Dispersion and blockage effects in the flow over a sill

The problem of a homogeneous heavy liquid flow over a local obstacle is considered in the long-wave approximation. The steady and unsteady waves in the vicinity of the obstacle are described by second-order models of the shallow-water theory and their hyperbolic approximations. The flow in the vicinity of the leading and trailing edges of bluff bodies (sills and steps) is studied. The solution of the problem of the blocked zone upstream of the step is constructed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 1, pp. 45–58, January–February, 2008.     

The numerical study of roll‐waves in inclined open channels and solitary wave run‐up

In this paper, we introduce a finite‐volume kinetic BGK scheme and its applications to the study of roll and solitary waves. The current scheme is based on the numerical solution of the gas‐kinetic Bhatnagar–Gross–Krook model in the flux evaluation across each cell interface. An intrinsic connection between the BGK model and time‐dependent, non‐linear, non‐homogeneous shallow‐water equations enables us to solve shallow‐water equations automatically with our kinetic scheme. The analytical solution, experimental measurements, and numerical calculations for problems associated with roll‐waves down an inclined open channel and solitary waves incident on a sloped beach are also presented. Copyright © 2005 John Wiley & Sons, Ltd.     

Dispersion of Surface Waves in a Periodically Laminated Piezoelectric Half-Space with Liquid Upper Layer

An approach is proposed to set up the dispersion equations for surface waves propagating through a periodically laminated piezoelectric medium, with the upper layer being a perfect compressible fluid. The approach is based on the formalism of Hamiltonian periodic systems. The dispersion equations derived are valid for an arbitrary law of variation in properties with periodicity coordinate. The influence of the liquid layer and inhomogeneity of the piezoelectric medium on the dispersion spectra of surface waves is studied__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 3, pp. 55–61, March 2005.     

基于Boussinesq方程的波浪模型

从欧拉方程出发,提供了另一种推导完全非线性Boussinesq方程的方法,并对方程的 线性色散关系和线性变浅率进行了改进. 改进后方程的线性色散关系达到了一阶Stokes波 色散关系的Pad\'{e}[4,4]近似,在相对水深达1.0的强色散波浪时仍保持较高的准确性,并且方程的非线性和线性 变浅率都得到了不同程度的改善. 方程的水平一维形式用预估-校正的有限差分格式求解, 建立了一个适合较强非线性波浪的Boussinesq波浪数值模型. 作为验证,模拟了波浪在潜 堤上的传播变形,计算结果和实验数据的比较发现两者符合良好.     

A comparison of the GWCE and mixed P − P1 formulations in finite‐element linearized shallow‐water models

The appearance of spurious pressure modes in early shallow‐water (SW) models has resulted in two common strategies in the finite element (FE) community: using mixed primitive variable and generalized wave continuity equation (GWCE) formulations of the SW equations. One FE scheme in particular, the P ? P₁ pair, combined with the primitive equations may be advantageously compared with the wave equation formulations and both schemes have similar data structures. Our focus here is on comparing these two approaches for a number of measures including stability, accuracy, efficiency, conservation properties, and consistency. The main part of the analysis centres on stability and accuracy results via Fourier‐based dispersion analyses in the context of the linear SW equations. The numerical solutions of test problems are found to be in good agreement with the analytical results. Copyright © 2011 John Wiley & Sons, Ltd.     

Similarities between the quasi‐bubble and the generalized wave continuity equation solutions to the shallow water equations

Two common strategies for solving the shallow water equations in the finite element community are the generalized wave continuity equation (GWCE) reformulation and the quasi‐bubble velocity approximation. The GWCE approach has been widely analysed in the literature. In this work, the quasi‐bubble equations are analysed and comparisons are made between the quasi‐bubble approximation of the primitive form of the shallow water equations and a linear finite element approximation of the GWCE reformulation of the shallow water equations. The discrete condensed quasi‐bubble continuity equation is shown to be identical to a discrete wave equation for a specific GWCE weighting parameter value. The discrete momentum equations are slightly different due to the bubble function. In addition, the dispersion relationships are shown to be almost identical and numerical experiments confirm that the two schemes compute almost identical results. Analysis of the quasi‐bubble formulation suggests a relationship that may guide selection of the optimal GWCE weighting parameter. Copyright © 2004 John Wiley & Sons, Ltd.     

Flooding in urban drainage systems: coupling hyperbolic conservation laws for sewer systems and surface flow

In this paper, we propose a model for a sewer network coupled to surface flow and investigate it numerically. In particular, we present a new model for the manholes in storm sewer systems. It is derived using the balance of the total energy in the complete network. The resulting system of equations contains, aside from hyperbolic conservation laws for the sewer network and algebraic relations for the coupling conditions, a system of ODEs governing the flow in the manholes. The manholes provide natural points for the interaction of the sewer system and the runoff on the urban surface modeled by shallow‐water equations. Finally, a numerical method for the coupled system is presented. In several numerical tests, we study the influence of the manhole model on the sewer system and the coupling with 2D surface flow. Copyright © 2014 John Wiley & Sons, Ltd.     

方柱尾流流态对床面冲淤形态的影响

应用平面二维悬沙数学模型对方柱尾流区的泥沙输运及床面调整进行了数值模拟，并对重要物理参数做了分析说明。数值计算采用时间分裂一全隐式有限差分格式，流场的计算基于沿水深平均的RANS方程。通过系统的数值模拟，揭示了不同来流情况下，方柱尾流区的流态变化，及与之相应的床面变化规律，并对不同流态下的泥沙运动作了分析。计算结果表明在方柱尾流区，不同的流场流态对尾流区的床面调整有明显影响。流场较弱的情况下，尾流区中湍动强度相应较弱，此时悬浮泥沙由于流速减小而普遍落淤，床面应力的减小也致使床面冲刷量减小。随着流场强度的逐渐增大，尾流区中湍动强度相应增强，床面应力增大，同时湍流的强烈交换作用增强了对泥沙的输运作用，床面变化不再是简单的淤积状态，部分区域出现了冲刷。     

NUMERICAL SIMULATION OF SHALLOW WATER WAVE PROPAGATION USING A BOUNDARY ELEMENT WAVE EQUATION MODEL

The present paper makes use of a wave equation formulation of the primitive shallow water equations to simulate one-dimensional free surface flow. A numerical formulation of the boundary element method is then developed to solve the wave continuity equation using a time-dependent fundamental solution, while an explicit finite difference scheme is used to derive velocities from the primitive momentum equation. One-dimensional free surface flows in open channels are treated and the results compared with analytical and numerical solutions. © 1997 John Wiley & Sons, Ltd.     

Dispersion and Attenuation of Surface Waves in a Fluid-Saturated Porous Medium

An investigation is conducted of propagation of surface waves in a porous medium consisting of a microscopically incompressible solid skeleton in which a microscopically incompressible liquid flows within the interconnected pores, and particularly the case where the solid skeleton deforms linear elastically. The frequency equations of Rayleigh- and Love-type waves are derived relating the dependence of wave numbers, being complex quantities, on frequency, as a result those waves are dispersive as well as inhomogeneous. Nevertheless, the amplitudes of both surface waves attenuate along the surface of the porous medium, whereas they decay exponentially receding from the surface of the medium.     

Optimization of the ADER‐DG method in GPU applied to linear hyperbolic PDEs

We optimized the Arbitrary accuracy DErivatives Riemann problem (ADER) ‐ Discontinuous Galerkin (DG) numerical method using the CUDA‐C language to run the code in a graphic processing unit (GPU). We focus on solving linear hyperbolic partial–differential equations where the method can be expressed as a combination of precomputed matrix multiplications becoming a good candidate to be used on the GPU hardware. Moreover, the method is arbitrarily high order involving intensive work on local data, a property that is also beneficial for the target hardware. We compare our GPU implementation against CPU versions of the same method observing similar convergence properties up to a threshold where the error remains fixed. This behavior is in agreement with the CPU version, but the threshold is slightly larger than in the CPU case. We also observe a big difference when considering single and double precisions where in the first case, the threshold error is significantly larger. Finally, we did observe a speed‐up factor in computational time that depends on the order of the method and the size of the problem. In the best case, our novel GPU implementation runs 23 times faster than the CPU version. We used three partial–differential equation to test the code considering the linear advection equation, the seismic wave equation, and the linear shallow water equation, all of them considering variable coefficients. Copyright © 2015 John Wiley & Sons, Ltd.     

Dispersion Phenomena in Thermal Diffusion and Modelling of Thermogravitational Experiments in Porous Media

In a TIPM paper published in 1992, the authors presented a simple model of thermogravitational diffusion in packed columns (TPC). Though qualitatively in agreement with the experimental results, this model exhibited a systematic discrepancy with respect to the magnitude of the permeability of maximum separation in the TPC experiments. Here, the results of a re-examination of the classical phenomenology of irreversible thermodynamics in porous media, applied to TPC, are described. Through the interpretation of additional TPC experiments, we show that the effective thermal diffusion coefficient in TPC includes a dependency upon the fluid velocity. This dependency is consistent with a nonlinear extension of irreversible thermodynamics, and the model so amended accounts for a correct re-interpretation of the experiments.     

破碎带波浪的数值模拟

基于一组色散关系得到改进的完全非线性Boussinesq方程建立了一个波浪模型可以模拟近岸水域的波浪变浅、破碎以及在海滩上的爬高等多种变形。波浪破碎引起的能量衰减是在动量方程中引入一个在空间和时间上都只作用于波前的涡粘项来模拟。动海岸线边界用窄缝法处理。波浪爬高用非线性浅水方程推导的非破碎波浪在斜坡上爬高的解析解来验证。本模型还模拟了波浪在斜坡上不同类型的破碎变形过程,并将其波高和平均水位的沿程变化和物理模型实验的结果比较,两者符合良好。     

Numerical simulation of tidal bore in Hangzhou Gulf and Qiantangjiang

In this paper, a new numerical model, based on a set of non‐linear shallow water equations is developed for the simulation of the formation and evolution of tidal bore in the Hangzhou Gulf and Qiantangjiang river of China. The numerical method and boundary conditions are described in detail. The method is validated against analytical solutions and experimental data. Simulation of the actual tidal bore in Hangzhou Gulf and its propagation in the Qiantangjiang river are performed. Numerical results show that this proposed method is effective for the prediction of tidal bore and current flow at the entrance of Qiantangjiang river. Copyright © 2001 John Wiley & Sons, Ltd.     

Flux and source term discretization in two‐dimensional shallow water models with porosity on unstructured grids

Two‐dimensional shallow water models with porosity appear as an interesting path for the large‐scale modelling of floodplains with urbanized areas. The porosity accounts for the reduction in storage and in the exchange sections due to the presence of buildings and other structures in the floodplain. The introduction of a porosity into the two‐dimensional shallow water equations leads to modified expressions for the fluxes and source terms. An extra source term appears in the momentum equation. This paper presents a discretization of the modified fluxes using a modified HLL Riemann solver on unstructured grids. The source term arising from the gradients in the topography and in the porosity is treated in an upwind fashion so as to enhance the stability of the solution. The Riemann solver is tested against new analytical solutions with variable porosity. A new formulation is proposed for the macroscopic head loss in urban areas. An application example is presented, where the large scale model with porosity is compared to a refined flow model containing obstacles that represent a schematic urban area. The quality of the results illustrates the potential usefulness of porosity‐based shallow water models for large scale floodplain simulations. Copyright © 2005 John Wiley & Sons, Ltd.     

Dynamics and wave propagation in dilatant granular materials

The equations of motion for dilatant granular material are obtained from a Hamiltonian variational principle of local type in the conservative case. The propagation of nonlinear waves in a region with uniform state is studied by means of an asymptotic approach that has already appeared useful in an investigation on wave propagation in bubbly liquids and in fluid mixtures. When the grains are assumed to be incompressible, it is shown that the material behaves as a continuum with latent microstructure.

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Sommario Si ricavano le equazioni di moto per i materiali granulari dilatanti da un principio variazionale Hamiltoniano di tipo locale nel caso conservativo. Si studia la propagazione delle onde non lineari in una regione di stato costante per mezzo di un approccio asintotico già rivelatosi utile nello studio della propagazione di onde nei liquidi con bolle e nelle miscele di fluidi. Quando si supponga che i granuli siano incomprimibili, si dimostra che il materiale si comporta come un continuo con microstruttura latente.

     

Explicit solution to the exact Riemann problem and application in nonlinear shallow‐water equations

The Riemann solver is the fundamental building block in the Godunov‐type formulation of many nonlinear fluid‐flow problems involving discontinuities. While existing solvers are obtained either iteratively or through approximations of the Riemann problem, this paper reports an explicit analytical solution to the exact Riemann problem. The present approach uses the homotopy analysis method to solve the nonlinear algebraic equations resulting from the Riemann problem. A deformation equation defines a continuous variation from an initial approximation to the exact solution through an embedding parameter. A Taylor series expansion of the exact solution about the embedding parameter provides a series solution in recursive form with the initial approximation as the zeroth‐order term. For the nonlinear shallow‐water equations, a sensitivity analysis shows fast convergence of the series solution and the first three terms provide highly accurate results. The proposed Riemann solver is implemented in an existing finite‐volume model with a Godunov‐type scheme. The model correctly describes the formation of shocks and rarefaction fans for both one and two‐dimensional dam‐break problems, thereby verifying the proposed Riemann solver for general implementation. Copyright © 2007 John Wiley & Sons, Ltd.     

浅水域中多船编队航行时的船波干涉

以Green-Naghdi(GN)方程为基础,采用波动方程和运动网格的有限元法研究多船在浅水域中集体航行时的波浪干涉特性。把运动船舶对水面的扰动作为移动压强直接加在GN方程里,以描述运动船体和水面的相互作用。GN方程合理地考虑非线性和频率散射对浅水船波的影响。以Series60 CB=0.6船体为算例,给出两船并行、前后跟随、三船品字形编队航行时的波浪干涉图形,波浪阻力及侧向力的数值分析结果。计算结果表明:1)当两船并行时,两船承受侧向吸引力,同时波浪阻力稍有增加。2)当两船前后跟随时,两船的波浪阻力都减小。3)当三船品字形航行时,前船的阻力减小,后船的阻力增加,同时后面两船的吸力减小。     

在哈密顿体系下水容器内非线性浅水波理论和实验方法

本文通过引进哈密顿体系非线性浅水波理论,建立一套数值计算方法,并设计一套实验方案和装置,将理论与实验结果相互验证,研究水容器中的水在倾斜和振动过程中的非线性浅水波表明,实验与理论结果基本吻合。同时揭示了该类问题非线性水波波动的一些机理,从而也为解决工程实际问题提供可靠的依据。     

Adjoint Methods in Data Assimilation for Estimating Model Error

Data assimilation aims to incorporate measured observations into a dynamical system model in order to produce accurate estimates of all the current (and future) state variables of the system. The optimal estimates minimize a variational principle and can be found using adjoint methods. The model equations are treated as strong constraints on the problem. In reality, the model does not represent the system behaviour exactly and errors arise due to lack of resolution and inaccuracies in physical parameters, boundary conditions and forcing terms. A technique for estimating systematic and time-correlated errors as part of the variational assimilation procedure is described here. The modified method determines a correction term that compensates for model error and leads to improved predictions of the system states. The technique is illustrated in two test cases. Applications to the 1-D nonlinear shallow water equations demonstrate the effectiveness of the new procedure. This revised version was published online in July 2006 with corrections to the Cover Date.