一类含源Boussinesq系统解的数值分析及仿真

对于一类含源的高阶非线性波动方程Boussinesq方程的初边值问题,利用D1Q5模型的格子Boltzmann方程,通过选取不同的演化方程和局部平衡态分布函数及修正函数,应用ChapmanEnskog多尺度技术和Taylor展开技术,提出了具有五阶高精度带修正函数的非标准格子Boltzmann模型.应用所提出的模型,仿真模拟了几个具有精确解的Boussinesq方程初边值系统,并与传统的修正有限差分法(MFDM)进行了对比,结果表明该文模型所得的数值解与精确解吻合,其模误差小于MFDM.此外,还针对精确解未知的Boussinesq方程初边值系统进行了数值仿真,并与MFDM进行了对比.数值结果表明,两种计算格式的数值解比较吻合,进一步证明了文中所构造模型的有效性和稳定性.     

垂直与水平渗透作用下潜水非稳定渗流运动规律

对河渠边界控制的半无限含水层,建立垂向入渗与河渠水平渗透共同作用下的潜水非稳定渗流模型;利用Boussinesq第一线性化方法,通过Laplace变换,给出模型的解析解. 证明相关经典公式与模型特定解之间的转换关系,分析经典公式适用范围．根据模型解,逐一定量研究下述变量,如垂向入渗强度、河渠水位变动幅度、含水层结构参数如给水度和导压系数、计算点与边界之间的距离,对渗流过程的影响．这些变量的变化,对潜水位获最大上升速度的时间产生延迟效应;论证一些变量间产生等效延迟效应的条件．根据解的数学特征,讨论其对应的物理意义和潜水位变动规律．     

连铸中的自由边界问题的边界元解法

本文应用边界元法求解钢铁生产中连铸工艺出现的自由边界问题。首先,对较一般的连铸过程的数学模型进行简化并给出相应的边界积分方程,以及叙述了用边界元法求解该问题的步骤。然后,我们给出了一个计算实例,并对该方法的收敛快慢、对初值的敏感性和对区域形状的适应性等问题进行了探讨。最后,针对一种简化的模型,将数值解与解析解进行比较,两者吻合较好。     

时变垂向入渗影响下河渠-潜水非稳定渗流模型的解及应用

在河渠边界控制下的半无限含水层中,对时变垂向入渗影响下的潜水非稳定渗流模型,利用Boussinesq第一线性化方法,通过Laplace变换并结合卷积原理,导出模型的解析解．根据不同水文地质条件下解的数学特征,建立相应的含水层参数求解方法;在此基础上,建立计算河渠与含水层之间水量交换的公式,以及计算潜水蒸发强度的递推公式．以安徽淮北平原某河流-潜水含水层系统为例,阐述上述方法的计算过程与步骤．     

有限土层轴对称Biot固结的一个新的解析解

提出一个新的解析方法来研究有限土层的轴对称Biot固结.从轴对称Biot固结的控制方程出发,结合Laplace变换的微分性质,建立了Laplace和Hankel变换域内有限土层地基表面(z=0)和任意深度z处基本变量之间的关系.然后结合有限土层的边界条件,推导出Laplace和Hankel变换域内任意一点的解析解.通过进行Laplace逆变换和Hankel逆变换得到了物理域内的解.编制了计算程序,并对有限土层轴对称固结进行了数值分析.     

有限深两层流中内孤立波的高阶解*

本文研究有限水深两层流中孤立波的三阶近似理论,并考虑了自由表面对孤立波的影响,运用坐标变形方法得到了三阶内孤立波的发展方程,求得波速的解析表达式。对方程进行了数值计算,得到了几种参数下三阶解曲线,指出自由表面对波型和波速的影响是二阶的。计算表明三阶解对一阶、二阶解有明显的改进,使其更加接近试验结果。     

无限域拉普拉斯方程问题的高阶边界条件及其应用

对无限域Laplace方程问题,推导出了高阶边界条件.在采用数值方法的有限域的外边界上应用高阶边界条件,可以在保证计算精度的前提下缩小数值求解域,从而减小计算工作量和少占用计算机内存.数值算例表明,一阶边界条件近似于精确边界条件,它明显地优于经典边界条件和二阶边界条件.     

精确计算Bingham流体圆管层流压降及速度分布规律的新方法

Bingham(宾汉)模型情况下,多采用通用公式进行圆管层流压降的解析计算,即将Bingham模型本构方程代入粘性流体圆管层流流动通用公式进行计算,仅能得到压降的解析解.新方法结合Bingham流体本构方程与运动方程,建立有关力学平衡方程,并运用代数方程的根式解理论对圆管层流流动时的非线性方程进行求解,可直接求得Bingham流体圆管层流压降及速度流核区半径的解析解,进一步可求得圆管层流速度解析解;Bingham流体圆管层流速度的直接影响因素为流量、塑性粘度和屈服值,研究发现速度流核宽度与屈服值成正比,与流量及塑性粘度成反比,且流核的宽度越大,流核区的速度越小.     

混合微通道中不稳定的广义Couette流

在一个平行板通道中,部分充满了均匀的多孔介质,部分为纯流体的流动区,对其微通道中完全发展的不稳定层流进行了数值分析,流动由其中一块板的运动和压力梯度所引起.多孔介质区域的流动,采用扩展的Brinkman模型,即Darcy模型,纯净流动区域的流动,采用Stokes方程.还对稳定的完全发展流进行了理论分析,给出了分界面速度、边界板处的速度和表面摩擦的闭式解.通过数值计算发现,稳定完全发展流的闭式解,和不稳定流动的数值解,在所有时间点上得到很好地吻合.     

任意厚度具有自由边叠层板的精确解析解

自由边问题一直是三维弹性力学中的难题,通常很难满足自由边上一个正应力和两个剪应力都等于0．基于三维弹性力学基本方程和状态空间方法,引入自由边界位移函数并考虑全部弹性常数,建立了正交异性具有自由边单层和叠层板的状态方程．对状态方程中的变量以级数形式展开,通过边界条件的满足精确求解任意厚度具有自由边叠层板的位移和应力,此解满足层间应力和位移的连续条件．算例计算表明,采用引入的位移函数形式,简化了计算过程并且采用较少的级数项可以获得收敛解．与有限元方法计算结果进行了对比,可以得到较高精度的数值结果．其解可以作为其它数值方法和半解析方法的参考解．     

Homogenization Method in the Problem of Long Wave Propagation from a Localized Source in a Basin over an Uneven Bottom

In the framework of the linearized shallow water equations, the homogenization method for wave type equations with rapidly oscillating coefficients that generally cannot be represented as periodic functions of the fast variables is applied to the Cauchy problem for the wave equation describing the evolution of the free surface elevation for long waves propagating in a basin over an uneven bottom. Under certain conditions on the function describing the basin depth, we prove that the solution of the homogenized equation asymptotically approximates the solution of the original equation. Model homogenized wave equations are constructed for several examples of one-dimensional sections of the real ocean bottom profile, and their numerical and asymptotic solutions are compared with numerical solutions of the original equations.     

Numerical modelling of surface water waves generated by moving underwater landslide on irregular bottom

The numerical modelling of surface water waves generated by a moving underwater landslide on irregular bottom is considered. The non-linear shallow water model is used with taking into account bottom mobility. The equations are obtained for an underwater landslide movement under the action of gravity force, buoyancy force, friction force and water resistance force. The predictor-corrector scheme [5], preserving the monotonicity of the numerical solution profiles in a linear case, is used on adaptive grids. The scheme is validated on the problem with a known analytical solution. The analysis is done for the dependencies of wave regimes on bottom slope, initial landslide depth, its length and width. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Long Wave Approximation for Capillary-Gravity Waves and an Effect of the Bottom

The Korteweg–de Vries (KdV) equation is known as a model of long waves in an infinitely long canal over a flat bottom and approximates the 2-dimensional water wave problem, which is a free boundary problem for the incompressible Euler equation with the irrotational condition. In this article, we consider the validity of this approximation in the case of the presence of the surface tension. Moreover, we consider the case where the bottom is not flat and study an effect of the bottom to the long wave approximation. We derive a system of coupled KdV like equations and prove that the dynamics of the full problem can be described approximately by the solution of the coupled equations for a long time interval. We also prove that if the initial data and the bottom decay at infinity in a suitable sense, then the KdV equation takes the place of the coupled equations.     

Water waves diffraction by a circular plate

The interaction of water waves with circular plate within the framework of a linear theory is considered. The plate lies on the free surface in water of finite depth. The integral transform technique is used to solve this problem. The problem is reduced to a system of dual integral equations for a spectral function. The way to solve these equations consists in converting them into Fredholm integral equation of the second kind. The asymptotic solutions of this equation are obtained. Representations for diffraction field and for the forces on the plate are given.     

Flexural‐Gravity Wave Scattering by a Circular‐Arc‐Shaped Porous Plate

The interaction of flexural‐gravity waves with a thin circular‐arc‐shaped permeable plate submerged beneath the ice‐covered surface of water with uniform finite depth is considered under the assumption of linear theory. The problem is reduced to a second kind hypersingular integral equation for the potential difference across the plate which is solved approximately by an expansion–collocation method. Utilizing the solution, the reflection and the transmission coefficients and the hydrodynamic forces are evaluated numerically. The focus of the paper is to illustrate the effect of a porous curved plate submerged in finite depth water with an ice‐cover on the normally incident waves. Numerical results for a circular‐arc‐shaped plate for different configurations are derived and represented graphically. Also, by choosing an appropriate set of parameters, the known results for a circular‐arc‐shaped rigid plate submerged in deep water and a semicircular porous plate submerged in finite depth water with a free surface are recovered as special cases.     

轴对称体大角度斜出水的非线性摄动解

本文处理的是轴对称体大角度斜出水三元非线性问题,以出水角的余角α为小参数进行摄动展开,化为二维非线性问题求解.给出了零阶、一阶和二阶解的积分形式.其中零阶解对应于轴对称体垂直出水的情况,仍是非线性的.数值结果给出了不同Froude数、物体不同长细比情况下各阶自由面形状及各阶力的变化过程.     

Water waves over a rough bottom in the shallow water regime

This is a study of the Euler equations for free surface water waves in the case of varying bathymetry, considering the problem in the shallow water scaling regime. In the case of rapidly varying periodic bottom boundaries this is a problem of homogenization theory. In this setting we derive a new model system of equations, consisting of the classical shallow water equations coupled with nonlocal evolution equations for a periodic corrector term. We also exhibit a new resonance phenomenon between surface waves and a periodic bottom. This resonance, which gives rise to secular growth of surface wave patterns, can be viewed as a nonlinear generalization of the classical Bragg resonance. We justify the derivation of our model with a rigorous mathematical analysis of the scaling limit and the resulting error terms. The principal issue is that the shallow water limit and the homogenization process must be performed simultaneously. Our model equations and the error analysis are valid for both the two- and the three-dimensional physical problems.     

Constructing zeros of accretive operators II

In this paper we study a homogenization problem for a time periodic boundary value problem concerning the quasi-stationary Maxwell equations with a non linear monotone magne tic characteristic. The main features of the problem are related to the vanishing of the conductivity inside each period so that the type of the equations is rapidly oscillating. The unknowns are a vector potential and a scalar potential. We show that the first one converges to zero up to terms of second order, while the second one converges to the solution of a suitable homogenized stationary equation (with time as a parameter). We show moreover that when the magnetic characteristic is linear and symmetric the second order terms in the asymptotic expansion of the vector potential can be identified and related to the time derivative of the limit scalar potential.     

Quasistationary Approximation in the Rotating Ring Problem

We consider the planar rotation-symmetric motion by inertia of a viscous incompressible fluid in a ring with free boundary. We reduce the corresponding initial-boundary value problem for the Navier–Stokes equations to some problem for a coupled system of one parabolic equation and two ordinary differential equations. We suppose that the coefficient of the derivatives of the sought functions with respect to time (the quasistationary parameter) is small; so the system is singularly perturbed. In this article we construct an asymptotic expansion for a solution to the rotating ring problem in a small quasistationary parameter and obtain a smallness estimate for the difference between the exact and approximate solutions.     

The exact solution of a non-linear boundary-value problem of the theory of waves on the surface of a liquid of finite depth

A non-linear boundary-value problem of the theory of waves on the surface of a heavy ideal incompressible liquid, which arises as a result of the expansion of the required functions in amplitude, taking quadratic terms into account, is investigated. A solution is constructed, on the one hand, suitable for describing long waves, and on the other, matched to the Stokes expansion (i.e., with the expansion in amplitude of the first order of infinitesimals). A function is sought which conformally maps a strip into the plane of the complex potential in the flow region. An exact solution is obtained for this problem, defined by fairly simply formulae. This solution, in the limit of long and short waves, gives linear sinusoidal waves and cnoidal waves respectively.