Groundwater Fluctuations in Sloping Aquifers Induced by Time-Varying Replenishment and Seepage from a Uniformly Rising Stream

This paper analyses the classical problem of transient surface?Cgroundwater interaction in a stream?Caquifer system under rather realistic conditions. The downward sloping unconfined aquifer is in contact with a constant water level at one end, and a fully penetrating stream at the other end whose water level is rising at a uniform speed. Furthermore, the aquifer is replenished by a vertical time-varying recharge. Closed form analytical expressions for hydraulic head and flow rate in the aquifer are obtained by solving the linearized Boussinesq equation using Laplace transform method. Effects of aquifer parameters on transient water table and flow rate are illustrated with a numerical example. To assess the efficiency of the linearization method, analytical solutions are compared with numerical solutions of the corresponding non-linear equation.     

Nonlinear vibration of a two-mass system with nonlinear stiffnesses

An analytical approach is developed for nonlinear free vibration of a conservative, two-degree-of-freedom mass–spring system having linear and nonlinear stiffnesses. The main contribution of the proposed approach is twofold. First, it introduces the transformation of two nonlinear differential equations of a two-mass system using suitable intermediate variables into a single nonlinear differential equation and, more significantly, the treatment a nonlinear differential system by linearization coupled with Newton’s method and harmonic balance method. New and accurate higher-order analytical approximate solutions for the nonlinear system are established. After solving the nonlinear differential equation, the displacement of two-mass system can be obtained directly from the governing linear second-order differential equation. Unlike the common perturbation method, this higher-order Newton–harmonic balance (NHB) method is valid for weak as well as strong nonlinear oscillation systems. On the other hand, the new approach yields simple approximate analytical expressions valid for small as well as large amplitudes of oscillation unlike the classical harmonic balance method which results in complicated algebraic equations requiring further numerical analysis. In short, this new approach yields extended scope of applicability, simplicity, flexibility in application, and avoidance of complicated numerical integration as compared to the previous approaches such as the perturbation and the classical harmonic balance methods. Two examples of nonlinear two-degree-of-freedom mass–spring system are analyzed and verified with published result, exact solutions and numerical integration data.     

Numerical method for unsteady compressible boundary layers driven by compression or expansion wave

A numerical analysis is presented for the unsteady compressible laminar boundary layer driven by a compression or expansion wave. Approximate or series expansion methods have been used for the problems because of the characteristics of the governing equations, such as non-linearity, coupling with the thermal boundary layer equation and initial conditions. Here a transformation of the governing equations and the numerical linearization technique are introduced to deal with the difficulties. First, the governing equations are transformed for the initial conditions by Howarth and semisimilarity variables. These transformations reduce the number of independent variables from three to two and the governing equations from partial to ordinary differential equations at the initial point. Next, the numerical linearization technique is introduced for the non-linearity and the coupling with the thermal boundary layer equation. Because the non-linear terms are linearized without sacrifice of numerical accuracy, the solutions can be obtained without numerical iterations. Therefore the exact numerical solution, not approximate or series expansion, can be obtained. Compared with the approximate or series expansion method, this method is much improved. Results are compared with the series expansion solutions.     

Construction of Basis Functions and Their Application to Boundary-Value Problems of Mechanics of Continuous Media

A unified method for constructing basis (eigen) functions is proposed to solve problems of mechanics of continuous media, problems of cubature and quadrature, and problems of approximation of hypersurfaces. Numerical-analytical methods are described, which allow obtaining approximate solutions of internal and external boundary-value problems of mechanics of continuous media of a certain class (both linear and nonlinear). The method is based on decomposition of the sought solutions of the considered partial differential equations into series in basis functions. An algorithm is presented for linearization of partial differential equations and reduction of nonlinear boundary-value problems, which are reduced to systems of linear algebraic equations with respect to unknown coefficients without using traditional methods of linearization.     

Modeling Three-Dimensional Groundwater Flows by the Body-Fitted Coordinate (BFC) Method: II. Free and Moving Boundary Problems

This paper presents a methodology and solution procedure of the time-dependent body-fitted coordinate (BFC) method for the analysis of transient, three-dimensional groundwater flow problems characterized by free and moving boundaries. The technique consists of numerical grid generation, time-dependent body-fitted coordinate transformation, and application of the finite difference method (FDM) to the transformed partial differential equations. Based on the time-dependent BFC method, a three-dimensional finite-difference computer code, BFC3DGW, was developed and used to solve two unconfined flow problems. The code was verified by comparing numerical results with analytical solutions for a steady-state seepage problem. In order to demonstrate capability of the method in dealing with flow problems with irregular and moving boundary surfaces, an unconfined well-flow problem was solved by the developed code. Difficulties associated with the free and moving irregular boundary have been successfully overcome by employing this method.     

A semi-analytical solution for linearized multicomponent cation exchange reactive transport in groundwater

Cation exchange in groundwater is one of the dominant surface reactions. Mass transfer of cation exchanging pollutants in groundwater is highly nonlinear due to the complex nonlinearities of exchange isotherms. This makes difficult to derive analytical solutions for transport equations. Available analytical solutions are valid only for binary cation exchange transport in 1-D and often disregard dispersion. Here we present a semi-analytical solution for linearized multication exchange reactive transport in steady 1-, 2- or 3-D groundwater flow. Nonlinear cation exchange mass–action–law equations are first linearized by means of a first-order Taylor expansion of log-concentrations around some selected reference concentrations and then substituted into transport equations. The resulting set of coupled partial differential equations (PDEs) are decoupled by means of a matrix similarity transformation which is applied also to boundary and initial concentrations. Uncoupled PDE’s are solved by standard analytical solutions. Concentrations of the original problem are obtained by back-transforming the solution of uncoupled PDEs. The semi-analytical solution compares well with nonlinear numerical solutions computed with a reactive transport code (CORE^2D) for several 1-D test cases involving two and three cations having moderate retardation factors. Deviations of the semi-analytical solution from numerical solutions increase with increasing cation exchange capacity (CEC), but do not depend on Peclet number. The semi-analytical solution captures the fronts of ternary systems in an approximate manner and tends to oversmooth sharp fronts for large retardation factors. The semi-analytical solution performs better with reference concentrations equal to the arithmetic average of boundary and initial concentrations than it does with reference concentrations derived from the arithmetic average of log-concentrations of boundary and initial waters.     

非定常不定边界问题边界无法的若干进展

对于复杂的非线性工程问题的数值模拟,边界元法（ＢＥＭ）日益显示出优于区域解法的长处,特别是时间相关（需按时段逐步迭代推进）和含各种不定边界（造成可变区域,网络需不断重分）的情形,ＢＥＭ可显著减少存贮要求与计算量．针对非线性问题数值模拟的主要难点,即微分算子线性化,时间相关项与可动边界（非线性边条）的处理等,综述了国内外边界无法学术界的近期研究进展,总体目标是寻求一种适应多种微分算子、非线性迭代和时段推进计算效能高的稳定数值模式．     

The analysis of groundwater flows in unconfined aquifers with nonuniform hydraulic conductivity

Horizontal groundwater flows in unconfined aquifers with horizontal lower boundaries can be found exactly by the seepage analysis that allows the hydraulic conductivity to vary in the vertical direction. The exact analysis of flows when the lower boundary of the aquifer is not a horizontal plane, requires the soil-water pressure on this boundary to be known, and this is not generally the case except in the situation of a freshwater aquifer overlying saline water fed from the sea. For aquifers with spatial variations of hydraulic conductivity in both the vertical and horizontal directions, the seepage analysis can be modified to give groundwater flows in situations where the hydraulic conductivity can be represented by the product of independent functions of the three spatial coordinates. Different forms of three-dimensional variation are generated from suitably chosen functions. The use of such forms in calculations of equivalent uniform hydraulic conductivities of some groundwater flow regions demonstrates the dependence of equivalent hydraulic conductivity values on the flow boundary conditions. The exact groundwater flows calculated for particular groundwater situations by the seepage analysis provide results that are useful in validating numerical methods for solving groundwater problems in heterogeneous soils.     

Modeling full-tensor anisotropy in groundwater flow via an iterative scheme for mixed finite elements

This paper presents an iterative scheme for the efficient simulation of groundwater flow in a two-dimensional, heterogeneous aquifer in which the hydraulic conductivity is anisotropic. The scheme is applicable to matrix equations arising from both mixed finite-element and cell-centered finite-difference approximations to the flow equations, and it extends readily to three space dimensions. The scheme, which generalizes an earlier technique for isotropic aquifer, admits a fast multigrid solver for hydraulic heads. Numerical experiments illustrate both the effectiveness of the scheme and the importance of accurately treating anisotropy: Small changes in the off-diagonal terms in the conductivity tensor cause relatively large changes in both the predicted heads and the Darcy velocities.     

Falkner-Skan方程的近似解析解

研究了粘性流体绕流楔型物体的Falkner-Skan边界层方程求解问题.利用Adomian拆分方法,通过引入Crocco变量变换将无穷区间的边界值问题转为初值问题并利用Padé逼近技巧确定初值,给出了一种有效的解析分解方法.进一步,本文设计了一种数值解法,将本文得到的近似解析解及数值结果与早期研究者Hartree等人的结果进行了比较,证明了本文提出的解法的有效性和可靠性.     

Nonsteady flow of groundwater in aquifer system with viscoelastic properties

The transient flow behavior of groundwater in aquifer-aquitard system with viscoelastic properties is studied. On the basis of previous works (Hantush, Neuman, Brutsaert, Corapcioglu), the new partial differential-integral equations are derived. The well-known equations (Hantush, Brutsaert) are the special cases of the new equations. The new equations describe the flow of a slightly compressible groundwater in layers with viscoelastic properties.Analytical solutions of the partial differential-integral equations are obtained by using the method of Laplace transform. The viscoelastic properties enhance the heterogeneities of elastic aquifer system which have delay and feed qualities. The agreements between the numericl inversion results of Laplace transform and the analytical solutions are good. The formulae predict the transient flow behavior of groundwater in this heterogeneous layers.     

On the Numerical-Analytic Investigation of Parametrized Problems with Nonlinear Boundary Conditions

We consider a parametrized boundary-value problem containing an unknown parameter both in nonlinear ordinary differential equations and in nonlinear boundary conditions. By using a suitable change of variables, we reduce the original problem to a family of problems with linear boundary conditions plus certain nonlinear algebraic determining equations. We construct a numerical-analytic scheme suitable for studying the solutions of the transformed boundary-value problem.     

An explicit formulation for the evolution of nonlinear surface waves interacting with a submerged body

An explicit formulation to study nonlinear waves interacting with a submerged body in an ideal fluid of infinite depth is presented. The formulation allows one to decompose the nonlinear wave–body interaction problem into body and free‐surface problems. After the decomposition, the body problem satisfies a modified body boundary condition in an unbounded fluid domain, while the free‐surface problem satisfies modified nonlinear free‐surface boundary conditions. It is then shown that the nonlinear free‐surface problem can be further reduced to a closed system of two nonlinear evolution equations expanded in infinite series for the free‐surface elevation and the velocity potential at the free surface. For numerical experiments, the body problem is solved using a distribution of singularities along the body surface and the system of evolution equations, truncated at third order in wave steepness, is then solved using a pseudo‐spectral method based on the fast Fourier transform. A circular cylinder translating steadily near the free surface is considered and it is found that our numerical solutions show excellent agreement with the fully nonlinear solution using a boundary integral method. We further validate our solutions for a submerged circular cylinder oscillating vertically or fixed under incoming nonlinear waves with other analytical and numerical results. Copyright © 2007 John Wiley & Sons, Ltd.     

RULE OF TRANSIENT PHREATIC FLOW SUBJECTED TO VERTICAL AND HORIZONTAL SEEPAGE

In a semi-infinite aquifer bounded by a channel,a transient flow model is constructed for phreatic water subjected to vertical and horizontal seepage.Based on the first linearized Boussinesq equation,the analytical solution of the model is obtained by Laplace transform.Having proven the transformation between the analytical solution and some relevant classic formulas,suitable condition for each of these formulas is demon- strated.On the base of the solution,the variation of transient flow process caused by the variables,such as vertical infiltration intensity,fluctuation range of river stage,aquifer parameters such as transmissivity and specific yield,and the distance from calculating point to channel boundary,are analyzed quantitatively one by one.Lagging effect will happen to the time,when phreatic water gets its maximum fluctuation velocity,response to the varying of the variables stated above.The condition for some variables to form equivalent lagging effect is demonstrated.Corresponding to the mathematical charac- teristics of the analytical solution,the physical implication and the fluctuation rule of groundwater level are discussed.     

Computer simulation of third‐order nonlinear singular problems arising in draining and coating flows

In this paper, the nonlinear third‐order singular ordinary differential equations that arise in draining and coating flows are studied and computationally solved by employing the modified Adomian decomposition method, and the series solutions of the governing nonlinear problems are developed. Because of singularity, it is not possible to move a contact line over a no‐slip surface, so we cannot find the solution of such problems. To overcome this difficulty, we modify boundary condition to apply the modified Adomian decomposition method with Padé approximants. Copyright © 2011 John Wiley & Sons, Ltd.     

轮对非线性动力学系统蛇行运动的解析解

在对铁路车辆系统的极限环幅值和非线性临界速度进行分析时通常采用数值方法, 不便于研究其随系统参数的变化规律. 轮对系统保留了影响车辆系统动力学性能的几个关键要素: 如轮轨几何非线性约束、轮轨接触蠕滑关系和悬挂系统等, 可以反映铁路车辆系统蛇行运动的本质特性. 轮对系统自由度少、参数少, 可以采用解析方法进行分析. 本文选取合适的特征量把轮对非线性动力学方程无量纲化, 得到了带有小参数的两自由度微分方程; 采用多尺度方法对该方程进行了解析求解; 给出了轮对系统极限环幅值的解析表达式并对其稳定性进行了判定; 给出了轮对系统的分岔速度解析表达式, 并进而获得系统的非线性临界速度的解析表达式. 在对得到的解析解用数值结果进行验证后, 用得到的解析解进行了系统参数影响分析. 传统的分岔图计算方法(如降速法、路径跟踪法等)需对微分方程进行大量数值积分计算方可求解系统的非线性临界速度值, 而通过本文获得的解析表达式可直接给出系统的非线性临界速度值和极限环幅值, 便于研究轮对系统动力学特性随参数的变化规律,进行快速方案比对和筛选, 为转向架结构优化设计提供参考.      

Unsteady flow of ground-water in an aquifer system with viscoelastic properties

The transient flow behaviour of groundwater in an aquifer-aquitard system with viscoelastic properties is studied. On the basis of previous work (Hantush, Neuman, Brutsaert, Corapcioglu), new partial-differential integral equations are assembled. The well-known equations (Hantush, Brutsaert) are special cases of these equations, which describe the flow of slightly compressible groundwater in aquifer-aquitard layers with viscoelastic properties.Analytical solutions of the coupled partial differential-integral equations are obtained by Laplace transform. The viscoelastic properties enhance the heterogeneities of an elastic aquifer system with delay and feed. Numerical inversion of the Laplace transformed drawdown is in good agreement with the analytical solutions obtained.     

Analytical solutions of mixed mode plane strain crack problems in elastic perfectly plastic materials

In the present paper analytical solutions concerning the stress state at the tip of a crack in an elastic-perfectly plastic body, subjected to mixed mode loadings under plane strain conditions, are presented. Analytical solutions of the nonlinear ordinary differential equations are obtained and the dominant singularity is completely determined with the aid of suitable boundary conditions. The obtained results are in perfect agreement with those given by other investigators, both analytical and numerical. The novel aspect here is the methodology used for the solution, as well as the direct determination of the plastic zones. As a consequence, the resulting analytical solutions cover many more problems in the mathematical theory of plasticity compared to similar existing methods and they may be proved of importance in various applications.     

通过Adomian分解法求解二维Helmholtz方程

提出基于Adomian分解法求解二维Helmholtz方程。通过Adomian分解法可以把Helmholtz微分方程和边界条件分别转换成递归代数公式和适用符号计算的简单代数公式。利用边界条件可以很容易得到方程的解析解表达式。Adomian分解法的主要特点在于计算简单快速,并且不需要进行线性化或离散化。最后给出数值实例以验证Adomian分解法求解二维Helmholtz方程的有效性。通过数值计算可以发现,基于Adomian分解法的计算结果非常接近精确解,并且该方法具有良好的收敛性。这表明Adomian分解法能够快速有效求解Helmholtz方程。     

Calculating parameters for infiltration equations from soil hydraulic functions

Simple equations for predicting infiltration of water into soil are valuable both for hydrological application and for investigating soil hydraulic properties. Their value is greatly enhanced if they involve parameters that can be related to more basic soil hydraulic properties. In this paper we extend infiltration equations developed previously for positive surface heads to negative heads. The equations are then used to calculate infiltration into a sand and a clay for a range of initial and surface conditions. Results show errors of less than three percent compared with accurate numerical solutions. Analytical approximations to parameters in the equations are developed for a Brooks and Corey power law hydraulic conductivity-water content relation combined with either a Brooks and Corey or a van Genuchten water retention function. These are compared with accurate numerical values for a range of hydraulic parameters encompassing the majority of soil types and a range of initial and boundary conditions. The approximations are excellent for a wide range of soil parameters.An important attribute of the infiltration equations is their use of dimensionless parameters that can be calculated from normalised water retention and hydraulic conductivity functions. These normalised functions involve only parameters that it may be possible to estimate from surrogate data such as soil particle size distribution. Application of the equations for predicting infiltration, or their use in inferring hydraulic properties, then involves only simple scaling parameters.