Self-similar solutions of the problem of formation of a hydraulic fracture in a porous medium

The problem of hydraulic fracture formation in a porous medium is investigated in the approximation of small fracture opening and inertialess incompressible Newtonian fluid fracture flow when the seepage through the fracture walls into the surrounding reservoir is asymptotically small or large. It is shown that the system of equations describing the propagation of the fracture has self-similar solutions of power-law or exponential form only. A family of self-similar solutions is constructed in order to determine the evolution of the fracture width and length, the fluid velocity in the fracture, and the length of fluid penetration into the porous medium when either the fluid flow rate or the pressure as a power-law or exponential function of time is specified at the fracture entrance. In the case of finite fluid penetration into the soil the system of equations has only a power-law self-similar solution, for example, when the fluid flow rate is specified at the fracture entrance as a quadratic function of time. The solutions of the self-similar equations are found numerically for one of the seepage regimes.     

壁面加热边界条件对湍流扩展的自相似性的影响

用热线风速仪测量了风洞中壁面加热和不加热平板湍流边界层不同法向位置的瞬时流向速度的时间序列,研究了壁面加热的边界条件对流向速度增量的p阶结构函数的扩展的自相似性和层次结构模型的影响。实验结果表明,壁面加热的边界条件对流向速度增量的结构函数扩展的自相似标度律的相对标度指数δ（p,3）和层次结构模型中间歇参数β、最奇异指数γ的影响明显存在。     

湍流火焰自相似临界微尺度分析

自相似临界尺度表征了湍流火焰内保持自相似特性的最大尺度，是确定湍流各相同性的特征尺度．本文利用光学显微技术和计算机图像处理技术，提取并分析了湍流火焰微尺度结构的曲线度和线性回归均方差在不同放大倍率下的变化规律，提出了利用相似度偏差和临界放大倍率来确定自相似临界尺度的思路，并借助于Ｋｏｌｍｏｇｏｒｏｖ微尺度η的定义，推导出临界尺度η＇的数学关联式．     

Log-Poisson statistics and extended self-similarity in driven dissipative systems

The Bak–Chen–Tang forest fire model [Phys. Lett. A 147 (1990) 297] was proposed as a toy model of turbulent systems, where energy (in the form of trees) is injected uniformly and globally, but is dissipated (burns) locally. We review our previous results on the model [Phys. Rev. E 62 (2000) 1613; Phys. Rev. Lett. 86 (2000) 4215] and present our new results on the statistics of the higher-order moments for the spatial distribution of fires. We show numerically that the spatial distribution of dissipation can be described by Log-Poisson statistics which leads to extended self-similarity [Phys. Rev. E. 48 (1993) R29; Phys. Rev. Lett. 73 (1994) 959]. Similar behavior is also found in models based on directed percolation; this suggests that the concept of Log-Poisson statistics of (appropriately normalized) variables can be used to describe scaling not only in turbulence but also in a wide range of driven dissipative systems.     

On the existence of conically self-similar free-vortex solutions to the Navier-Stokes equations

In this paper a proof of existence and non-existence of theconically self-similar free-vortex solutions to the Navier-Stokesequations, originally found by Yih et al. (1982, Phys. Fluids.25, 2147-2158), is presented. This proof clearly establishesthat these solutions do not have any kind of singularity atthe symmetry axis. This analysis gives considerably improvedexistence and non-existence bounds and it is shown that thesebounds are close to optimal in the low-swirling limit. Thisapproach links the questions of existence and non-existencefor the swirling case and for the non-swirling case. The proof,which is an extension of techniques developed by Serrin (1972,Phil. Trans. R. Soc. Lond. 271, 325-360), is based on Schauder'sFixed Point Theorem and is, therefore, non-constructive. Therefore,the paper ends with a brief discussion of the question of howto compute the conically self-similar free-vortex solutionsto the Navier-Stokes equations.     

Exact Soliton Solutions to a Generalized Nonlinear Schrodinger Equation

The （1＋1）-dimensional F-expansion technique and the homogeneous nonlinear balance principle have been generalized and applied for solving exact solutions to a general （3＋1）-dimensional nonlinear Schr6dinger equation （NLSE） with varying coefficients and a harmonica potential. We found that there exist two kinds of soliton solutions. The evolution features of exact solutions have been numerically studied. The （3＋1）D soliton solutions may help us to understand the nonlinear wave propagation in the nonlinear media such as classical optical waves and the matter waves of the Bose-Einstein condensates.     

Diffraction Fields of Fractally Bounded Apertures

Experimental study is performed of optical diffractions by two-dimensional, self-similar mass and surface fractal apertures as well as mass fractal apertures bounded by surface fractals. Self-similar intensity distributions are observed in the Fraunhofer diffraction fields from surface and mass fractal apertures. Power law decays in average intensities of the diffraction patterns are also investigated for the mass, surface and combined fractal apertures in connection with the fractal dimensions of the objects.     

Macroscopic models for long-range dependent network traffic

A common way to inject long-range dependence in a stochastic traffic model possessing a weak regenerative structure is to make the variance of the underlying period infinite (while keeping the mean finite). This method is supported both by physical reasoning and by experimental evidence. We exhibit the long-range dependence of such a process and, by studying its second-order properties, we asymptotically match its correlation structure to that of a fractional Brownian motion. By studying a certain distributional limit theorem associated with such a process, we explain the emergence of an extremely skewed stable Lévy motion as a macroscopic model for the aforementioned traffic. Surprisingly, long-range dependence vanishes in the limit, being “replaced” by independent increments and highly varying marginals. The marginal distribution is computed and is shown to match the one empirically obtained in practice. Results on performance of queueing systems with Lévy inputs of the aforementioned type are also reported in this paper: they are shown to be in agreement with pre-limiting models, without violating experimental queueing analysis. This revised version was published online in June 2006 with corrections to the Cover Date.     

Self-Similar Parabolic Optical Solitary Waves

We study solutions of the nonlinear Schrödinger equation (NLSE) with gain, describing optical pulse propagation in an amplifying medium. We construct a semiclassical self-similar solution with a parabolic temporal variation that corresponds to the energy-containing core of the asymptotically propagating pulse in the amplifying medium. We match the self-similar core through Painlevé functions to the solution of the linearized equation that corresponds to the low-amplitude tails of the pulse. The analytic solution accurately reproduces the numerically calculated solution of the NLSE.     

Twenty-species and 15-species chemical kinetic mechanisms for cyclohexane using the local self-similarity tabulation method

The 1081 species cyclohexane-oxidation elementary reaction mechanism of Silke et al. (DOI: 10.1021/jp067592d ) is reduced in the number of species by a factor using the local self-similarity tabulation (LS2T) method. Reduced-species mechanisms of both 20 (R20) and 15 (R15) species are created in the high-pressure combustion regime typical of diesel engines. To evaluate the performance of R20 and R15 against the elementary kinetics, simulations are performed for cyclohexane/air mixtures at initial temperatures of 1150, 900, 750, and 680 K and constant pressures of 20 and 40 bar for a variety of equivalence ratios (, 1.0, and 2 for 1150 and 900 K; for 750 K; for 680 K). Very good agreement between R20 and R15 with the elementary kinetics mechanism is demonstrated at 1150 and 900 K for which the self-similarity is very well obeyed; however, only fair agreement is obtained at 750 and 680 K, a fact which is traced to the less faithful adherence to the self-similarity due to the one order of magnitude increase in ignition time over the range 750-680 K. These results are found to be quasi-independent of the tabulation grid. Future work is proposed to improve the reduction in the cold-ignition, high-pressure regime.