Weighted averaged equations for modeling velocity profiles of 1D steady open channel flows

A new mathematical algorithm is proposed to address the essential details of vertical distributions of horizontal velocity for one‐dimensional steady open‐channel flow. This new algorithm comprises a system of weighted averaged equations developed from corresponding Reynolds equations by performing weighted average operations instead of conventional depth average operations. It is the system of weighted averaged equations, instead of the vertical grids, that allows for more hydraulic coefficients identifiable. It can be thought of as an extension of the St. Venant equations to address the vertical distributions of horizontal velocities, as well as the water surface profiles. To avoid the difficult expansion of governing partial differential equations in high order, an indirect scheme is proposed to solve hydraulic variables through their weighted average values. The governing partial differential equations are generated by using a variety of weight functions, and the weighted averages of relevant hydraulic variables are taken as the unknown independent variables to be solved first. Then, on the basis of the values and polynomial expansions of these weighted averaged velocities, a system of linear algebraic equations is generated and the unknown hydraulic variables or their coefficients are easily solved. Note that the new model is not proposed to compete with any three‐dimensional models in modeling accuracy or accommodation ability to all conditions. It just provides a valuable option to study the vertical structure of flow in open channels where only essential detail and reasonable accuracy of vertical distributions are required, and the data availability and other conditions limit the application of fully three‐dimensional models. The performance of the model is evaluated with experimental data of flows in two different flumes. It is shown that the model well predicted the velocity profiles of sections along the centerlines of these flumes with reasonable accuracy and essential details of vertical distributions of horizontal velocity. Copyright © 2013 John Wiley & Sons, Ltd.     

A practical implementation of high‐order RKDG models for the 1D open‐channel flow equations

This paper comprises an implementation of a fourth‐order Runge–Kutta discontinuous Galerkin (RKDG4) scheme for computing the open‐channel flow equations. The main features of the proposed methodology are simplicity and easiness in the implementation, which may be of possible interest to water resources numerical modellers. A version of the RKDG4 is blended with the Roe Riemann solver, an adaptive high‐order slope limiting procedure, and high‐order source terms approximations. A comparison of the performance of the proposed method with lower‐order RKDG models is performed showing a benefit of considering the RKDG4 model. The scheme is applied to computerize the Saint Venant system by considering benchmark tests that have exact solutions. Finally, numerical results are illustrated discussing the performance of the proposed high‐order model. Copyright © 2008 John Wiley & Sons, Ltd.     

Application of a second‐order Runge–Kutta discontinuous Galerkin scheme for the shallow water equations with source terms

The present work addresses the numerical prediction of discontinuous shallow water flows by the application of a second‐order Runge–Kutta discontinuous Galerkin scheme (RKDG2). The unsteady flow of water in a one‐dimensional approach is described by the Saint Venant's model which incorporates source terms in practical applications. Therefore, the RKDG2 scheme is reformulated with a simple way to integrate source terms. Further, an adequate boundary conditions handling, by the theory of characteristics, was overviewed to be adapted to the external points of the mesh, as well as to some points of local invalidity of the Saint Venant's model. To validate the proposed technique, steady and transient test problems (all having a reference solution) were considered and computed by means of the overall method. The results were illustrated jointly with the reference solution and the results carried out by a traditional second‐order finite volume (FV2) scheme implemented with the same techniques as the RKDG2. The proposed method has proven its practical consideration when solving discontinuous shallow water flow involving: non‐prismatic channels, various cross‐sections, smoothly varying bed topography and internal boundary conditions. Copyright © 2007 John Wiley & Sons, Ltd.     

Hamilton体系与与弹性力学Saint－Venant问题

本文一改传统的在Ｌａｇｒａｎｇｅ体系欧几里德空间中用半道法讨论Ｓａｉｎｔ－Ｖｅｎａｎｔ问题的方法，而在具有守恒性的Ｈａｍｉｌｔｏｎ体系中辛空间里研究该问题．通过讨论Ｈａｍｉｌｔｏｎ算子矩阵的零本征值及其Ｊｏｒｄａｎ型，直接求解出全部Ｓａｉｎｔ－Ｖｅｎａｎｔ问题的解．     

New insights on free energies and Saint-Venant’s principle in viscoelasticity

Explicit expressions for the minimum free energy of a linear viscoelastic material and Noll’s definition of state are used here to explore spatial energy decay estimates for viscoelastic bodies, in the full dynamical case and in the quasi-static approximation.In the inertial case, Chirita et al. obtained a certain spatial decay inequality for a space–time integral over a portion of the body and over a finite time interval of the total mechanical energy. This involves the work done on histories, which is not a function of state in general. Here it is shown that for free energies which are functions of state and obey a certain reasonable property, the spatial decay of the corresponding space–time integral is stronger than the one involving the work done on the past history. It turns out that the bound obtained is optimal for the minimal free energy.Two cases are discussed for the quasi-static approximation. The first case deals with general states, so that general histories belonging to the equivalence class of any given state can be considered. The continuity of the stress functional with respect to the norm based on the minimal free energy is proved, and the energy measure based on the minimal free energy turns out to obey the decay inequality derived Chirita et al. for the quasi-static case.The second case explores a crucial point for viscoelastic materials, namely that the response is influenced by the rate of application of loads. Quite surprisingly, the analysis of this phenomenon in the context of Saint-Venant principles has never been carried out explicitly before, even in the linear case. This effect is explored by considering states, the related histories of which are sinusoidal. The spatial decay parameter is shown to be frequency-dependent, i.e. it depends on the rate of load application, and it is proved that of those considered, the most conservative estimate of the frequency-dependent decay is associated with the minimal free energy. A comparison is made of the results for sinusoidal histories at low frequencies and general histories.     

Discharge estimation under uncertainty using variational methods with application to the full Saint‐Venant hydraulic network model

Estimating river discharge from in situ and/or remote sensing data is a key issue for evaluation of water balance at local and global scales and for water management. Variational data assimilation (DA) is a powerful approach used in operational weather and ocean forecasting, which can also be used in this context. A distinctive feature of the river discharge estimation problem is the likely presence of significant uncertainty in principal parameters of a hydraulic model, such as bathymetry and friction, which have to be included into the control vector alongside the discharge. However, the conventional variational DA method being used for solving such extended problems often fails. This happens because the control vector iterates (i.e., approximations arising in the course of minimization) result into hydraulic states not supported by the model. In this paper, we suggest a novel version of the variational DA method specially designed for solving estimation‐under‐uncertainty problems, which is based on the ideas of iterative regularization. The method is implemented with SIC², which is a full Saint‐Venant based 1D‐network model. The SIC² software is widely used by research, consultant and industrial communities for modeling river, irrigation canal, and drainage network behavior. The adjoint model required for variational DA is obtained by means of automatic differentiation. This is likely to be the first stable consistent adjoint of the 1D‐network model of a commercial status in existence. The DA problems considered in this paper are offtake/tributary estimation under uncertainty in the cross‐device parameters and inflow discharge estimation under uncertainty in the bathymetry defining parameters and the friction coefficient. Numerical tests have been designed to understand identifiability of discharge given uncertainty in bathymetry and friction. The developed methodology, and software seems useful in the context of the future Surface Water and Ocean Topography satellite mission. Copyright © 2016 John Wiley & Sons, Ltd.     

Central schemes for open‐channel flow

The resolution of the Saint‐Venant equations for modelling shock phenomena in open‐channel flow by using the second‐order central schemes of Nessyahu and Tadmor (NT) and Kurganov and Tadmor (KT) is presented. The performances of the two schemes that we have extended to the non‐homogeneous case and that of the classical first‐order Lax–Friedrichs (LF) scheme in predicting dam‐break and hydraulic jumps in rectangular open channels are investigated on the basis of different numerical and physical conditions. The efficiency and robustness of the schemes are tested by comparing model results with analytical or experimental solutions. Copyright © 2003 John Wiley & Sons, Ltd.     

坡面流运动方程和有支流入汇时的一维明渠流方程形式

从带源项的圣维南方程出发,根据坡面流特点,分析给出了坡面水流运动方程的基本形式,通过基本的力学分析,指出了通常应用于有支流放汇时的一维明渠流方程存在的问题,并得到了新的方程形式。     

自然弯扭梁广义翘曲坐标的求解

提出了自然弯扭梁受复杂载荷作用时静力分析的一种理论方法,重点在于对控制方程的求解,其中考虑了与扭转有关的翘曲变形和横向剪切变形的影响．在特殊的情况下,可以比较容易地得到这些方程的解答,包括各种内力、应力、应变和位移的计算．算例给出了平面曲梁受水平和垂直分布载荷作用时广义翘曲坐标的求解方法．计算结果表明,求得的应力和位移的理论值和三维有限元结果非常接近．此外,该理论不限于具有双对称横截面的自然弯扭梁,同样可推广至具有一般横截面形状的情况．     

Numerical solution of the St. Venant equations with the MacCormack finite-difference scheme

This paper describes the use of the MacCormack explicit time-spilitting scheme in the development of a two-dimensional (in plan) hydraulic simulation model that solves the St. Venant equations. Various tests devised to assess the performance of the method have been performed and the results are reported. Finally, two industrial applications of the model are presented. The method has been found to be computationally efficient and warrants further development.