Transport of phosphate in a heterogeneous field

A model for the transport of P in an ensemble of vertical homogeneous columns is given. For a single column, the dimensionless concentration of P sorbed to the solid phase, , as a function of dimensionless depth, is approximated with a piston profile. The velocity of the P-front within a column depends on the application rate of P and the retention capacity of the soil. For a field, represented by an ensemble of columns differing with respect to P applied (A _T ) and retention capacity (F _T ), the field average dimensionless concentration , at fixed depth and time, is related to A _T and F _T using probability theory. F _T and A _T are expressed in terms of easily measured variables: oxalate extractable P and Fe + Al. With the probability density functions of these random variables the field-averaged profile is calculated. Experimental and computed profiles are reasonably in agreement and differences can be explained by assuming correlation of F _T and A _T . A parameter analysis shows the increase in field-scale dispersion if the coefficients of variation of the random variables are increased. Negative correlation of A _T and F _T or a positive correlation of successive applications A _i cause an increase in field-scale dispersion. Trends observed for A _T and F _T must be taken into consideration if the model is used for predictive purposes.Notation of variables and parameters A _T Total amount of phosphate (P) applied [mmol/m²] - A Annually applied amount of P [mmol/m²] - C Covariance - CV Coefficient of variation - D Coefficient of molecular diffusion and hydrodynamic dispersion [m²/yr] - D ^* Effective dispersion coefficient for adsorbing solute, D/R [m²/yr] - E Expectation - F Functional relationship of sorption with time and concentration [mmol/xxx - F _m Maximal sorption based on (Fe + Al) _ox [mmol/xxx - F _T Total sorption capacity for P of soil layer with thickness L [mmol/m²] - K Kolmogorov-Smirnov statistic - L Length of column [m] - M Number of applications - N Number of sample locations - P _m Maximal sorption based on P _ox + P _R [mmol/xxx - P _R Extrapolated measured sorption [mmol/xxx - Q Adsorbed amount [mmol/xxx - R Retardation factor - S Precipitated amount [mmol/xxx - VAR Variance - X Generic notation of a random variable - c Concentration of P in solution [mmol/m³] - c ⁰ Feed solution concentration of P [mmol/m³] - f Probability density function - h Separation vector - l _infD ^supf Field scale dispersivity [m] - m Mean, - m Ratio of means, m/m ^R with m ^R for reference situation - t Time [yr] - t ₁ Period in time between successive applications [yr] - t ^* Time required to dissolve solid P [yr] - v Interstitial water velocity [m/yr] - v ^* Effective interstitial water velocity for adsorbing solute, v/R [m/yr] - Time averaged propagation velocity of front [m/yr] - x, y, z Coordinates [m] - Ratio between P _m and (Fe + Al) _OX - Semivariogram - Dimensionless concentration of P in the solid phase - Variable - Coefficient of correlation - _s Soil bulk density [kg/m³] - Standard deviation, (VAR)^1/2 - Ratio of standard deviations, / _R with _R for reference situation - Total time [yr] - Volumetric water content - Coordinate [m] - Dimensionless depth - _p Dimensionless front penetration depth     

Monte Carlo simulation of self-avoiding lattice chains under slit flow

This paper proposes a pseudo-potential describing slit flow in the formalism of lattice Monte Carlo simulation with the bond fluctuation algorithm as the unique basic micro-relaxation mode. The main characteristics of slit flow, such as the parabolic velocity profile and the pressure-flux relationship, are successfully reproduced in a three-dimensional self-avoiding multi-chain system. Both Newtonian and non-Newtonian regimes are revealed. The chain conformation and nonlinear rheological behavior are investigated. The simulation results agree with experimental measurements. This method can be used to investigate the viscoelastic properties at different layers as well as the global properties. Some peculiar phenomena in inhomogeneous flow are found to be consistent with previous theoretical predictions by others. Received: 20 April 1999/Accepted: 22 July 1999     

Monte Carlo simulation of self-avoiding lattice chains subject to oscillatory shear flow in polymer solution

 This paper has introduced a pseudo-potential in bond-fluctuation model to simulate oscillatory shear flow of multiple self-avoiding chains in three dimensions following our previous work under simple shear flow. The oscillatory flow field was reasonably reproduced by lattice Monte Carlo simulation using this pseudo-potential neglecting hydrodynamic interaction. By sampling the configuration distribution functions, the macroscopic viscoelasticity of semi-concentrated polymer solution was determined. Both Newtonian and non-Newtonian regimes were studied. The complex modulus and dynamic viscosity exhibit a reasonable power relation with oscillatory frequency, which is consistent with present theories and experiments. Consequently, lattice Monte Carlo simulation has been extended to model free-draining self-avoiding multi chains subject to oscillatory shear flow and to investigate associated viscoelasticity on the molecular level. Received: 1 October 1999 Accepted: 19 October 1999     

Monte Carlo simulations on the thermodynamic properties of high‐density gases

The thermodynamic properties of Ar, H₂ and CH₄ at high‐density conditions are studied using Monte Carlo simulations. The isotherms of Ar at 500K, H₂ at 1000K and CH₄ at 500K are obtained respectively. To validate the accuracy of the simulation results, the thermodynamic properties of these gases are also studied with van der Waals equation and compared with the reference data. The agreement with reference shows that Monte Carlo method can produce reliable thermodynamic properties of high‐density gases based on the accurate intermolecular potential model. Therefore, the accuracy of the simulations depends primarily on the accuracy of the potential model, and this dependence is also discussed. Copyright © 2012 John Wiley & Sons, Ltd.     

Interaction of a centered isentropic compression wave with the bow shock ahead of a body in a rarefied gas

Rarefied gas flow with a centered isentropic compression wave is investigated using direct Monte Carlo simulation of the solution of the Boltzmann equation. For monatomic gas flow the pattern of formation of a suspended compression shock near the geometric center of the compression wave is considered. The flow pattern is compared with the results obtained within the framework of gas dynamics. For a diatomic gas the interference of a centered compression wave with the bow shock ahead of a cylinder is investigated. The dependence of the pressure and the heat transfer to the surface on the Reynolds number and the wave center position relative to the cylinder center is analyzed. The results are compared with those of numerical simulation of the Euler and boundary-layer equations.     

Possibility of acceleration of the threshold processes for multi-component gas in the front of a shock wave

Studies of translational nonequilibrium in the front of a shock wave propagating in a three-component gas were performed by the Monte Carlo simulation method. Simulations were performed for mixtures of components with molecular mass ratios , and shock Mach number . The distribution of relative velocities for pairs of molecules of heavy low-concentration additives 2 and 3 substantially exceeded, in the front, its equilibrium values behind the wave at high values of . The maximum value of this superequilibrium was about for the numerical density ratio: 1000:1:1 and . Calculations showed that high values of the effect of superequilibrium take place up to a ratio of densities 200:1:1. Simulations performed for and a mixture of He, molecular oxygen and Xe with the numerical density ratio 200:1:1 showed also the high value of the superequilibrium effect at corresponding to dissociation threshold of oxygen. Thus, dissociation of oxygen by collisions with Xe in the front of a wave may have a considerably higher rate than total dissociation behind the wave. Received 4 August 1995 / Accepted 25 April 1996     

Stochastic thermal stresses in an FGM annular disc of variable thickness with spatially random heat transfer coefficients

The second-order statistics (i.e. mean and standard deviation) of the temperature and thermal stresses are evaluated in an axisymmetrically heated functionally graded annular disc of variable thickness with spatially random heat transfer coefficients (HTCs) on the major surfaces of the disc. This annular disc is assumed to have arbitrary variations in the HTCs and material composition along the radial direction only. The randomness in the HTCs is considered to be a random field. The stochastic temperature field is analysed by considering the annular disc to be a multilayered one with stepwise thickness variation, where each layer is assumed to have constant deterministic material properties and random HTCs. In order to evaluate the statistics, the Monte Carlo simulation method is applied to analytical solutions for the deterministic temperature and thermal stresses. The analytical solution for the thermal stresses is obtained through the use of a piecewise power function approximation for Young’s modulus. Numerical results demonstrate the effects of the magnitude of the HTC means, volume fraction distributions of the constitutive materials and thickness variation on the statistics of the temperature and thermal stresses.     

两体磨料磨损的三维动态模拟

概述了目前存在的几种两体磨料磨损的数学模型。将Ｍｏｎｔｅ Ｃａｒｌｏ方法怀数值模拟相结合,提出了一个新的三维仿真模型。运用Ｍｏｎｔｅ Ｃａｒｌｏ方法随机选取磨粒的形状参数,采用打靶法生成随机的磨粒表面,采用网格剖分法记忆被磨表面的外形,为脊的处理以及磨损量的计算提供了一种的新方法,在三维空间内模拟了多个磨粒同时参与磨损的随机的动态磨损过程。将模拟结果和碳钢系列材料的试验结果以及其他研究者的模拟结果     

An elementary molecular-statistical basis for the Mooney and Rivlin-Saunders theories of rubber elasticity

By relaxing the assumption that the end-to-end vectors of molecules transform as macroscopic material line elements, we arrive at a generalization of the molecular-statistical theory of rubber elasticity. This generalization includes as special cases continuum-mechanical theories proposed by Mooney and by Rivlin and Saunders as improvements upon the classical neo-Hookean theory.     

Monte Carlo analysis of dissociation and recombination behind strong shock waves in nitrogen

Computations are presented for the relaxation zone behind strong, one-dimensional shock waves in nitrogen. The analysis is performed with the direct simulation Monte Carlo method (DSMC). The DSMC code is vectorized for efficient use on a supercomputer. The code simulates translational, rotational and vibrational energy exchange and dissociative and recombinative chemical reactions. A new model is proposed for the treatment of three body recombination collisions in the DSMC technique which usually simulates binary collision events. The new model represents improvement over previous models in that it can be employed with a large range of chemical rate data, does not introduce into the flow field troublesome pairs of atoms which may recombine upon further collision (pseudo-particles) and is compatible with the vectorized code. The computational results are compared with existing experimental data. It is shown that the derivation of chemical rate coefficients must account for the degree of vibrational nonequilibrium in the flow. A nonequilibrium chemistry model is employed together with equilibrium rate data to compute successfully the flow in several different nitrogen shock waves.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1990.     

Nonlinear stochastic analysis of subharmonic response of a shallow cable

The paper deals with the subharmonic response of a shallow cable due to time variations of the chord length of the equilibrium suspension, caused by time varying support point motions. Initially, the capability of a simple nonlinear two-degree-of-freedom model for the prediction of chaotic and stochastic subharmonic response is demonstrated upon comparison with a more involved model based on a spatial finite difference discretization of the full nonlinear partial differential equations of the cable. Since the stochastic response quantities are obtained by Monte Carlo simulation, which is extremely time-consuming for the finite difference model, most of the results are next based on the reduced model. Under harmonical varying support point motions the stable subharmonic motion consists of a harmonically varying component in the equilibrium plane and a large subharmonic out-of-plane component, producing a trajectory at the mid-point of shape as an infinity sign. However, when the harmonical variation of the chordwise elongation is replaced by a narrow-banded Gaussian excitation with the same standard deviation and a centre frequency equal to the circular frequency of the harmonic excitation, the slowly varying phase of the excitation implies that the phase difference between the in-plane and out-of-plane displacement components is not locked at a fixed value. In turn this implies that the trajectory of the displacement components is slowly rotating around the chord line. Hence, a large subharmonic response component is also present in the static equilibrium plane. Further, the time variation of the envelope process of the narrow-banded chordwise elongation process tends to enhance chaotic behaviour of the subharmonic response, which is detectable via extreme sensitivity on the initial conditions, or via the sign of a numerical calculated Lyapunov exponent. These effects have been further investigated based on periodic varying chord elongations with the same frequency and standard deviation as the harmonic excitation, for which the amplitude varies in a well-defined way between two levels within each period. Depending on the relative magnitude of the high and low amplitude phase and their relative duration the onset of chaotic vibrations has been verified.     

Reliability of aerodynamic analysis using a moment method

In this study, a reliability analysis was performed for the aerodynamic analysis. Among various reliability analysis methods, the moment method was used and the results were compared with other methods. The reliability of aerodynamic analysis of a 2D airfoil, a 3D wing and a wing body configuration were considered. In the case of 2D airfoil, the reliability was computed by parameterising an airfoil shape using the PARSEC function and considering uncertainties of its shape. In the case of 3D wing and wing body configuration, it was computed by considering uncertainties of flow conditions.     

Deterministic and Stochastic Response of Nonlinear Coupled Bending-Torsion Modes in a Cantilever Beam

The purpose of this study is to understand the main differences between the deterministic and random response characteristics of an inextensible cantilever beam (with a tip mass) in the neighborhood of combination parametric resonance. The excitation is applied in the plane of largest rigidity such that the bending and torsion modes are cross-coupled through the excitation. In the absence of excitation, the two modes are also coupled due to inertia nonlinearities. For sinusoidal parametric excitation, the beam experiences instability in the neighborhood of the combination parametric resonance of the summed type, i.e., when the excitation frequency is in the neighborhood of the sum of the first bending and torsion natural frequencies. The dependence of the response amplitude on the excitation level reveals three distinct regions: nearly linear behavior, jump phenomena, and energy transfer. In the absence of nonlinear coupling, the stochastic stability boundaries are obtained in terms of sample Lyapunov exponent. The response statistics are estimated using Monte Carlo simulation, and measured experimentally. The excitation center frequency is selected to be close to the sum of the bending and torsion mode frequencies. The beam is found to experience a single response, two possible responses, or non-stationary responses, depending on excitation level. Experimentally, it is possible to obtain two different responses for the same excitation level by providing a small perturbation to the beam during the test.     

Multiscale mechanics of macromolecular materials with unfolding domains

We propose a general multiscale approach for the mechanical behavior of three-dimensional networks of macromolecules undergoing strain-induced unfolding. Starting from a (statistically based) energetic analysis of the macromolecule unfolding strategy, we obtain a three-dimensional continuum model with variable natural configuration and an energy function analytically deduced from the microscale material parameters. The comparison with the experiments shows the ability of the model to describe the complex behavior, with residual stretches and unfolding effects, observed in different biological materials.     

Acceleration of chemical reactions within a shock wave front

The aim of the present numerical study was to illustrate the possible influence of translational nonequilibrium in the front of a shock wave on the rate of the threshold chemical reaction. The Monte Carlo method of nonstationary statistical simulation with variable weighting factors was used. Gas mixtures which contained, ahead of the front, two chemically interacting small additives , and an inert light main component were considered. A chemical reaction of the additives started in the front of a shock wave and led to formation of two new low-concentration components and . It was shown that for the ratio of molecular number densities of the additives , and an inert component of 1:10:200 and for the molecular mass ratio of components , , , , of 34.5:8:38.5:4:1 the value of the direct reaction rate obtained in the front exceeds its equilibrium value behind the wave by more than 100 times. As a result, the reaction occurs more intensively in the zone of translational nonequilibrium. It was also shown that for the cases of an exothermic reaction and a weak endothermic reaction, a small amount of the light reaction product has the velocity of the shock wave and is carried by the front. Received 13 June 1997/ Accepted 13 July 1998     

Non‐equilibrium behaviour of equilibrium reservoirs in molecular simulations

We explore two widely used algorithms for fluid reservoirs in molecular simulations and demonstrate that they may induce non‐physical non‐equilibrium effects, even in systems that should be at equilibrium. For example, correlations of momentum and density fluctuations lead to a bias in the mean fluid velocity when measured as the mean over samples of instantaneous fluid velocity. The non‐physical behaviour is entirely computational in origin and is an instance of a more general issue in molecular simulations: a failure to correctly model stochastic properties may induce non‐equilibrium behaviour that does not exist in the corresponding physical system. Finally, we demonstrate that simple algorithm corrections eliminate this artifact. Copyright © 2005 John Wiley & Sons, Ltd.     

Effective Diffusivity Tensors of Point-Like Molecules in Anisotropic Porous Media by Monte Carlo Simulation

Monte Carlo simulations of random walks in anisotropic structured media are performed to determine the dependence of effective diffusivities on geometrical properties. The anisotropic media used in this study are periodic systems, which are generated by extending primitive, face-centered, and body-centered unit cells indefinitely in all axial directions. Results of simulations compare well with published experimental data and the calculations by the volume averaging method. In addition, these results suggest that if the 2D media with percolation thresholds subtantially differ from those of 3D, 2D approximations of 3D media are not satisfactory. When percolation thresholds are the same, the effective diffusivity tensors depend solely on the porosity. This fact has been suggested for isotropic media and it seems to hold for anisotropic media.     

Hypersonic rarefied flows DSMC analysis by a simplified chemical model

A simplified model for the analysis of molecular and reaction dynamics in hypersonic flows of rarefied gases is used to investigate the influence of oxygen dissociation in the prediction of aerothermal coefficients. The model is a direct simulation Monte Carlo method derived from the Larsen-Borgnakke method for the analysis of inelastic collisions. Tests performed in conditions near to re-entry conditions at an altitude of 115 km show significant effects of chemical reactions on the evaluation of the heat-transfer coefficient, while no real influence is found on lift and drag coefficients. Good agreement is found with results achieved using more sophisticated chemical models, as a 4 % maximum difference arises in the determination of surface quantities in re-entry conditions at 160 km.

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Sommario Si studiano gli effetti delle reazioni chimiche sui coefficienti aerodinamici di un' ala piana priva di spessore in regime di moto ipersonico in un gas rarefatto. A tal fine, si sono ottenute delle soluzioni numeriche della equazione di Boltzmann con reazioni chimiche mediante il cosiddetto metodo di Simulazione Diretta. Le collisioni reattive sono state trattate con un modello fortemente semplificato che però fornisce risultati molto vicini a quelli ottenuti da modelli più sofisticati.

     

Monte Carlo simulation of stage separation dynamics of a multistage launch vehicle

This paper provides the formulation used for studing the cold and hot separating stages of a multistage launch vehicle. Monte Carlo simulation is employed to account for the off nominal design parameters of the bodies undergoing separation to evaluate the risk of failure for the separation event. All disturbances, effect of dynamic unbalance, residual thrust, separation disturbance caused by the separation mechanism and misalignment in cold and hot separation are analyzed to find out nonoccurrence of collision between the separation bodies. The results indicate that the current design satisfies the separation requirements.     

Reconstruction of probabilistic S-N curves under fatigue life following lognormal distribution with given confidence

When the historic probabilistic S-N curves are given under special survival probability and confidence levels and there is no possible to re-test, fatigue reliability analysis at other levels can not be done except for the special levels. Therefore, the wide applied curves are expected. Monte Carlo reconstruction methods of the test data and the curves are investigated under fatigue life following lognormal distribution. To overcome the non-conservative assessment of existent man-made enlarging the sample size up to thousands, a simulation policy is employed to address the true production where the sample size is controlled less than 20 for material specimens, 10 for structural component specimens and the errors matching the statistical parameters are less than 5 percent. Availability and feasibility of the present methods have been indicated by the reconstruction practice of the test data and curves for 60Si2Mn high strength spring steel of railway industry.