Measurements of flame-front curvature based on Fourier transformation

An optimized image processing method for calculating the local curvature of two-dimensional digitized flame-front contours is proposed. This contour-smoothing method is based on Fourier transformation of the entire contour and is different from conventional methods in which a chosen number of neighbouring pixel points is locally fitted to an ad hoc smooth function along the contour on a point-by-point basis. The maximum curvature that can be measured with the proposed method is determined by the inner cut-off scale from a fractal length measure of the flame-front contour and is limited by image spatial resolution only. Curvature measurements are also made in highly stretched turbulent premixed methane–air jet flames. The measured probability density functions of local flame curvature are consistent with the generalized form of a symmetric Gaussian distribution suggested by Bradley et al. (Combustion and Flame, 135, 503–523 (2003)). Advantages and limitations of the proposed method are also discussed.     

Fractional motions

Brownian motion is the archetypal model for random transport processes in science and engineering. Brownian motion displays neither wild fluctuations (the “Noah effect”), nor long-range correlations (the “Joseph effect”). The quintessential model for processes displaying the Noah effect is Lévy motion, the quintessential model for processes displaying the Joseph effect is fractional Brownian motion, and the prototypical model for processes displaying both the Noah and Joseph effects is fractional Lévy motion. In this paper we review these four random-motion models–henceforth termed “fractional motions” –via a unified physical setting that is based on Langevin’s equation, the Einstein–Smoluchowski paradigm, and stochastic scaling limits. The unified setting explains the universal macroscopic emergence of fractional motions, and predicts–according to microscopic-level details–which of the four fractional motions will emerge on the macroscopic level. The statistical properties of fractional motions are classified and parametrized by two exponents—a “Noah exponent” governing their fluctuations, and a “Joseph exponent” governing their dispersions and correlations. This self-contained review provides a concise and cohesive introduction to fractional motions.     

A fractal interpretation of size effects on the strength of metallic glasses

Yielding strength of metallic glasses in the uniaxial tensile and compressive tests is scale-dependent, which is attributed to the self-similar distribution of atomic cluster and free volume in the work. In contrast with the Weibull statistical theory previously employed in scaling phenomena of metallic glasses, fractal scaling laws are for the first time applied to describe the size effect inherent to the material disorder. Especially, the Multifractal Scaling Law (MFSL) originally proposed for quasi-brittle materials is used to interpret some experimental data in the literature. The best-fitted parameters (_fy$f_{\mathrm{y}}$ and l_ch) from the MFSL are in good consistency with the bulk yielding strength and the shear band size of metallic glasses observed in the alternative approaches or experiments. The fractal size effect laws provide insight into not only the scaling phenomena, but also further engineering strength predictions and designs.     

Spatial and Spatiotemporal Karhunen-Loève-Type Representations on Fractal Domains

Abstract

We study the spectral properties of spatial and spatiotemporal Gaussian random fields defined as the solutions to stochastic elliptic, parabolic, and hyperbolic fractional pseudodifferential equations on compact fractal domains. The fractal dimension of the domain modifies the asymptotic properties of the eigenvalues that define the pure point spectra of the covariance functions of the solutions and their Karhunen-Loève-type expansions. The eigenfunction systems involved constitute orthogonal bases of the corresponding trace spaces on fractal sets. The Hölder exponent of the sample paths of the random fields is computed in terms of the fractional order of mean-quadratic variation on their increments. Such an exponent also depends on the Hausdorff dimension of the domain.     

Exploration of fractal nature of WO3 nanowire aggregates

The morphology of WO₃ aggregates formed by irregular nanoparticles (D∼40 nm) and nanowires of different aspect ratios (2, 4, 6, and 10 μm nominal lengths) dispersed in commonly used polar solvents without dispersant agents is investigated using a small-angle light scattering technique and by means of fractal theory. Nanoparticles form compact spherical aggregates (D_f∼2.6), whereas 2 μm nanowires with low aspect ratio (L/D∼10) follow a slow cluster-cluster aggregation mechanism with no discernable change in fractal dimension (D_f=2.1) monitored in an extended period of 6 months, despite a notable growth in size (R_g=2.3-3.1 μm). For higher aspect ratio nanowires, scattered intensity profiles, which migrate towards the Porod regime, qualitatively obey the Lorenz-Mie theory predictions. The 10 μm nanowires with very high aspect ratio (L/D∼250) are observed to form stable dispersions in a time span of 6 days. Analytical methods based on spherical primary particle formulations predict D_f=1.9, 1.7, and 1.4 for 4, 6, and 10 μm nanowires, respectively.     

A study of radiative properties of fractal soot aggregates using the superposition T-matrix method

We employ the numerically exact superposition T-matrix method to perform extensive computations of scattering and absorption properties of soot aggregates with varying state of compactness and size. The fractal dimension, D_f, is used to quantify the geometrical mass dispersion of the clusters. The optical properties of soot aggregates for a given fractal dimension are complex functions of the refractive index of the material m, the number of monomers N_S, and the monomer radius a. It is shown that for smaller values of a, the absorption cross section tends to be relatively constant when D_f<2 but increases rapidly when D_f>2. However, a systematic reduction in light absorption with D_f is observed for clusters with sufficiently large N_S, m, and a. The scattering cross section and single-scattering albedo increase monotonically as fractals evolve from chain-like to more densely packed morphologies, which is a strong manifestation of the increasing importance of scattering interaction among spherules. Overall, the results for soot fractals differ profoundly from those calculated for the respective volume-equivalent soot spheres as well as for the respective external mixtures of soot monomers under the assumption that there are no electromagnetic interactions between the monomers. The climate-research implications of our results are discussed.     

Extended calculation of polarization and intensity of fractal aggregates based on rigorous method for light scattering simulations with numerical orientation averaging

The intensity and polarization of fractal aggregates have been investigated using both rigorous and approximate methods for light scattering simulations. However, previous studies using the analytical orientation averaging version of the rigorous method were generally limited to a few hundred monomers when the monomer size parameter was around 1.7. In this study, we propose using numerical orientation averaging instead of analytical orientation averaging. The numerical averaging is performed together with a fixed orientation version of the rigorous T-matrix method for clusters of spheres. This approach enables increasing the number of monomers by a factor of 2-7 or the size of monomers by a factor of 8-10 compared to the analytical orientation averaging version.We investigated the influence of monomer size and the number of monomers on the light scattering of silicate aggregates (refractive index m=1.68+0.03i) for incident light with a wavelength of . We considered ballistic particle-cluster aggregates (BPCA) and ballistic cluster-cluster aggregates (BCCA) composed of 128, 256, 512, and 1024 monomers with radii between 0.11 and .Our results show that the size of monomers plays an important role in reproducing the negative polarization branch for all the BPCA and BCCA. Silicate aggregates with the monomer radius of less than contribute to reproducing the negative polarization branch, while aggregates with monomers larger than do not have the negative polarization branch. Polarization oscillation with scattering angle occurs for larger monomers (i.e., monomer radius ).The maximum polarization decreases for increasing monomer radius between 0.11 and . However, the negative polarization branch is generally enhanced for monomer radii up to around , and reduced for further increase of monomer size.The number of monomers also has a large influence on the negative polarization branch in the case of BPCA. The increase in the number of monomers from 128 to 1024 shifts the scattering angle of minimum polarization to larger angles for BPCA. In addition, the increase in the number of monomers reduces the values of negative polarization for BPCA while the variation with the number of monomers for BCCA is small and is not monotonic.     

Scaling reduction of the contrast of fractal speckles detected with a finite aperture

It is known that speckle patterns with fractal properties, called fractal speckles, are produced by illuminating a diffuser with the coherent light having the intensity distribution obeying a negative power law. One of key properties of fractal speckles is the spatial correlation function obeying a negative power law, which implies that such speckles have scaling properties. In detecting fractal speckles, the effect of the spatial integration is inevitable in most cases since they have speckle grains of various scales including very fine ones. To evaluate this effect, in this paper, the contrast of spatially integrated intensity distributions is investigated theoretically and experimentally for fractal speckles. The results show that the contrast reduction with the size of the detector aperture obeys a negative power function related with the exponent of the intensity correlation coefficient of fractal speckles.     

一种抗噪分形维计算方法及其在声纳图像识别中的应用

用于计算图像分形维的差分盒计数法(DBC)和鲁棒差分盒计数法(RDBC)都对脉冲噪声和斑点噪声较敏感,为此本文提出一种抗噪差分盒计数法(NRDBC),利用剪切局部标准差(TLSD)来计算图像分形维,由于TLSD可有效滤除脉冲噪声和斑点噪声、且对高斯噪声敏感性小,因此NRDBC能对有噪图像进行可靠的分形维估计。利用多分辨率的DBC、RDBC和NRDBC对混有高斯噪声、脉冲噪声和斑点噪声的7种Brodatz纹理以及3种海底的侧扫声纳图像进行了分类实验,结果表明,本文提出的NRDBC可获得更高的识别率和更好的抗噪性。     

Fractal dimension and phase transition of graphene oxide (GO) doped polyacrylamide

Graphene Oxide (GO)- Polyacrylamide composites prepared between 5 and 50 μl GO were performed by Fluorescence Spectroscopy. The phase transition performed on the composites was measured by calculating the critical exponents, β and γ, respectively. In addition, fractal analysis of the composites was calculated by a fluorescence intensity of 427 nm. The geometrical distribution of GO in the composites was calculated based on the power law exponent values using scaling models. While the gelation proceeded GO plates first organized themselves into a 3D percolation cluster with the fractal dimension (D_f) of the composite, D_f = 2.63, then After it goes to diffusion limited clusters with D_f = 1.4, its dimension lines up to a Von Koch curve with a random interval of D_f = 1.14.