A shell-model approach to fractal-induced turbulence

We present a shell-model of fractal induced turbulence which predicts that structure function scaling exponents decrease in absolute value as the fractal dimension of the turbulence-inducing fractal object increases. This qualitative prediction is in agreement with laboratory measurements. Finer details of the fractal induced turbulence statistics and dynamics depend on the fractal force's phases, i.e. on the detailed construction of the fractal stirrer. In a case of deterministic forcing phases, a critical fractal dimension exists below which the average rate of inter-scale energy transfer <T _n> is a decreasing function of the wavenumber k_n and the structure function scaling exponents take close to Kolmogorov values. Above this critical fractal dimension, <T _n> is an increasing function of k_n and the structure function scaling exponents deviate significantly from Kolmogorov values. Received 25 June 2001 / Received in final form 5 April 2002 Published online 19 July 2002     

Comparison of fractal analysis methods in the study of fractals with independent branching

The notion of dimension as a quantitative characteristic of space geometry is discussed. It is supposed that hadrons created in interactions between particles and nuclei can be considered sets of points possessing fractal properties in the three-dimensional phase space (p _T , η, ?). The Hausdorff-Besicovich dimension D _F is considered the most natural characteristic for determining the fractal dimension. Different methods for determining the fractal dimension are compared: box counting (BC), P-adic coverage (PaC), and system of equations of P-adic coverage (SePaC). A procedure for choosing optimum values of parameters of the considered methods is presented. These parameters are shown to be able to reconstruct the fractal dimension D _F , number of levels N _lev, and fractal structure with maximal efficiency. The features of the PaC- and SePaC-methods in the analysis of fractals with independent branching are noted.     

Monte Carlo simulation to study surface diffusion and reaction processes on a fractal catalyst

The surface of a catalyst is a fractal, and its fractal dimension can strongly influence the performance of a catalytic reaction. By means of the Monte Carlo method, the decompositions of N₂O on a square lattice and a fractal surface are investigated in this paper. From the simulation it is found that the fractal surface can change both the reaction rate constant K and the reaction order α in the N₂O decomposition, and all of these changes are caused by the abnormal diffusion of the surface species on the fractal surface.     

Dynamic cracking resistance of metallic materials during rapid propagation of a self-similar crack

A model is proposed to determine the dynamic cracking resistance K _ID of metals and alloys for the case of a rapidly moving fractal or self-affine crack. The values of this characteristic correlate with the fractal dimension D _f of the future contour of a crack surface profile. K _ID is lower or higher than K _IC depending on the fractal dimension.     

Formation of different surface reliefs of metallic glasses under mechanical stress

The micro- and nanoreliefs of loaded lateral surfaces and fracture surfaces of foils of the Fe₇₇Ni₁Si₉B₁₃, Fe₅₈Ni₂₀Si₉B₁₃, and Fe₇₀Cr₁₅B₁₅ amorphous alloys have been investigated using scanning tunneling and atomic force microscopy. The isotropic and anisotropic surface reliefs have been examined. The fractal dimensions of the surfaces of loaded specimens and the fracture surfaces along and across the direction of crack propagation have been estimated using the box counting method. Fractal characteristics of the surfaces, such as the Hölder exponent and the half-width of the singularity spectrum, have been calculated using the wavelet transform method. It has been found that, on the topographies with a clearly pronounced anisotropy of the relief, the surface is fractal in only one direction, and the surface is fractal in two directions on the topographies with a less pronounced anisotropy of the relief. The fractal characteristics of the lateral surfaces and the fracture surfaces with allowance made for their anisotropy have close values. It has been shown that the formation of two types of fracture surfaces is adequately described in terms of the model of a cellular automaton.     

Analysis of fractals with dependent branching by box counting, P-adic coverages, and systems of equations of P-adic coverages

Concept of the dimension of space-time in the general relativity theory and quantum theory is discussed. It is emphasized that the dimension of a discrete space can be defined based on the Hausdorff measure. The noninteger dimension is a typical characteristic of a fractal. The process of hadron formation in interactions between high-energy particles and nuclei is supposed to possess fractal properties. The following methods for analyzing fractals are considered: box counting (BC), method of P-adic coverages (PaC), and method of systems of equations of P-adic coverages (SePaC), for determining the fractal dimension. A comparative analysis of fractals with dependent branching is performed using these methods. We determine the optimum values of parameters permitting one to determine the fractal dimension D _F , number of levels N _lev, and the fractal structure with maximal efficiency. It is noted that the SePaC method has advantages in analyzing fractals with dependent branching.     

Fractal model of the atom and some properties of the matter through an extended model of scale relativity

The scale relativity model was extended for the motions on fractal curves of fractal dimension D _F and third order terms in the equation of motion of a complex speed field. It results that, in a fractal fluid, the convection, dissipation and dispersion are compensating at any scale (differentiable or non-differentiable), whereas a generalized Schrödinger type equation is obtained for an irrotational movement of the fractal fluid. For D _F = 2 and the dissipative approximation of the motions, the fractal model of atom is build: the real part of the complex speed field describes the electron motion on stationary orbits according to a quantification condition, while the imaginary part of the complex speed field gives the electron energy quantification. For D _F = 3 and the dispersive approximation of motions, some properties of the matter are explained: at the differentiable scale the flowing regimes (non-quasi-autonomous and quasi-autonomous) of the fractal fluids are separated by the experimental “0.7 structure”, while for the non-differentiable scale the fractal potential acts as an energy accumulator and controls through coherence the transport phenomena. Moreover, the compatibility between the differentiable and non-differentiable scales implies a Cantor space-time, and consequently a fractal at any scale. Thus, some properties of the matter (the anomaly of nano-fluids thermal conductivity, the superconductivity etc.) can be explained by this model.     

Application of fractal concepts to polymer statistics and to anomalous transport in randomly porous media

We consider the application of fractal concepts to polymer statistics and to anomalous transport in randomly porous media. It is found that answers to interesting physics questions can be expressed in terms of several new fractal dimensions (in addition to the fractal dimensiond _f ): (1)d _f ^BB , the fractal dimension of the backbone, arises in connection with electric current flow, (2)d _red, the fractal dimension of the singly connected bonds in the backbone, arises in connection with its equivalence to the thermal scaling power, (3)d _E, the fractal dimension of the of the elastic backbone, (4)d _u, the fractal dimension of the unscreened perimeter, arises in connection with the viscosity singularity at the gelation threshold, (5)d _min the fractal dimension of the minimum path (or chemical distance) between two sites, arises in co-nnection with the Aharony-Stauffer conjecture, (6)dw, the fractal dimension of a random walk, (7)d _G, the fractal dimension of growth sites that arise as a random walk creates a cluster. Relations among these fractal dimensions are discussed, some of which can be proved and others of which are conjectures whose validity has been established only in certain limiting cases.Supported in part at the Center for Polymer Studies by grants from ONR and NSF.     

Some fractal properties of the percolating backbone in two dimensions

A new algorithm is presented, based on elements of artificial intelligence theory, to determine the fractal properties of the backbone of the incipient infinite cluster. It is found that the fractal dimensionality of the backbone isd _f ^BB =1.61±0.01, the chemical dimensionality isd _t=1.40±0.01, and the fractal dimension of the minimum pathd _min=1.15 ± 0.02 for the two-dimensional triangular lattice.     

基于分形多孔介质三维网络模型的非混溶两相流驱替数值模拟

提出了一个描述多孔介质孔隙尺寸分布的三维分形网络模型,利用该模型对多孔介质中的非混溶两相流驱替进行了数值模拟,研究了孔隙尺寸分布分维D_f和两相流黏滞比M对驱替前沿指进型的影响,结果表明指进型容量维数D_h随着孔隙尺寸分布分维D_f以及黏滞比M的增大而减少,并通过曲线拟合得到了它们之间的定量关系. 关键词： 多孔介质 三维网络 黏滞指进 非混溶两相流     

沉积在α-Al２Ｏ3分形基底上Ag薄膜的特性

报道了采用磁控溅射法在α－Ａｌ_２Ｏ₃分形基底上沉积Ａｇ薄膜表面的形貌、结晶状态以及其Ｖ－Ｉ特性．结果表明：分形的Ａｌ_２Ｏ₃基底导致Ａｇ薄膜具有起伏不平的结构、较差的结晶状态并且存在大量的孔洞，它们同样受基底温度和薄膜厚度的影响．在一定的厚度范围内，Ａｇ薄膜呈现反常的非线性Ｉ（Ｖ）特性，其行为也受薄膜厚度、基底温度和测试环境的强烈影响． 关键词：     

Measurement of Aggregate Fractal Dimensions Using Static Light Scattering

Aggregates formed from colloidal particles will vary in shape according to the aggregation regime prevalent. Compact structures are formed when the aggregation is slow, whilst loose tenuous structures are formed when rapid (or diffusion limited) aggregation prevails. These structures can be fractal in nature, that is, there is a relationship between porosity and the number of primary particles making up the aggregate, and is described by the fractal dimension, d_F. Fractal dimensions of hematite aggregates have been measured experimentally using the static light scattering technique. Fractal dimensions varied with aggregation regimes; for the rapid aggregation regime, d_F was found to be 2.8, whilst for conditions in which aggregation was slow (retardation forces prevail), d_F's of 2.3 were measured. For conditions which lead to aggregation in which both diffusion and retardation forces play a part, structures with fractal dimensions such that 2.3 < d_F < 2.8 were found. The effects of adsorbed fulvic acid, a naturally occuring organic acid, on the kinetics of hematite aggregation and on the resulting structure of hematite aggregates were also investigated. The study of aggregate structure shows that the fractal dimensions of hematite aggregates which are partially coated with fulvic acid molecules are higher than those obtained with no adsorbed fulvic acid. The scattering exponents obtained from static light scattering experiments of these aggregates range from 2.83 ± 0.08 to 3.42 ± 0.1. The scattering exponents of greater than 3 indicate that the scattering is the result of objects that contains pores which are bounded by surfaces with a fractal structure, and can be related only to surface fractal dimension. The high fractal dimensions are due to restructuring within the aggregates, which only occured at low coverage by the organic acid.     

Effect of uniaxial tension and hydrostatic compression on the geometry and morphology of amorphous Fe<Subscript>77</Subscript>Ni<Subscript>1</Subscript>Si<Subscript>9</Subscript>B<Subscript>13</Subscript> alloy ribbon surface

The effect of hydrostatic pressure and uniaxial compression on the relief of an amorphous Fe₇₇Ni₁Si₉B₁₃ alloy ribbon surface was studied using scanning tunneling and atomic-force microscopy. The fracture surfaces of samples were also studied. It is found that both the initial surfaces and the surfaces of samples subjected to hydrostatic compression or tension, as well as fracture surfaces, are fractal or multifractal, but their fractality parameters are different. Hydrostatic pressure decreases the surface roughness and the average fractal dimension of the surface on both sides of the ribbons. The dependence of the surface fractal characteristics on tension is more complex. Prior to the occurrence of a “critical event” on the surface (formation of a deformation band or a through crack), the Hölder index and the half-width of the singularity spectrum decrease. The correlation is discussed between the fractal characteristics of the ribbon surface and those of a fracture surface, and the role of an excess free volume in the initiation of fracture of amorphous alloys is analyzed.     

An analysis of the radial flow in the heterogeneous porous media based on fractal and constructal tree networks

In this paper, an analysis of the radial flow in the heterogeneous porous media based on fractal and constructal tree networks is presented. A dual-domain model is applied to simulate the heterogeneous porous media embedded with a constructal tree network based on the fractal distribution of pore space and tortuosity nature of flow paths. The analytical expressions for seepage velocity, pressure drop, local and global permeability of the network and binary system are derived, and the transport properties for the optimal branching structure are discussed. Notable is that the global permeability (K_n) of the network and the volume fraction (f_n) occupied by the network exhibit linear scaling law with the fractal dimension (D_p) of channel diameter bylogK_n∼0.46D_p and logf_n∼1.03D_p, respectively. Our analytical results are in good agreement with the available numerical results for steady-state soil vapor extraction and indicate that the fractal dimension for pore space has significant effect on the permeable properties of the media. The proposed dual-domain model may capture the characteristics of heterogeneous porous media and help understanding the transport mechanisms of the radial flow in the media.     

Special electronic structures of inverse spinels LiMVO<Subscript>4</Subscript>(M = Ni and Cu): a first-principles study

We use the Ulam method to study spectral properties of the Perron-Frobenius operators of dynamical maps in a chaotic regime. For maps with absorption we show numerically that the spectrum is characterized by the fractal Weyl law recently established for nonunitary operators describing poles of quantum chaotic scattering with the Weyl exponent ν = d-1, where d is the fractal dimension of corresponding strange set of trajectories nonescaping in future times. In contrast, for dissipative maps we numerically find the Weyl exponent ν = d/2 where d is the fractal dimension of strange attractor. The Weyl exponent can be also expressed via the relation ν = d₀/2 where d₀ is the fractal dimension of the invariant sets. We also discuss the properties of eigenvalues and eigenvectors of such operators characterized by the fractal Weyl law.     

火灾烟颗粒分形模型和球形模型光散射的比较研究

对烟颗粒的光散射进行模拟计算是研究火灾烟颗粒光散射特性的重要手段,目前对于火灾烟颗粒光散射的数值计算多采用球形或椭球模型.实际上,火灾烟颗粒的形貌与球形和椭球均存在着显著差异.扫描电子显微镜图像表明,烟颗粒具有近似分形的结构.本文利用离散偶极近似方法计算了随机取向的火灾烟颗粒分形凝团以及同体积的球形颗粒的光散射Muller矩阵,并对两者的归一化Muller矩阵元素随散射角的分布进行了比较.研究表明:火灾烟颗粒分形模型和球形模型的归一化矩阵元素F₁₁(θ)/     

Event structure investigation of AuAu interactions from HIJING model using fractal dimensions

This work presents an implementation of fractal geometry methods in the study of event structure for AuAu interactions at collision energies √s _NN = 9.2, 62 and 200 GeV for different interaction dynamics. The events are generated by using the HIJING model. It is shown, that the fractal dimension of events in phase space projections rapidity-transverse momentum (y - p_t) and azimuthal angle-transverse momentum (φ - p_t) are sensitive to the interaction dynamics.     

Fractal growth at dendrite tips in an ion-irradiated thin solid film

Summary Experimental observations of fractal growth at dendrite tips during crystalline-to-amorphous phase transition in an ion-irradiated Mo₆₅Ni₃₅ thin film are reported. It was found that the observed anisotropic fractal patterns were of a DLA type but not exactly self-similar. The possible mechanism of this exception to the ordinary parabolic tip growth of the dendrites is discussed.     

超薄膜多中心生长过程的计算机模拟

利用计算机模拟了不同的允许扩散步数下超薄膜的多中心分形生长和团状生长现象,研究了成核及长大的动力学过程．分形生长时分形维数随团簇大小的增大而增加．分形生长和团状生长时成核率随扩散步数的增大而减小,随时间的增大而急速下降．团簇长大过程可用团簇大小S和生长时间t-t₀的幂函数(t-t₀)^κ描述．由于团簇间的分流作用,生长指数κ比经典理论值1略小,并且存在着非线性现象,即长得较大的团簇的生长指数Κ也较大. 关键词：     

Synthesis, structure, and dielectric properties of KH2PO4 fractal aggregates

Macroscopic fractal aggregates of KH₂PH₄ (KDP) measuring up to 500 μm have been obtained. The fractal structure forms as a result of the precipitation of KDP particles from a supersaturated aqueous solution in the presence of a temperature gradient followed by a diffusioncontrolled mechanism of aggregation. The electron-microscopic analysis performed has shown that the fractals are formed predominantly from crystallites of the tetragonal modification measuring ∼1 μm. The dielectric constant (ɛ) of fractal KH₂PO₄ has been measured in the temperature range 80–300 K. A characteristic anomaly has been discovered on the ɛ(T) curve in the vicinity of 122 K, which attests to a ferroelectric phase transition. The absolute value of ɛ is significantly smaller than the components ɛ ₁₁ and ɛ ₃₃ for KH₂PO₄. Fiz. Tverd. Tela (St. Petersburg) 41, 2059–2061 (November 1999)