale and the exponential decay of the turbulent kinetic energy (see [2, 3]) fall into this new class of solutions. The latter new scaling law may have been the first clear indication towards the existence of the extended statistical scaling group.
Fractal-Markovian Scaling of Turbulent Bursting Process In Open Channel Flow
Chaos, Solitons & …, 2005
The turbulent coherent structure of flow in open channel is a chaotic and stochastic process in nature. The coherence structure of the flow or bursting process consists of a series of eddies with a variety of different length scales and it is very important for the entrainment of sediment particles from the bed. In this study, a fractal-Markovian process is applied to the measured turbulent data in open channel. The turbulent data was measured in an experimental flume using three-dimensional acoustic Doppler velocity meter (ADV). A fractal interpolation function (FIF) algorithm was used to simulate more than 500,000 time series data of measured instantaneous velocity fluctuations and Reynolds shear stress. The fractal interpolation functions (FIF) enables to simulate and construct time series of u 0 , v 0 , and u 0 v 0 for any particular movement and state in the Markov process. The fractal dimension of the bursting events is calculated for 16 particular movements with the transition probability of the events based on 1st order Markov process. It was found that the average fractal dimensions of the streamwise flow velocity (u 0) are; 1.