基于非结构网格求解二维浅水方程的高精度有限体积方法

采用HLL格式,在三角形非结构网格下采用有限体积离散,建立了求解二维浅水方程的高精度的数值模型.本文采用多维重构和多维限制器的方法来获得高精度的空间格式以及防止非物理振荡的产生,时间离散采用三阶Runge-Kutta法以获得高阶的时间精度.基于三角形网格,底坡源项采用简单的斜底模型离散,为保证计算格式的和谐性,对经典的HLL格式计算的数值通量中的静水压力项进行了修正.算例证明本文提出的方法的和谐性并具有高精度的间断捕捉能力和稳定性.     

弯道水流二维浅水模型的修正研究

借鉴有关弯道水流流速分布的研究成果,计入深度平均流速与真实流流速分布差值引起的扩散效应,在正交曲线坐标系下建立了平面二维浅水模型.采用以标准κ-ε模型为基础的曲率效应修正紊流模型模拟紊动应力项,在一定程度上考虑了流线弯曲水流紊动应力的各向异性.应用控制体积法和交错网格法离散方程,并用SIMPLEC算法求解离散方程;同时采用修正后的模型对90°弯道水流进行了数值模拟,并与原模型的计算结果及实测资料进行了比较,结果表明该模型能够有效地模拟流线弯曲水流的水力特性.     

一维孔隙率浅水方程及其数值离散

通过孔隙率方法来描述挡水物对过水能力的影响建立了一维孔隙率浅水方程. 采用有限体积方法和Roe格式的近似Riemann解建立了孔隙率浅水方程的离散模式. 对底坡和孔隙率源项采用特性方向分解的方法进行处理,使模型精确满足C(Conservative)特性,增加了模型的稳定性. 通过算例模拟证明了模型可以对河道中的挡水物作用进行模拟,且计算结果表明模型具有和谐、稳定、分辨率高等优点.      

复杂地形条件下WENO-Roe方法在浅水模拟中的改进

在分析浅水方程与二维空气运动方程差异的基础上,在不改变原有浅水方程形式的前提下,提出了局部水位法离散连续方程,并针对动量方程中底坡项提出了更具普遍意义的离散方法,保证了浅水方程离散后的平衡性。通过不规则地形下潮流以及混合流流动的模拟,得到的计算结果符合物理实际,与精确解符合良好,最大相对误差不超过4%,验证了此方法在复杂地形上的平衡性,同时本方法又具有良好的间断捕捉能力与稳定性。     

曲线坐标下连通域河道平面二维水流的数值模拟

在边界拟合曲线坐标系下,运用B型交错网格模式和动边界扫描技术建立了基于连通域的二维水流数学模型,并提出了模型中有关参数的处理方法.采用贴体坐标变换将复杂的物理域变换成规则的计算域,在计算域上采用控制容积法离散方程,应用SIMPLEC算法计算速度-压力耦合.研究结果表明:采用控制容积法和SIMPLEC算法离散求解方程,具有良好的守恒性和稳定性; 该模型能够较准确地模拟连通域河段的流场变化、水位变化等过程,可供实际工程应用.     

用离散速度方法计算浅水长波方程

用离散速度法计算浅水波方程，将空气动力学方程和浅水波方程作了比较，用Nadiga提出的近平衡流动方法模拟浅水波方程的连续和间断解。计算了一维的溃坝波问题和Thacker提出的连续解问题，结果与精确解作了比较，并且计算了水流跃过障碍物的问题。     

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应用一维水沙耦合数学模型研究了可冲刷坡面上滚波的水动力学特性. 模型的基本控制方程 采用完整的、基于守恒定律的一维浅水动力学方程,运用能够捕捉激波和泥沙运动不连续性 的WAF TVD二阶数值格式离散控制方程. 通过复演定床滚波的运动特点,对模型进行了验证. 可冲刷坡面的滚波数值模拟结果表明,床面形态对滚波水动力学特性有显著的影响.     

二维双曲型守恒律的一种五阶松弛格式

松弛格式是Jin和Xin提出的无振荡有限差分方法,其主要思想是将守恒律转化为松弛方程组进行求解.本文用逐维五阶WENO重构和显隐式Runge-Kutta方法对松弛方程组的空间和时间进行离散,得到了一种求解二维双曲型守恒律五阶松弛格式.所得格式保持了松弛格式简单的优点,不用求解Riemann问题和计算通量函数的雅可比矩阵.通过二维Burgers方程和二维浅水方程的数值算例验证了格式的有效性.     

关于采用高阶Boussinesq方程求解波浪散射问题的注记

本文讨论了采用高阶Boussinesq方程模拟波浪散射时对基本速度变量位置的局部光滑处理方法。通过光滑局部基本速度变量的取值深度，减小其高阶导数项的量值、加快级数收敛速度进而改善模型方程求解深水波浪散射问题的能力。对于底部边界具有一阶导数不连续的情况，通过局部光滑．可以将基本速度变量取值深度尖角转化为圆角过渡，从而改善速度分布。对于其它任意变化的底部边界，为了减少高阶底坡导数项的影响，在曲率和高阶底坡导数项与斜率具有相同量级的情况下亦需要对基本速度变量的取值深度局部光滑。数值计算结果表明本文提出的光滑技术可以很好地改善Boussinesq方程模拟浅水波和深水波在斜坡地形上散射问题的能力。     

求解双曲守恒律方程的熵相容格式

为了更好地求解流体动力学中的双曲守恒律方程,本文提出了一种熵相容格式。通过分析单元内跨越激波时熵的产生情况,得到熵产的显式表达式。在熵守恒通量中加入耗散项与熵增项获得熵相容格式的通量,并在此基础上加入限制器构造出高分辨率熵相容格式。在一维浅水波方程与一维相对论力学方程基础上对新格式进行检验,数值模拟结果表明:这种新的格式能准确捕捉解的结构,具有稳定、无振荡、高分辨率特性。因此,本文方法是求解双曲守恒律方程的较为理想的方法。     

A finite volume solver for 1D shallow‐water equations applied to an actual river

This paper describes the numerical solution of the 1D shallow‐water equations by a finite volume scheme based on the Roe solver. In the first part, the 1D shallow‐water equations are presented. These equations model the free‐surface flows in a river. This set of equations is widely used for applications: dam‐break waves, reservoir emptying, flooding, etc. The main feature of these equations is the presence of a non‐conservative term in the momentum equation in the case of an actual river. In order to apply schemes well adapted to conservative equations, this term is split in two terms: a conservative one which is kept on the left‐hand side of the equation of momentum and the non‐conservative part is introduced as a source term on the right‐hand side. In the second section, we describe the scheme based on a Roe Solver for the homogeneous problem. Next, the numerical treatment of the source term which is the essential point of the numerical modelisation is described. The source term is split in two components: one is upwinded and the other is treated according to a centred discretization. By using this method for the discretization of the source term, one gets the right behaviour for steady flow. Finally, in the last part, the problem of validation is tackled. Most of the numerical tests have been defined for a working group about dam‐break wave simulation. A real dam‐break wave simulation will be shown. Copyright © 2002 John Wiley & Sons, Ltd.     

The effect of numerical methods on the simulation of mid-ocean ridge hydrothermal models

This work considers the effect of the numerical method on the simulation of a 2D model of hydrothermal systems located in the high-permeability axial plane of mid-ocean ridges. The behavior of hot plumes, formed in a porous medium between volcanic lava and the ocean floor, is very irregular due to convective instabilities. Therefore, we discuss and compare two different numerical methods for solving the mathematical model of this system. In concrete, we consider two ways to treat the temperature equation of the model: a semi-Lagrangian formulation of the advective terms in combination with a Galerkin finite element method for the parabolic part of the equations and a stabilized finite element scheme. Both methods are very robust and accurate. However, due to physical instabilities in the system at high Rayleigh number, the effect of the numerical method is significant with regard to the temperature distribution at a certain time instant. The good news is that relevant statistical quantities remain relatively stable and coincide for the two numerical schemes. The agreement is larger in the case of a mathematical model with constant water properties. In the case of a model with nonlinear dependence of the water properties on the temperature and pressure, the agreement in the statistics is clearly less pronounced. Hence, the presented work accentuates the need for a strengthened validation of the compatibility between numerical scheme (accuracy/resolution) and complex (realistic/nonlinear) models.     

NEAR-WAKE FLOW CALCULATIONS BY AN IMPLICIT 2D NAVIER-STOKES ROE RIEMANN SOLVER

Abstract

A flux formulation using a projected 2D Roe Riemann solver on unstructured grids (R2D Solver) is introduced for solving the Navier-Stokes equations and is applied to calculations of axisymmetric laminar near-wake flows behind a spherical-conical body. The numerical framework was first developed by P. L. Roe et al, in the late eighties. They looked for unsteady solutions to Euler's equations using a rather simple but exact three state linearization on triangular grids and decomposing the solution using some effective wave models. Our approach differs from their techniques by constructing a second order accurate and conservative flux functions under the well-known classical finite volume formulation. However, our Riemann Solver is obtained by a suitable linearization procedure upon all three prescribed nodal values given on each triangle. Our numerical method is applied to a Mach 4.3 flow problem for refined unstructured triangular grid behind the body. Numerical results indicate that our technique is stable, accurate and converges successfully to a stationary solution as the cell size is reduced from the coarse lo the finest grid.     

流固耦合地震波动问题的显式谱元模拟方法

流固耦合地震波动问题主要研究由流体和固体构成的复杂系统中地震波传播特性及其规律. 传统模拟方法中一般以声波方程、弹性波方程的数值解分别描述理想流体和弹性固体中的波动, 并实时地处理两种不同性质介质之间的相互耦合作用, 数值格式复杂且限制数值模拟精度与计算效率. 本文采用谱元法结合多次透射公式人工边界条件实现了一种流固耦合地震波动问题的高阶显式数值计算方法. 该方法利用了流固耦合问题统一计算框架,可将饱和多孔介质的Biot波动方程分别退化为理想流体的声波方程和弹性固体的弹性波方程. 通过P波垂直入射的水平成层理想流体-饱和多孔介质-弹性固体场地模型、P波斜入射的不规则层状界面以及任意形状界面的理想流体-饱和多孔介质-弹性固体场地模型等三个算例, 与传递函数法解析解以及集中质量有限元法计算结果进行对比分析, 证明了本文方法的正确性与有效性. 数值模拟结果表明, 本文方法相较传统有限元法可以少得多的节点数量获得更高的数值精度, 并且在较宽的频率范围内都能可靠地模拟出流固耦合系统的动力响应, 充分体现出本文方法兼顾高精度、计算效率和复杂场地建模灵活的特点.      

Incompressible Roe‐like scheme for turbulent flows

In the current study, numerical investigation of incompressible turbulent flow is presented. By the artificial compressibility method, momentum and continuity equations are coupled. Considering Reynolds averaged Navier–Stokes equations, the Spalart–Allmaras turbulence model, which has accurate results in two‐dimensional problems, is used to calculate Reynolds stresses. For convective fluxes a Roe‐like scheme is proposed for the steady Reynolds averaged Navier–Stokes equations. Also, Jameson averaging method was implemented. In comparison, the proposed characteristics‐based upwind incompressible turbulent Roe‐like scheme, demonstrated very accurate results, high stability, and fast convergence. The fifth‐order Runge–Kutta scheme is used for time discretization. The local time stepping and implicit residual smoothing were applied as the convergence acceleration techniques. Suitable boundary conditions have been implemented considering flow behavior. The problem has been studied at high Reynolds numbers for cross flow around the horizontal circular cylinder and NACA0012 hydrofoil. Results were compared with those of others and a good agreement has been observed. Copyright © 2012 John Wiley & Sons, Ltd.     

Finite volume solution to integrated shallow surface–saturated groundwater flow

This paper describes development of an integrated shallow surface and saturated groundwater model (GSHAW5). The surface flow motion is described by the 2‐D shallow water equations and groundwater movement is described by the 2‐D groundwater equations. The numerical solution of these equations is based on the finite volume method where the surface water fluxes are estimated using the Roe shock‐capturing scheme, and the groundwater fluxes are computed by application of Darcy's law. Use of a shock‐capturing scheme ensures ability to simulate steady and unsteady, continuous and discontinuous, subcritical and supercritical surface water flow conditions. Ground and surface water interaction is achieved by the introduction of source‐sink terms into the continuity equations. Two solutions are tightly coupled in a single code. The numerical solutions and coupling algorithms are explained. The model has been applied to 1‐D and 2‐D test scenarios. The results have shown that the model can produce very accurate results and can be used for simulation of situations involving interaction between shallow surface and saturated groundwater flows. Copyright © 2005 John Wiley & Sons, Ltd.     

3D frictional contact of anisotropic solids using BEM

A numerical method to study three-dimensional (3D) contact problems in solids with anisotropic elastic behavior is developed in this work. This formulation is based on the Boundary Element Method (BEM) for computing the elastic influence coefficients and on projection functions over the augmented Lagrangian for contact restrictions fulfillment. The constitutive equations of the potential contact zone are Signorini’s contact conditions and Coulomb’s law of friction. The formulation uses a recently introduced explicit approach for fundamental solutions evaluation, which are valid for general anisotropic behavior meanwhile mathematical degeneracies are allowed. The accuracy and robustness of the proposed method is illustrated by solving some examples previously presented in the literature. This approach is further applied to study the influence of solids anisotropy on the contact problem.     

Effect of quadratic pressure gradient term on a one-dimensional moving boundary problem based on modified Darcy’s law

A relatively high formation pressure gradient can exist in seepage flow in low-permeable porous media with a threshold pressure gradient, and a significant error may then be caused in the model computation by neglecting the quadratic pressure gradient term in the governing equations. Based on these concerns, in consideration of the quadratic pressure gradient term, a basic moving boundary model is constructed for a one-dimensional seepage flow problem with a threshold pressure gradient. Owing to a strong nonlinearity and the existing moving boundary in the mathematical model, a corresponding numerical solution method is presented. First, a spatial coordinate transformation method is adopted in order to transform the system of partial differential equations with moving boundary conditions into a closed system with fixed boundary conditions; then the solution can be stably numerically obtained by a fully implicit finite-difference method. The validity of the numerical method is verified by a published exact analytical solution. Furthermore, to compare with Darcy’s flow problem, the exact analytical solution for the case of Darcy’s flow considering the quadratic pressure gradient term is also derived by an inverse Laplace transform. A comparison of these model solutions leads to the conclusion that such moving boundary problems must incorporate the quadratic pressure gradient term in their governing equations; the sensitive effects of the quadratic pressure gradient term tend to diminish, with the dimensionless threshold pressure gradient increasing for the one-dimensional problem.     

二维洪水演进数值模拟

利用非结构化的有限体积方法,建立了二维浅水方程高精度、高分辨率模型。以Roe类型的近似Rie-mann解计算界面通量,通过MUSCL和两步TVD Runge-Kutta法获得了空间和时间都具有二级精度的TVD格式。采用特征分解的方法处理底坡源项和采用半隐式方法处理摩擦源项均能保证了格式的稳定性与和谐性。通过水滴算例对模型进行验证,并应用此模型对98年胖头泡分滞洪区分洪过程进行模拟,获得滞洪区不同时段的淹没范围和淹没水深,为防洪救灾提供了依据。     

UNSTEADY／STEADY NUMERICAL SIMULATION OF THREE-DIMENSIONAL INCOMPRESSIBLE NAVIER-STOKES EQUATIONS ON ARTIFICIAL COMPRESSIBILITY

A mixed algorithm of central and upwind difference scheme for the solution of steady/unsteady incompressible Navier-Stokes equations is presented. The algorithm is based on the method of artificial compressibility and uses a third-order flux-difference splitting technique for the convective terms and the second-order central difference for the viscous terms. The numerical flux of semi-discrete equations is computed by using the Roe approximation. Time accuracy is obtained in the numerical solutions by subiterating the equations in pseudotime for each physical time step. The algebraic turbulence model of Baldwin-Lomax is ulsed in this work. As examples, the solutions of flow through two dimensional flat, airfoil, prolate spheroid and cerebral aneurysm are computed and the results are compared with experimental data. The results show that the coefficient of pressure and skin friction are agreement with experimental data, the largest discrepancy occur in the separation region where the lagebraic turbulence model of Baldwin-Lomax could not exactly predict the flow.