Suspended load and bed-load transport of particle-laden gravity currents: the role of particle–bed interaction

The development of particle-enriched regions (bed-load) at the base of particle-laden gravity currents has been widely observed, yet the controls and relative partitioning of material into the bed-load is poorly understood. We examine particle-laden gravity currents whose initial mixture (particle and fluid) density is greater than the ambient fluid, but whose interstitial fluid density is less than the ambient fluid (such as occurs in pyroclastic flows produced during volcanic eruptions or when sediment-enriched river discharge enters the ocean, generating hyperpycnal turbidity currents). A multifluid numerical approach is employed to assess suspended load and bed-load transport in particle-laden gravity currents under varying boundary conditions. Particle-laden flows that traverse denser fluid (such as pyroclastic flows crossing water) have leaky boundaries that provide the conceptual framework to study suspended load in isolation from bed-load transport. We develop leaky and saltation boundary conditions to study the influence of flow substrate on the development of bed-load. Flows with saltating boundaries develop particle–enriched basal layers (bed-load) where momentum transfer is primarily a result of particle–particle collisions. The grain size distribution is more homogeneous in the bed-load and the saltation boundaries increase the run-out distance and residence time of particles in the flow by as much as 25% over leaky boundary conditions. Transport over a leaky substrate removes particles that reach the bottom boundary and only the suspended load remains. Particle transport to the boundary is proportional to the settling velocity of particles, and flow dilution results in shear and buoyancy instabilities at the upper interface of these flows. These instabilities entrain ambient fluid, and the continued dilution ultimately results in these currents becoming less dense than the ambient fluid. A unifying concept is energy dissipation due to particle–boundary interaction: leaky boundaries dissipate energy more efficiently at the boundary than their saltating counterparts and have smaller run-out distance.

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A Statistical Representation of the Matrix�CFracture Transfer Function for Porous Media

In this article, the authors present a matrix?Cfracture transfer function where the statistical variation in geometric properties of the matrix blocks is considered. Several particular representations with hypothetical probability density functions (PDFs) for matrix block size distributions are presented, including: (a) the single-value distribution (the limiting case); (b) the uniform distribution; (c) the Gamma distribution; and (d) an approximate representation for arbitrary PDFs. An example using experimental data from the literature, along with the single-block based transfer function developed in this study, is presented demonstrating how the statistical procedure proposed in this text can be applied in practice. It is shown with this example that significant relative errors can be introduced when the statistical variance is ignored. Furthermore, two existing dual-porosity models, the Lim and Aziz model and the Zimmerman et al. model, are also considered using the experimental data. It is shown that considerable relative errors can be introduced with these two models when the effect of statistical variance is not taken into account.     

New framework for Bayesian statistical analysis and interpolation of residual stress measurements

In the present study, different residual stress and strain data measured from various techniques are analyzed using a Bayesian statistical approach and finally interpolated utilizing modified Shepard method. This research is carried out to compare the capability, simplicity and accuracy of Bayesian approach with different probability density functions. Three different probability density functions: Gaussian, Cauchy and Sivia's distribution are studied and compared here. Finally the modified Shepard method is utilized with new interpolant and weight functions, to interpolate the scattered measured data. The proposed framework is then applied to two sets of measured residual data obtained from various experimental techniques.     

Probability analysis of random plane waves in gas dynamics

A statistical analysis is made of random nonlinear plane waves in a gas with polytropic exponent = 3 by reduction of the original problem to an auxiliary Cauchy boundary-value problem for a system of stochastic ordinary differential equations. The probability distribution is found for the velocity and density of the gas in the case when at the initial time the gas density is constant and the velocity field Gaussian and statistically homogeneous. It is noted that there exists a finite time of statistical nonlinear interaction of colliding waves during which the probability distribution of the velocity and density of the gas can be essentially non-Gaussian.Translated from Izvesitya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 99–104, September–October, 1982.I thank A. N. Malakhov and S. N. Gurbatov for helpful discussions.     

A 3D smoothed particle hydrodynamics model for erosional dam-break floods

A mesh-less smoothed particle hydrodynamics (SPH) model for bed-load transport on erosional dam-break floods is presented. This mixture model describes both the liquid phase and the solid granular material. The model is validated on the results from several experiments on erosional dam breaks. A comparison between the present model and a 2-phase SPH model for geotechnical applications (Gadget Soil; TUHH) is performed. A demonstrative 3D erosional dam break on complex topography is investigated. The present 3D mixture model is characterised by: no tuning parameter for the mixture viscosity; consistency with the Kinetic Theory of Granular Flow; ability to reproduce the evolution of the free surface and the bed-load transport layer; applicability to practical problems in civil engineering. The numerical developments of this study are represented by a new SPH scheme for bed-load transport, which is implemented in the SPH code SPHERA v.8.0 (RSE SpA), distributed as FOSS on GitHub.     

A new approximation for the three-point probability function

Statistical continuum theory based approaches are commonly used for the computation of the effective properties of heterogeneous materials. Statistical distribution and morphology of the microstructure are represented by n-point probability function. One-point probability function statistical representation of the microstructure leads to volume fraction dependent homogenization. However, second and higher order probability functions include the information of phase distribution and morphology. Most statistical based homogenization methods are limited to two-point probability function due to the lack of simple approximation of higher order probability functions that can be easily exploited. In this paper, a new approximation of the three-point probability function is proposed and discussed. The new approximation results are compared to existing approximations from the literature and to the real probability functions calculated from a computer generated two-phase micrographs.     

Investigation of bed shear stresses subject to external turbulence

The fluctuating bed shear stress has largely been investigated only for uniform channel flows and boundary layers. In practical engineering, the flow conditions are often modified due to the presence of various hydraulic structures. To what extent the modification affects the known characteristics of the bed shear stress is not clear. This study presents experimental results of the bed shear stress fluctuations, which are obviously subjected to external turbulence generated by superimposing artificial structures in the open channel flows. The statistical analysis of the measurements shows that the probability density function of the bed shear stress can be described by the lognormal function. Some associated relations concerning higher-order moments, skewness and kurtosis, which are derived from the lognormal function, are further compared with the experimental data. Physical implication of the skewed probability density distribution is finally discussed.     

A Novel Framework for Integration of Random-Walk Particle-Tracking Simulation in Subsurface Multi-phase Flow Modeling

Coarse-scale models are generally preferred in the numerical simulation of multi-phase flow due to computational constraints. However, capturing the effects of fine-scale heterogeneity on flow and isolating the impacts of numerical (artificial) dispersion, which increases with scale, are not trivial. In this paper, a particle-tracking method is devised and integrated in a scale-up workflow to estimate the conditional probability distributions of multi-phase flow functions, which can be considered as inputs in coarse-scale simulations with existing commercial packages. First, a novel particle-tracking method is developed to solve the saturation transport equation. The transport calculation is coupled with a velocity update, following the implicit pressure, explicit saturation framework, to solve the governing equations of two-phase immiscible flow. Each phase particle is advanced in a deterministic convection step according to the phase velocity, as well as in a stochastic dispersion step based on the random Brownian motion. A kernel-based formulation is proposed for computation of fluid saturation in accordance with the phase particle distribution. A novel aspect is that this method employs the kernel approach to construct saturation from phase particle distribution, which is an important improvement to the conventional box method that necessitates a large number of particles per grid cell for consistent saturation interpolation. The model is validated against various analytical solutions. Finally, the validated model is integrated in a statistical scale-up procedure to calibrate effective, or “pseudo,” multi-phase flow functions (e.g., relative permeability functions). The proposed scale-up framework does not impose any length scale requirement regarding the distribution of sub-grid heterogeneities.     

Transport and deposition of colliding particles in turbulent channel flows

The purpose of this paper is twofold: (i) to present statistical models that describe particle–turbulence interactions as well as particle–particle collisions and (ii) to gain a better understanding of the effect of inter-particle collisions on transport, deposition, and preferential concentration of heavy particles in turbulent channel flows. The models presented are based on a kinetic equation for the probability density function of the particle velocity distribution in anisotropic turbulent flow. The model predictions compare reasonable well with numerical simulations and properly reproduce the crucial trends of computations.     

Experimental determination of the concentration Probability Density Function for a saltating particle layerDétermination expérimentale de la densité de probabilité de concentration d'une couche de particules en saltation

A horizontal saltation layer of glass particles in air was investigated experimentally in a wind tunnel. Particle concentrations are measured by light scattering diffusion and image processing and all the statistical characteristics were evaluated and thus the probability density function. Our experimental results confirm that the mean concentration decreases exponentially with height, the mean Eulerian dispersion height H being a characteristic lengthscale and that the instantaneous concentration distribution $\stackrel{?}{c}(\overrightarrow{x},t)$ is a random variable following quite well a lognormal distribution. To cite this article: X. Zhang et al., C. R. Mecanique 334 (2006).     

Relative particle velocity in a turbulent flow

On the basis of a statistical approach using a probability density function for the coordinates of two particles in a turbulent flow, the parameters of the relative particle motion are investigated. For the functions describing particle entrainment in the turbulence, rigorous results are obtained using a 3D turbulence spectrum. A method of calculating the particle relative-velocity rate with account for particle trajectory correlation is presented. The effects of particle inertia and velocity slip on the parameters of the relative particle motion are studied. Simple approximating formulas for calculating the relative particle motion in a turbulent flow are proposed. The calculation results are compared with the data of direct numerical simulation of stochastic particle trajectories in an isotropic turbulent field.     

On the Calculation of Stationary Solutions of Multi-Dimensional Fokker–Planck Equations by Orthogonal Functions

In this paper, nonlinear stochastic systems are investigatedvia associated Fokker–Planck equations. Their stationary solutions arecalculated by expansions into orthogonal functions, e.g. especiallyadjusted polynomials and Fourier series. The weighting functions of thenew polynomials are obtained by the application of the stochasticaveraging method. The proposed analysis is demonstrated with severalexamples. The first one is a two-dimensional problem of nonlinearoscillators driven by white noise. The second one describes two-massoscillators with independent coloured noise excitations leading tosix-dimensional probability density functions. The next example ispresenting a system driven by both harmonic and stochastic excitationleading to three-dimensional probability density functions. Finally,oscillators with dry friction characteristics are examined.     

Characteristic lengths and maximum entropy estimation from probe signals in the ellipsoidal bubble regime

The bubble size, surface and volume distributions in two and three phase flows are essential to determine energy and mass transfer processes. The traditional approaches commonly use a conditional probability density function of chord-lengths to calculate the bubble size distribution, when the bubble size, shape and velocity are known. However, the approach used in this paper obtains the above distributions from statistical relations, requiring only the moments inferred from the measurements given by a sampling probe. Using image analysis of bubbles injected in a water tank, and placing an ideal probe on the image, a sample of bubble diameter, shape factor and velocity angle are obtained. The samples of the bubble chord-length are synthetically generated from these variables. Thus, we propose a semi-parametric approach based on the maximum entropy (MaxEnt) distribution estimation subjected to a number of moment constraints avoiding the use of the complex backward transformation. Therefore, the method allows us to obtain the distributions in close form. The probability density functions of the most important length scales (D, D₂₀, D₃₀, D₃₂), obtained applying the semi-parametric approach proposed here in the ellipsoidal bubble regime, are compared with experimental measurements.     

随机结构非线性动力响应的概率密度演化分析

提出了随机结构非线性动力响应分析的概率密度演化方法．根据结构动力响应的随机状态方程，利用概率守恒原理，建立了随机结构非线性动力响应的概率密度演化方程．结合Newmark-Beta时程积分方法与Lax-Wendroff差分格式，提出了概率密度演化方程的数值分析方法．通过与Monte Carlo分析方法对比，表明所给出的概率密度演化方法具有良好的计算精度和较小的计算工作量．研究表明：随机结构非线性动力响应概率密度具有典型的演化特征，随着时间增长，概率密度曲线分布趋于复杂．     

脆性断裂的微观机理和非平衡统计特性

Ⅰ.引言如何才能将断裂的微观机理与宏观特性结合起来,把断裂理论建立于微裂纹演化的微观动力学基础上,从而统一导出所有重要的宏观力学量并以某些更基本的物理量表示之?这是人们为实现材料的强度和韧性设计必需解决的一个重要理论课题。就脆性断裂来说,尽管现有几个主要代表性的理论如断裂力学理论、位错理论和统计理论都各取得一定成就,但就其理论框架来说,由于明显的局限性,却难以发展成可供指导设计的理论。因此,人们在探索微观与宏观相结合的断裂理论。最近的工作表明:从微裂纹演      

轿车钢板表面形貌特征参数的提取

通过对典型轿车钢板表面形貌的分析,以形貌与其摩擦学性能的关系为出发点,对粗糙表面、粗糙峰和粗糙谷的分布特性进行了统计分析。结果表明：粗糙表面、粗糙峰和粗糙谷数的统计分布特性与润滑及接触等性能密切相关;针对轿车钢板表面、粗糙峰和粗糙谷数在高度方向概率分布曲线的非Gauss特性,提出采用Weibull概率密度函数的非线性曲线拟合对分布特征进行分析处理。所得拟合参数具有非常明确的物理意义,所此可以采用少量参数来较全面地描述其对应的轿车钢板表面形貌特征分布。     

Large eddy simulation (2D) of spatially developing mixing layer using vortex‐in‐cell for flow field and filtered probability density function for scalar field

A large eddy simulation based on filtered vorticity transport equation has been coupled with filtered probability density function transport equation for scalar field, to predict the velocity and passive scalar fields. The filtered vorticity transport has been formulated using diffusion‐velocity method and then solved using the vortex method. The methodology has been tested on a spatially growing mixing layer using the two‐dimensional vortex‐in‐cell method in conjunction with both Smagorinsky and dynamic eddy viscosity subgrid scale models for an anisotropic flow. The transport equation for filtered probability density function is solved using the Lagrangian Monte‐Carlo method. The unresolved subgrid scale convective term in filtered density function transport is modelled using the gradient diffusion model. The unresolved subgrid scale mixing term is modelled using the modified Curl model. The effects of subgrid scale models on the vorticity contours, mean streamwise velocity profiles, root‐mean‐square velocity and vorticity fluctuations profiles and negative cross‐stream correlations are discussed. Also the characteristics of the passive scalar, i.e. mean concentration profiles, root‐mean‐square concentration fluctuations profiles and filtered probability density function are presented and compared with previous experimental and numerical works. The sensitivity of the results to the Schmidt number, constant in mixing frequency and inflow boundary conditions are discussed. Copyright © 2005 John Wiley & Sons, Ltd.     

Conditional Fluid-Particle Accelerations in Turbulence

Simple closures for average fluid-particle accelerations, conditional on fixed local fluid velocity, are considered in isotropic, homogeneous and stationary turbulence using exact probability density transport equations and are compared with direct numerical simulations (DNS). Such accelerations are common ingredients in Lagrangian stochastic models for fluid-particle trajectories in turbulence. One-particle accelerations are essentially trivial, so the focus is on two-particle relative accelerations, which are important in the relative dispersion process. The closure is simply a quadratic form in the velocity variable and this special form also defines the Eulerian velocity probability density function (pdf), and comparisons with DNS (for grids up to 512³) of both the acceleration closure and velocity pdf's are encouraging. Received 2 June 1997 and accepted 29 December 1997