Diffusion fields associated with size and shape coarsening of oblate spheroids

The diffusion field or solute concentration distributed around an oblate spheroidal particle simulating a disc-shaped precipitate has been solved for varying particle aspect ratios and varying concentrations along the precipitate surface because of the curvature effect. With oblate spheroidal coordinates, the principal curvatures of the oblate spheroidal surface are derived as functions of the angular variable, and the Laplace field equation is separated into two Legendre equations on the angular variable and on the radial variable. The analytical solution to the Laplace equation, fitting the present boundary conditions, is secured as the sum of a Legendre function and a Legendre series composed of Legendre functions of the second kind with imaginary arguments. The Legendre function gives the concentration distribution with an ignored curvature effect, whereas the series shows the contribution from the curvature effect. Numerical results of normalized concentrations are presented as functions of the radial and angular variables for selected aspect ratios. The concentration distributions around both oblate and prolate spheroidal particles are shown to reduce to the concentration distributed around a spherical particle when the aspect ratio of the spheroids approaches unity.     

Numerical Model for Local Scour under Offshore Pipelines

A numerical model, based on potential-flow theory is proposed for simulating the equilibrium scour hole formed by unidirectional flow underneath offshore pipelines. The model employs a finite-difference method to solve the Laplace equation in terms of velocity potential in a curvilinear coordinate system. A boundary adjustment technique based on the Newton-Raphson method is used to calculate the free boundary formed by the eroded seabed by means of the equilibrium of all forces acting on a sediment particle on a sloping bed. Because the solution of flow field and adjustment of the seabed topography are carried out in an iterative manner, the model takes into account the interactions between the flow, pipe, and the seabed. The comparison of the present model with empirical formulas on the prediction of the maximum scour depth indicates that the present model is useful for approximate estimation of scour depth at a pipeline on the seabed for the case of clear-water scour.     

RANS∕Laplace Calculations of Nonlinear Waves Induced by Surface-Piercing Bodies

An interactive zonal numerical method has been developed for the prediction of free surface flows around surface-piercing bodies, including both viscous and nonlinear wave effects. In this study, a Laplace solver for potential flow body-wave problems is used in conjunction with a Reynolds-averaged Navier-Stokes (RANS) method for accurate resolution of viscous, nonlinear free surface flows around a vertical strut and a series 60 ship hull. The Laplace equation for potential flow is solved in the far field to provide the nonlinear waves generated by the body. The RANS method is used in the near field to resolve the turbulent boundary layers, wakes, and nonlinear waves around the body. Both the kinematic and dynamic boundary conditions are satisfied on the exact free surface to ensure accurate resolution of the divergent and transverse waves. The viscous-inviscid interaction between the potential flow and viscous flow regions is captured through a direct matching of the velocity and pressure fields in an overlapping RANS and potential flow computational region. The numerical results demonstrate the capability of an interactive RANS∕Laplace coupling method for accurate and efficient resolution of the body boundary layer, the viscous wake, and the nonlinear waves induced by surface-piercing bodies.     

Viscoelastic State for the Solutions of Spherically Symmetric Viscoelasticity Problems

The solutions of the spherically symmetric, linear, isothermal, and transient viscoelasticity problems via reciprocity theorem have been investigated for a specific material. The integral form of stress–strain relations has been used. The Laplace transform of a viscoelastic state, which is necessary for the integral equation arising as a result of reciprocity theorem, has been derived. This integral equation has been solved by Laplace transform. A sample problem has been solved to test the presented formulation. A numerical application of the analytic solution of this problem has been given.     

Influence Zone of Recharging-Dewatering Actions in Unconfined Aquifer

This work deals with the recharging-dewatering problem for an unconfined aquifer. In particular, the evolution of the aquifer free surface consequent to changes in the water level in a trench near the aquifer itself has been investigated. The problem has been analyzed using a numerical approach so as not to eliminate the nonlinearities of the boundary conditions: the Laplace equation has been solved using the finite-element method coupled to an interpolation-collocation method for the solution of the free surface kinematic condition. Many configurations, which are most likely to cover cases of practical interest, have been studied, and the results have been summarized in a simple formula that permits the evaluation of the aquifer portion that is affected by recharging-dewatering actions.     

Consolidation of a Double-Layered Compressible Foundation Partially Penetrated by Deep Mixed Columns

Deep mixed columns often penetrate partially into the soft soil as floating columns due to the depth of the end-bearing layer. Partially penetrated soft soil by columns and the underlying compressible soft soil create a double-layered compressible foundation. So far, no reasonable solution is available to estimate the consolidation of such a double-layered foundation. This paper proposes an analytical solution for consolidation of a double-layered compressible foundation partially penetrated by deep mixed columns considering one-side or two-side vertical drainage The Laplace transform method was used to solve the consolidation equation for the double-layered system while Stehfest’s algorithm was used to solve the inverse Laplace transform for time-dependent loading. A consolidation algorithm was used to calculate the time-settlement relationship of an embankment constructed upon the double-layered foundation partially penetrated by deep mixed columns. The calculated settlements were compared well with field measurements.     

Flow Depletion Induced by Pumping Well from Stream Perpendicularly Intersecting Impermeable/Recharge Boundary

Analytical solutions for rate and volume of flow depletion induced by pumping a well from a stream that intersects an impermeable or a recharge boundary at right angles are derived using the basic flow depletion factor defined earlier by the author. A new concept of directly obtaining stream flow depletion using the method of images is proposed. The solutions are derived for five different management cases of a stream and boundary intersecting at right-angles, assuming the aquifer to be confined with semi-infinite areal extent. A computationally simple function is proposed for accurately approximating the error function. The existing analytical solution in the case of a right-angle bend of stream given by Hantush was obtained for unconfined aquifers using a linearization of the governing partial differential equation. The solution for this case obtained using the proposed method for confined aquifer is the same as obtained by Hantush for unconfined aquifers, which shows that the linearization adopted by Hantush does not actually solve this problem for unconfined aquifers.     

Capillary Fringe and Unsaturated Flow in a Porous Reservoir Bank

Steady, 2D Darcian seepage from a zero-depth reservoir into a homogeneous porous bank is studied analytically. Using the Green-Ampt assumption for hydraulic conductivity as a function of pressure and the Vedernikov model for the tension-saturated zone, a free boundary problem with the capillary fringe spread along the bank surface is solved. Accurate direct calculations are made for the extent of the capillary rise along a vertical and horizontal bank. Unsaturated flow from a vertical phreatic surface into a bank with either an impermeable or isobaric vertical soil slope is analyzed in terms of the Philip model. Explicit expressions for the Darcian velocity components, stream function, and/or Kirchhoff potential are presented. The flow topology and characteristics are shown to depend strongly on the boundary condition at the soil surface.     

Electrical impedance tomography of complex conductivity distributions with noncircular boundary

Electrical impedance tomography (EIT) uses low-frequency current and voltage measurements made on the boundary of a body to compute the conductivity distribution within the body. Since the permittivity distribution inside the body also contributes significantly to the measured voltages, the present reconstruction algorithm images complex conductivity distributions. A finite element model (FEM) is used to solve the forward problem, using a 6017-node mesh for a piecewise-linear potential distribution. The finite element solution using this mesh is compared with the analytical solution for a homogeneous field and a maximum error of 0.05% is observed in the voltage distribution. The boundary element method (BEM) is also used to generate the voltage data for inhomogeneous conductivity distributions inside regions with noncircular boundaries. An iterative reconstruction algorithm is described for approximating both the conductivity and permittivity distributions from this data. The results for an off-centered inhomogeneity showed a 35% improvement in contrast from that seen with only one iteration, for both the conductivity and the permittivity values. It is also shown that a significant improvement in images results from accurately modeling a noncircular boundary. Both static and difference images are distorted by assuming a circular boundary and the amount of distortion increases significantly as the boundary shape becomes more elliptical. For a homogeneous field in an elliptical body with axis ratio of 0.73, an image reconstructed assuming the boundary to be circular has an artifact at the center of the image with an error of 20%. This error increased to 37% when the axis ratio was 0.64. A reconstruction algorithm which used a mesh with the same axis ratio as the elliptical boundary reduced the error in the conductivity values to within 0.5% of the actual values.     

Coupled Surface–Subsurface Solute Transport Model for Irrigation Borders and Basins. II. Model Evaluation

The development of a coupled surface–subsurface solute transport model for surface fertigation management is presented in a companion paper (Part I). This paper discusses an evaluation of the coupled model. The numerical solution for pure advection of solute in the surface stream was evaluated using test problems with steep concentration gradients. The result shows that the model can simulate advection without numerical diffusion and oscillations, an important problem in the solution of the advection–dispersion equation in advection dominated solute transport. In addition, a close match was obtained between the numerical solution of the one-dimensional advection–dispersion equation and a simplified analytical solution. A comparison of field data and model output show that the overall mean relative residual between field observed and model predicted solute breakthrough curves in the surface stream is 16.0%. Excluding only two outlier (in the graded basin data) reduces the over all mean relative residual between field observed and model predicted breakthrough curves to 5.2%. Finally, potential applications of the model in surface fertigation and salinity management are highlighted.     

Analysis of Point-Source and Boundary-Source Solutions of One-Dimensional Groundwater Transport Equation

The solute transport equation is commonly used to describe the migration and fate of solutes in a groundwater flow system. Depending on the problem nature, the source of the solute may be represented as a point source term in the equation or specified as the first-type or third-type boundary condition. The solutions derived under the condition that the solute introduced into the flow system is from the boundary is herein considered as the boundary-source solutions. The solution obtained when solving the transport equation with a point-source term is considered as the point-source solution. The Laplace transform technique is employed to derive the formulas for those solutions expressed in terms of the normalized mass release rate. The underlying nature of different source release modes and the differences among those boundary-source solutions and the constant point-source solution can be easily and clearly differentiated based on the derived formulas for one-dimensional transport. The methodology could, however, be easily extended to two- and three-dimensional problems.     

Simulation of Free Surface Flow over Spillway

A numerical approach is proposed to simulate and study the effect of geometry on the free surface flow over a tunnel spillway. A three-step solution procedure is proposed to speed up the solution. The first step is to obtain an approximate free surface profile and mean velocity distribution, assuming 1D steady flow. Next, the 3D turbulent flow field is computed while the water surface profile is kept fixed. Finally, the water surface is set free to move and generate waves. The governing equations for weakly compressible flow (compressible hydrodynamic flow) are solved with an explicit finite volume method. A boundary fitted grid system is used to accurately resolve the flow near the free surface with steep waves. A mixed Lagrangian-Eulerian approach is proposed to calculate the new free surface position. The numerical results of a time-averaged free surface profile as well as pressure and velocity distribution have been compared with some experimental data.     

Earthquake Hydrodynamic Pressure on Dams

Hydrodynamic pressures on the vertical upstream face of straight dams during horizontal earthquakes were studied by Westergaad in 1933, and an analytical solution was obtained. Assuming that water is incompressible, an approximation can be made to reduce Westergaad’s mathematical formulation to the Laplace equation. The computer program SEEP2D, from the U.S. Army Corps of Engineers (COE), is available for the study of seepage flow in porous media; this flow can be expressed mathematically in a form of the Laplace equation. Therefore, we can use this computer program to study the hydrodynamic pressure on dams during a horizontal earthquake in the upstream/downstream direction. In practice, the proposed procedure is not limited to SEEP2D but can also be applied to any computer model capable of solving Laplace equations in bounded domains. Two examples are presented to show the application of the COE’s computer program, and the accuracy of the proposed method is discussed.     

Efficient Meshfree Computation with Fast Treatment of Essential Boundary Conditions for Industrial Applications

This paper presents a fast treatment of essential boundary conditions in three-dimensional (3D) meshfree computation for computational efficiency. Due to the loss of Kronecker delta properties in the meshfree shape functions, the imposition of essential boundary conditions is tedious, especially in 3D applications. The proposed boundary singular kernel (BSK) method introduces singularities to the kernel functions associated with the essential and kinematically constrained boundary nodes so that the corresponding coefficients of the singular kernel shape functions recover nodal values, and consequently constraints can be imposed directly. In this work, the recovery of nodal value properties on essential boundary nodes is proved for general n-dimensional geometries. The extension of previously proposed two-dimensional BSK method to 3D formulation thus becomes straightforward, and essential boundary treatment consumes almost no additional cost to meshfree computation and makes the method affordable for industrial applications. The effectiveness of the proposed method is demonstrated in 3D metal forming examples.     

Roll Flattening Analytical Model in Flat Rolling by Boundary Integral Equation Method

In order to improve rolled strip quality, precise plate shape control theory should be established. Roll flattening theory is an important part of the plate shape theory. To improve the accuracy of roll flattening calculation based on semi-infinite body model, especially near the two roll barrel edges, a new and more accurate roll flattening model is proposed. Based on boundary integral equation method, an analytical model for solving a finite length semi-infinite body is established. The lateral surface displacement field of the finite length semi-infinite body is simulated by finite element method (FEM) and lateral surface displacement decay functions are established. Based on the boundary integral equation method, the numerical solution of the finite length semi-infinite body under the distributed force is obtained and an accurate roll flattening model is established. Different from the traditional semi-infinite body model, the matrix form of the new roll flattening model is established through the mathematical derivation. The result from the new model is more consistent with that by FEM especially near the edges.     

Analytical Study of Turbulent Pollutant Dispersion near a Low Hill

An analytical methodology is developed to study the pollutant dispersion in a turbulent wind flow over a two-dimensional hill with a small slope. As in a typical boundary layer problem, the flow domain is divided into an inner and an outer region: the inviscid outer region is further subdivided into an upper and a middle layer while the viscous inner region is subdivided into a shear stress and an inner surface layer. Based on the Reynolds-averaged Navier–Stokes equations and the continuity equations, closed form analytical solutions of the stream functions and velocities are readily obtained for all regions in the domain. The velocity information is then imported into the diffusion equation, and the pollutant concentration distribution is readily solved. For reasons of turbulent shear, a variational method with adjustments to the streamline coordinate system is used to obtain an accurate solution of the pollutant concentration. Results show that when the source is located in the upper layer, the concentrations decrease with distance along the upwind side of the hill and tend to reach a constant value rapidly near the hilltop. Similar results are observed when the source is located in the middle layer. However, due to the reduction of wind speed in the middle layer, the concentrations become saturated at a later upslope position as compared to the source in the upper layer. This methodology is shown to be able to provide a quick and accurate estimate of local pollutant patterns and can be applied to any flow field provided that the streamlines can be specified through the velocities.     

Prediction of Velocity and Deformation Fields During Multipass Plate Hot Rolling by Novel Mixed Analytical-Numerical Method

 An integrated mathematical model is proposed to predict the velocity field and strain distribution during multi-pass plate hot rolling. This model is a part of the mixed analytical-numerical method (ANM) aiming at prediction of deformation variables, temperature and microstructure evolution for plate hot rolling. First a velocity field with undetermined coefficients is developed according to the principle of volume constancy and characteristics of metal flow during rolling, and then it is solved by minimizing the total energy consumption rate. Meanwhile a thermal model coupling with the plastic deformation is exploited through series function solution to determine temperature distribution and calculate the flow stress. After that, strain rate field is calculated through geometric equations and strain field is derived by means of difference method. This model is employed in simulation of an industrial seven-pass plate hot rolling process. The velocity field result and strain field result are in good agreement with that from FEM simulation. Furthermore, the rolling force and temperature agree well with the measured ones. The comparisons verify the validity of the presented method. The calculation of temperature, strain and strain rate are helpful in predicting microstructure. Above all, the greatest advantage of the presented method is the high efficiency, it only takes 12 s to simulate a seven-pass schedule, so it is more efficient than other numerical methods such as FEM.     

高炉炉底侵蚀监测的数学模型

本文建立了高炉炉底侵蚀监测的轴对称数字模型，即通过求解控制高炉炉底传热过程的热传导方程，解得在假想侵蚀边界条件下有限个边界点的温度值，再通过正交试验的方法确定满足实测边界温度分布的侵蚀边界。计算所采用的数值方法是边界元素法（ＢＥＭ），边界的离散采用常数元离散，代数方程组的求解采用主元素消去法，所得计算结果与通过水电模拟实验所得的实验结果相吻合。     

Classifying River Waves by the Saint Venant Equations Decoupled in the Laplacian Frequency Domain

The Saint Venant equations are often combined into a single equation for ease of solution. As a result however, this single equation gives rise to several redundant nonlinear terms that may impose significant limitations on model analyses. In order to avoid this, our paper employs a new procedure that separates, in the Laplace frequency domain, the governing equation of water depth from that of flow velocity and thus enables us to consider two independent equations rather than two coupled ones. The so-obtained analytical solutions are valid for prismatic channels of any shape. Solution validity is assured by repeated comparison with the corresponding numerical solutions based on Crump’s algorithm, which accelerates solution convergence. Utilizing this new procedure, this paper will construct a basic wave spectrum for classifying subcritical flow waves in a prismatic channel. The spectrum is basically a contour plot of the normalized specific energy loss for a small water wave moving in the channel for a finite distance of approximately 100?m. The distance is chosen so that four distinct regions with different contour patterns that represent kinematic, diffusion, gravity, and dynamic waves in a river are shown in the spectrum. By incorporating the spectrum with Ferrick’s criteria and Manning’s formula, a single contour line is also generated, which serves as the boundary of the four regions. Example computations show that the spectrum predicts a similar trend of wave attenuation for waves propagating in a trapezoidal channel. When the rising speed of a wave is of concern, the full Saint Venant equations are solved numerically to reconstruct a similar spectrum good for supercritical flow as well.     

Analytical Ground-Water Flow Solutions for Channel-Aquifer Interaction

A general solution scheme for determining ground-water levels for channel∕group-water systems with recharge is developed and verified. The analytical solution uses the Laplace transform method to solve a linearized form of the Boussinesq equation. Unlike other solutions, this scheme allows for both boundaries and sources∕sinks to vary as a function of time and space. To verify the analytical scheme, three one-dimensional case studies of flow between two line sources in an unconfined aquifer were explored through a base run and a set of sensitivity analyses. These runs involved comparisons to MODFLOW and changes in the boundary conditions and dimensions. As noted, the flow equations were linearized about a point called the representative flow depth. A value of havg, defined as the average water depth between the initial and steady flow conditions, was used as the representative flow depth. Results of the proposed method matched very well with MODFLOW solutions for all times and locations using an optimal linearization point. In addition, using havg improved the solutions compared to those obtained previously.