On the use of the Weierstrass-Mandelbrot function to describe the fractal component of turbulent velocity

Humphrey, J. A. C.; Schuler, Carlos; Rubinsky, B.

doi:10.1016/0169-5983(92)90059-6

Cited by 45 publications

(25 citation statements)

References 21 publications

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“…(39) needs to be convergent which yields α = 2 as for the Laplacian (17). To determine β we have to demand that u(x, t) is a Fourier transformable field 5 .…”

Section: The Physical Chain Modelmentioning

confidence: 99%

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Dispersion relations and wave operators in self-similar quasicontinuous linear chains

et al. 2009

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We construct self-similar functions and linear operators to deduce a self-similar variant of the Laplacian operator and of the D'Alembertian wave operator. The exigence of self-similarity as a symmetry property requires the introduction of non-local particle-particle interactions. We derive a self-similar linear wave operator describing the dynamics of a quasi-continuous linear chain of infinite length with a spatially self-similar distribution of nonlocal inter-particle springs. The self-similarity of the nonlocal harmonic particle-particle interactions results in a dispersion relation of the form of a Weierstrass-Mandelbrot function which exhibits self-similar and fractal features. We also derive a continuum approximation which relates the self-similar Laplacian to fractional integrals and yields in the low-frequency regime a power law frequency-dependence of the oscillator density.

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“…(39) needs to be convergent which yields α = 2 as for the Laplacian (17). To determine β we have to demand that u(x, t) is a Fourier transformable field 5 .…”

Section: The Physical Chain Modelmentioning

confidence: 99%

“…In this way we avoid the appearance of a characteristic length scale in our chain model. It seems there are analogue situations in turbulence [17] and other areas where the present interdisciplinary approach could be useful.…”

Section: Introductionmentioning

confidence: 99%

Dispersion relations and wave operators in self-similar quasicontinuous linear chains

et al. 2009

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“…Fractal geometric study of the Weierstrass function (as a mono-fractal) and its generalization has an interesting literature, both in theory and in applications [6,17,18,28]. Although in the present work, we have studied the local fractional derivative, non-local based fractional studies of f are reported [23].…”

Section: If F (α) Is Continuous Then It Does Not Vanish Aementioning

confidence: 98%

Non-differentiability and fractional differentiability on timescales

2017

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This work deals with concepts of non-differentiability and a non-integer order differential on timescales. Through an investigation of a local non-integer order derivative on timescales, a mean value theorem (a fractional analog of the mean value theorem on timescales) is presented. Then, by illustrating a vanishing property of this derivative, its objectivity is discussed. As a first-hand result, the potentials and capability of this fractional derivative connected to nonsmooth analysis, including non-differentiable paths and a class of self-similar fractals, are stated. It is stated that the non-integer order derivative never vanishes almost everywhere. It has been shown that with the help of changing the order of differentiability on a q-timescale, the non-differentiability disappears.

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“…The fractional Brownian motion simulation method can be further classified into the midpoint-displacement method, the Poisson Faulting method and Successive random additions method [28][29][30]. In particular, the midpoint displacement (MD) method and the Weierstrass-Mandelbrot (WM) fractal function method are the most commonly used fractal methods [31][32][33] by researchers. These fractal methods have often shown substantial differences in their form features and statistical properties [34,35].…”

Section: Introductionmentioning

confidence: 99%