Multifractal Description Research Articles

Multifractal subgrid-scale modeling within a variational multiscale method for large-eddy simulation of turbulent flow

Multifractal subgrid-scale modeling within a variational multiscale method is proposed for large-eddy simulation of turbulent flow. In the multifractal subgrid-scale modeling approach, the subgrid-scale velocity is evaluated from a multifractal description of the subgrid-scale vorticity, which is based on the multifractal scale similarity of gradient fields in turbulent flow. The multifractal subgrid-scale modeling approach is integrated into a variational multiscale formulation, which constitutes a new application of the variational multiscale concept. A focus of this study is on the application of the multifractal subgrid-scale modeling approach to wall-bounded turbulent flow. Therefore, a near-wall limit of the multifractal subgrid-scale modeling approach is derived in this work. The novel computational approach of multifractal subgrid-scale modeling within a variational multiscale formulation is applied to turbulent channel flow at various Reynolds numbers, turbulent flow over a backward-facing step and turbulent flow past a square-section cylinder, which are three of the most important and widely-used benchmark examples for wall-bounded turbulent flow. All results presented in this study confirm a very good performance of the proposed method. Compared to a dynamic Smagorinsky model and a residual-based variational multiscale method, improved results are obtained. Moreover, it is demonstrated that the subgrid-scale energy transfer incorporated by the proposed method very well approximates the expected energy transfer as obtained from appropriately filtered direct numerical simulation data. The computational cost is notably reduced compared to a dynamic Smagorinsky model and only marginally increased compared to a residual-based variational multiscale method.

Multiscale characterization of pore spaces using multifractals analysis of scanning electronic microscopy images of carbonates

Abstract. Pore spaces heterogeneity in carbonates rocks has long been identified as an important factor impacting reservoir productivity. In this paper, we study the heterogeneity of carbonate rocks pore spaces based on the image analysis of scanning electron microscopy (SEM) data acquired at various magnifications. Sixty images of twelve carbonate samples from a reservoir in the Middle East were analyzed. First, pore spaces were extracted from SEM images using a segmentation technique based on watershed algorithm. Pores geometries revealed a multifractal behavior at various magnifications from 800x to 12 000x. In addition, the singularity spectrum provided quantitative values that describe the degree of heterogeneity in the carbonates samples. Moreover, for the majority of the analyzed samples, we found low variations (around 5%) in the multifractal dimensions for magnifications between 1700x and 12 000x. Finally, these results demonstrate that multifractal analysis could be an appropriate tool for characterizing quantitatively the heterogeneity of carbonate pore spaces geometries. However, our findings show that magnification has an impact on multifractal dimensions, revealing the limit of applicability of multifractal descriptions for these natural structures.

Multifractal Analysis for the Teichmüller Flow

We present a multifractal description for Teichmuller flows. A key ingredient to do this is the Rauzy–Veech–Zorich reduction theory, which allows to treat the problem in the setting of suspension flows over subshifts. To perform the multifractal analysis we implement a thermodynamic formalism for suspension flows over countable alphabet subshifts a bit different from that developed by Barreira and Iommi.

Joint multifractal description of the relationship between wind patterns and land surface air temperature

Land surface air temperature (SAT), registered at 1.25–2m above the ground, is influenced by wind patterns. As a consequence, some phenomena such as urban heat islands and the formation of ground-level air pollutants are affected. Detailed understanding of the effects of wind circulations on SAT is convenient to improve the knowledge of these phenomena. Thus, the joint multifractal analysis has been applied to describe time series recorded at Cordoba (southern Spain) from 2001 to 2008 revealing the presence of seasonal patterns related with warm and cold winds blowing from the SW (later spring and summer) and NE (later autumn and winter), respectively, that provoke different heterogeneity in SAT values. In addition, the extreme high SAT values seem to be related with summer SW winds. However, the presence of rare low SAT values produced by NE winds is less relevant.

A universal multifractal description applied to precipitation patterns of the Ebro River Basin, Spain

Water supplies in the Ebro River Basin are increasingly stressed, especially during the summer season. The year-to-year fluctuations in rainfall over this area exert vital influence on the regional hydrology, agriculture and several related industries in the region. Repeated anomalous rainfall in recent decades has had a devastating impact on this region, both socially and economically. We characterised the change in the rainfall variability pattern in the Ebro River Basin using universal multifractal (UM) analysis, which estimates the concentration of the data around the precipitation average (C1, codimension average), the degree of multiscaling behaviour in time (α index) and the maximum probable singularity in the rainfall distribution (γs). Fourteen meteorological stations were selected based on the length of rainfall series and the climatic classification to obtain a representative untreated data set from the river basin. Daily rainfall series from 1957 to 2002 were obtained from each meteorological station and subdivided (1957–1980 and 1981–2002) to analyse the difference between the 2 periods. The general scenario observed in this study, through the UM parameters, can be summarised as follows: the range of variation of precipitation amounts was spatially more homogenous in 1980–2002 than in 1957–1979; at the same time, there is higher frequency of dry periods at different scales in 1980–2002; and in almost all of the stations, the range of precipitation over the years has been decreasing at a lower rate than the rainfall average. We then analysed the evolution of the UM parameters from 1957 to 2002. Continuous variations in C1 and α were found for 2 of the stations, indicating that a precipitation regime change has begun in the last few decades and should be considered in the agricultural development of the region.

СТіЙКіСТЬ МУЛЬТИФРАКТАЛЬНИХ СПЕКТРіВ ВіДНОСНО ЗМіНИ ЯСКРАВОСТі, ПОВОРОТУ і МАСШТАБУ ЗОБРАЖЕННЯ

Verification of multifractal spectra descriptions is conducted in relation to transformation of image: change a brightness, turn in relation to a center and down-scaling. The graphs of firmness are rotined in relation to each of transformations.

Short-term forecasts and scaling of intense events in turbulence

Extreme events such as intense tornadoes and huge floods, though infrequent, are particularly important because of their disproportionate impact. Our ability to forecast them is poor at present. Large events occur also in intermittent features of turbulent flows. Some dynamical understanding of these features is possible because the governing equations are known and can be solved with good accuracy on a computer. Here, we study large-amplitude events of turbulent vorticity using results from direct numerical simulations of isotropic turbulence in conjunction with the vorticity evolution equation. We show that the advection is the dominant process by which an observer fixed to the laboratory frame perceives vorticity evolution on a short time scale and that the growth of squared vorticity during large excursions is quadratic in time when normalized appropriately. This result is not inconsistent with the multifractal description and is simpler for present purposes. Computational data show that the peak in the viscous term of the vorticity equation can act as a precursor for the upcoming peak of vorticity, forming a reasonable basis for forecasts on short time scales that can be estimated simply. This idea can be applied to other intermittent quantities and, possibly, more broadly to forecasting other extreme quantities, e.g. in seismology.

Multifractal modeling of soil microtopography with multiple transects data

Soil complexity and environmental heterogeneity may be viewed as a consequence of deterministic chaotic dynamics and therefore highly irregular patterns with so-called multifractal behavior should be common. This approach introduces a distinct viewpoint as compared with fractal models for soil surface roughness based on fractional Brownian motion. It suggests that it would be useful to move away from the fractal geometry of sets towards the multifractal description of singular probability measures, as well as going beyond second order statistics. The goal of this study was to investigate the multifractal behavior of soil microtopography measured on transects. On rectangular 200 cm × 40 cm plots, point elevation values were obtained and soil microtopography was examined as two-dimensional probability measure. A well-defined multifractal behavior similar to multinomial measures was observed in all cases. The multinomial measures were simulated with a multifractal spectrum close to the spectra of the experimental plots to obtain the synthetic multifractals to evaluate the level of uncertainty in the estimates of the multifractal spectrum of natural roughness as a consequence of the transect rather than grid sampling. We found that the transect separation used to collect the experimental data in this work generates a realistic multifractal spectrum but it cannot precisely define its tails that correspond to asymptotic values of the singularity exponents.

The influence of the geometry of idealised porous media on the simulated flow velocity: A multifractal description

Soil geometry is complex, often multiscale, and flow models should better account for it. Previous works have shown that the flow velocity distribution simulated in idealised porous media with similar geometries exhibits a multifractal type of scaling. The aim of this work is to describe the influence of the porous media geometry on the flow velocity distributions by using multifractal analysis. The fluid velocity is simulated with a lattice model in two idealised 2D porous media with ordered and disordered geometry. The results obtained from the study of the mass exponent functions and the multifractal and Rényi spectra show that, in addition to the porosity, tortuosity is a main factor that has to be taken into account when analysing the flow velocity simulated for geometrically different porous media. The findings reported in this work show the information that can be obtained from a multifractal analysis in the description of the soil geometry effects on the flow velocity field.

Multifractal Description of the Maternal Surface of the Placenta

Background: Recently, multifractal analysis based on generalized concepts of fractals has been applied to biological tissues composed of complex structures. Methods: Using digitized images of the maternal surface of 278 placentas, multifractal parameters were measured with a fractal analysis software. Results: The values of α_min, α(0), α_max and the degree of multifractality given by the α_max – α_min difference calculated from 278 placentas were 1.840 ± 0.068, 2.089 ± 0.034, 2.856 ± 0.128 and 1.017 ± 0.136, respectively. A significant decrease of α_min and as a consequence a significant increase in the degree of multifractality were observed according to gestational age. The α(0) value of the placenta complicated by pregnancy-induced hypertension (PIH) was significantly higher than that without PIH. The α_min and α(0) values of the placenta having intrauterine growth restriction (IUGR) were significantly higher than those without IUGR. On the other hand, the presence of chorioamnionitis did not change multifractal properties of the maternal surface of the placenta. Conclusion: The multifractal parameters may be objective indices of the heterogeneity or complexity of the macroscopic morphology of the maternal surface of the placenta. Multifractal analysis holds a promise for quantitatively evaluating physiological and pathological development of the placenta.

Multifractal analysis of speckle intensities produced by power-law illumination of diffusers

Speckle patterns produced by computer simulations on the assumption of a power-law illumination of scattering objects are investigated on the basis of multifractal analysis. This analysis provides us with more detailed information on a fractal structure than conventional methods yielding a simple dimension. To examine multifractal properties of simulated speckle intensities, their generalized dimensions and singularity spectra are calculated by means of the method of moments. The results show that the multifractal description of the speckles characterizes clustering behaviour of the speckle intensities.

Homology at infinity; fractal geometry of limiting symbols for modular subgroups

In this paper we use fractal geometry to investigate boundary aspects of the first homology group for finite coverings of the modular surface. We obtain a complete description of algebraically invisible parts of this homology group. More precisely, we first show that for any modular subgroup the geodesic forward dynamic on the associated surface admits a canonical symbolic representation by a finitely irreducible shift space. We then use this representation to derive a complete multifractal description of the higher-dimensional level sets arising from the Manin–Marcolli limiting modular symbols.

Information of Structures in Galaxy Distribution

We introduce an information-theoretic measure, the Renyi information, to describe the galaxy distribution in space. We discuss properties of the information measure, and demonstrate its relationship with the probability distribution function and multifractal descriptions. Using the First Look Survey galaxy samples observed by the Infrared Array Camera onboard Spitzer Space Telescope, we present measurements of the Renyi information, as well as the counts-in-cells distribution and multifractal properties of galaxies in mid-infrared wavelengths. Guided by multiplicative cascade simulation based on a binomial model, we verify our measurements, and discuss the spatial selection effects on measuring information of the spatial structures. We derive structure scan functions at scales where selection effects are small for the Spitzer samples. We discuss the results, and the potential of applying the Renyi information to measuring other spatial structures.

Unified multifractal description of velocity increments statistics in turbulence: Intermittency and skewness

The phenomenology of velocity statistics in turbulent flows, up to now, relates to different models dealing with either signed or unsigned longitudinal velocity increments, with either inertial or dissipative fluctuations. In this paper, we are concerned with the complete probability density function (PDF) of signed longitudinal increments at all scales. First, we focus on the symmetric part of the PDFs, taking into account the observed departure from scale invariance induced by dissipation effects. The analysis is then extended to the asymmetric part of the PDFs, with the specific goal to predict the skewness of the velocity derivatives. It opens the route to the complete description of all measurable quantities, for any Reynolds number, and various experimental conditions. This description is based on a single universal parameter function D ( h ) and a universal constant R ∗ .

A multifractal detrended fluctuation description of Iranian rial–US dollar exchange rate

The miltifractal properties and scaling behaviour of the exchange rate variations of the Iranian rial against the US dollar from a daily perspective is numerically investigated. For this purpose the multifractal detrended fluctuation analysis (MF-DFA) is used. Through multifractal analysis, the scaling exponents, generalized Hurst exponents, generalized fractal dimensions and singularity spectrum are derived. Moreover, contribution of two major sources of multifractality, that is, fat-tailed probability distributions and nonlinear temporal correlations are studied.

Multifractal description of fracture morphology: theoretical basis

A theoretical basis is presented of the methodology of a fracture morphology description of a material using a mono- and multifractal analysis of a profile line. Much greater abilities of a multifractal description are shown. A simultaneous multifractal evaluation of a profile is proposed based on a real and simulated image analysis method as an idea leading to the usage of stereometric files of a fracture surface examined with a profilographometer.

Fractal description of rough surfaces.

The multifractal description of rough surfaces is discussed and the mechanisms for generation of fractal and multifractal height distributions of inhomogeneities for rough surfaces are simulated. The original technique for estimating the spectrum of singularities is proposed for the study of these distributions.

REACTION ENGINEERING PRINCIPLES OF PROCESSES CATALYZED BY FRACTAL SOLIDS

Fractals are fascinating objects that can serve as good models for highly porous materials or for materials of high surface areas, and several models of aggregation, leading to fractal objects, resemble the preparation methods of certain classes of catalysts. Evidence of fractal scaling in catalysts has been accumulating for the past two decades. This review focuses on processes that occur on or within fractal porous objects and addresses such issues as the scaling properties of such processes and their superiority over processes in random or nonfractal media. A plethora of reaction engineering problems has been studied. Fractal structures induce unique scaling properties: Diffusion on a fractal media is anomalous and has been extensively investigated; although it may have implications for certain problems of adsorption, surface diffusion, and reaction, and several continuum models for describing it were suggested, it probably does not describe reaction and pore diffusion in a porous pellet. Anomalous temporal or parameter scaling was shown to describe processes of diffusion toward a reactive or adsorbing corrugated fractal surface and, typically, a multifractal description is necessary to characterize the data. Fractional temporal scaling describes the adsorption process from a closed volume. Fractional scaling with respect to the thickness of a stagnant film was derived for instantaneous reaction limited by diffusion through such a film. Fractional scaling with respect to the rate constant (k) was shown to apply for several processes of moderate reaction velocities, such as diffusion toward a diffusion-limited aggregate. Etching of surface and mass fractals also exhibit fractional temporal scaling. Diffusion and reaction in a self-similar pore-fractal network exhibit a new intermediate asymptote, in which the rate is only weakly dependent on k, which lies between the known kinetics-limited (rate ∼ k) and diffusion-limited (k 1/2) asymptotes. Within this domain, the selectivity to an undesired slow side reaction, in a system of two parallel reactions, can be suppressed significantly when its order is higher than that of the main fast reaction. These results may have several technical implications. Fractal surface catalysts may be less or more sensitive to changes in the operating conditions than nonfractal surfaces. Reduced sensitivity to increasing rate constant is achieved, for example, in a corrugated fractal catalyst exposed to a fixed reaction concentration. Comparison of the rates in a pore-fractal catalyst with those in a uniform-pore object showed that the rate in the former is superior in the intermediate k-insensitive domain. Selectivities in a system of two parallel reactions may also be better in the pore fractal as described earlier. Also, the apparent rate of deactivation, when it is a uniform process, can also be suppressed. Future theoretical work should test the validity of these conclusions in stochastic three-dimensional fractals. Experimental verification is still lacking; the superiority of pore fractals for several processes of diffusion and reaction should serve as an incentive for well-designed experiments.

Intermittency of 1D velocity spatial profiles in turbulence: a magnitude cumulant analysis

We perform one- and two-points magnitude cumulant analysis of one-dimensional longitudinal velocity profiles stemming from three different experimental set-ups and covering a broad range of Taylor scaled Reynolds numbers from R λ = 89 to 2500. While the first-order cumulant behavior is found to strongly depend on Reynolds number and experimental conditions, the second-order cumulant and the magnitude connected correlation functions are shown to display respectively universal scale and space-lag behavior. Despite the fact that the Extended Self-Similarity (ESS) hypothesis is not consistent with these findings, when extrapolating our results to the limit of infinite Reynolds number, one confirms the validity of the log-normal multifractal description of the intermittency phenomenon with a well defined intermittency parameter C 2 = 0.025±0.003. But the convergence to zero of the magnitude connected correlation functions casts doubt on the asymptotic existence of an underlying multiplicative cascading spatial structure.

Multifractal returns and hierarchical portfolio theory

We extend and test empirically the multifractal model of asset returns based on a multiplicative cascade of volatilities from large to small time scales. Inspired by an analogy between price dynamics and hydrodynamic turbulence, it models the time scale dependence of the probability distribution of returns in terms of a superposition of Gaussian laws, with a log-normal distribution of the Gaussian variances. This multifractal description of asset fluctuations is generalized into a multivariate framework to account simultaneously for correlations across time scales and between a basket of assets. The reported empirical results show that this extension is pertinent for financial modelling. Two sources of departure from normality are discussed: at large time scales, the distinction between discretely and continuously discounted returns leads to the usual log-normal deviation from normality; at small time scales, the multiplicative cascade process leads to multifractality and strong deviations from normality. By perturbation expansions of the cumulants of the distribution of returns, we are able to quantify precisely the interplay and crossover between these two mechanisms. The second part of the paper applies this theory to portfolio optimization. Our multiscale description allows us to characterize the portfolio return distribution at all time scales simultaneously. The portfolio composition is predicted to change with the investment time horizon (i.e. the time scale) in a way that can be fully determined once an adequate measure of risk is chosen. We discuss the use of the fourth-order cumulant and of utility functions. While the portfolio volatility can be optimized in some cases for all time horizons, the kurtosis and higher normalized cumulants cannot be simultaneously optimized. For a fixed investment horizon, we study in detail the influence of the number of rebalancing of the portfolio. For the large risks quantified by the cumulants of order larger than two, the number of periods has a non-trivial influence, in contrast with Tobin's result valid in the mean-variance framework. This theory provides a fundamental framework for the conflicting optimization involved in the different time horizons and quantifies systematically the trade-offs for an optimal inter-temporal portfolio optimization.