Free-surface flow over curved surfaces: Part I: Perturbation analysis

The standard two-dimensional shallow water equation formulation assumes a mild bed slope and no curvature effect. These assumptions limit the applicability of these equations for some important classes of problems. In particular, flow over a spillway is affected by the bed curvature via a decidedly non-hydrostatic pressure distribution. A detailed derivation of a more general equation set is given here in Part I. The method relies upon a perturbation expansion to simplify a bed-fitted co-ordinate configuration of the three-dimensional Euler equations. The resulting equations are essentially the equivalent of the two-dimensional shallow water equations but with curvature included and without the mild slope assumption. A finite element analysis and flume result are given in Part II. © 1998 John Wiley & Sons, Ltd.     

简支梯形底扁球壳弯曲问题的准格林函数方法

以简支梯形底扁球壳的弯曲问题为例,详细阐明了准格林函数方法的思想.即利用问题的基本解和边界方程构造一个准格林函数,这个函数满足了问题的齐次边界条件,采用格林公式将简支扁球壳弯曲问题的控制微分方程化为两个互相耦合的第二类Fredholm积分方程.边界方程有多种选择,在选定一种边界方程的基础上,可以通过建立一个新的边界方程...     

Non-symmetrical large deformation of a shallow thin conical shell

I.IntroductionItisimportanttoresearchnon-symmetricalquestionsofshallowcollicalshellsintheoryoronapplication.Asonekindofpressurevessel'sparts,shallowconicalshellsareverycommonlyusedillellgineerillgpractice,becausethedifficultyofmanul\lctul.illgthemis'small.AlthotlghwelookupmanyChineseandtbreignperiodicalswhicharctlblctobefound,wehavenotyeth'ulldarticlesanddocumentsfornon-symmetricalandnolllinearquestionsofshitllowconictllshells.Oval'rccelltyears,ProlbssorWangXinzhiandhiscolleagueshavedonealot…     

An accurate and efficient semi‐implicit method for section‐averaged free‐surface flow modelling

An accurate, efficient and robust numerical method for the solution of the section‐averaged De St. Venant equations of open channel flow is presented and discussed. The method consists in a semi‐implicit, finite‐volume discretization of the continuity equation capable to deal with arbitrary cross‐section geometry and in a semi‐implicit, finite‐difference discretization of the momentum equation. By using a proper semi‐Lagrangian discretization of the momentum equation, a highly efficient scheme that is particularly suitable for subcritical regimes is derived. Accurate solutions are obtained in all regimes, except in presence of strong unsteady shocks as in dam‐break cases. By using a suitable upwind, Eulerian discretization of the same equation, instead, a scheme capable of describing accurately also unsteady shocks can be obtained, although this scheme requires to comply with a more restrictive stability condition. The formulation of the two approaches allows a unified implementation and an easy switch between the two. The code is verified in a wide range of idealized test cases, highlighting its accuracy and efficiency characteristics, especially for long time range simulations of subcritical river flow. Finally, a model validation on field data is presented, concerning simulations of a flooding event of the Adige river. Copyright © 2009 John Wiley & Sons, Ltd.     

Optimal boundary discretization by variational data assimilation

A variational data assimilation technique applied to the identification of the optimal discretization of interpolation operators and derivatives in the nodes adjacent to the boundary of the domain is discussed in frames of the linear shallow water model. The advantage of controlling the discretization of operators near the boundary rather than boundary conditions is shown. Assimilating data that have been produced by the same model on a finer grid, in a model on a coarse grid, we have shown that optimal discretization allows us to correct such errors of the numerical scheme as under‐resolved boundary layer and wrong wave velocity. Copyright © 2010 John Wiley & Sons, Ltd.     

4D variational data assimilation for locally nested models: Complementary theoretical aspects and application to a 2D shallow water model

We consider the application of a four‐dimensional variational data assimilation method to a numerical model, which employs local mesh refinement to improve its solution. We focus on structured meshes where a high‐resolution grid is embedded in a coarser resolution one, which covers the entire domain. The formulation of the nested variational data assimilation algorithm was derived in a preliminary work (Int. J. Numer. Meth. Fluids 2008; under review). We are interested here in complementary theoretical aspects. We present first a model for the multi‐grid background error covariance matrix. Then, we propose a variant of our algorithms based on the addition of control variables in the inter‐grid transfers in order to allow for a reduction of the errors linked to the interactions between the grids. These formulations are illustrated and discussed in the test case experiment of a 2D shallow water model. Copyright © 2010 John Wiley & Sons, Ltd.     

On the approximation of local efflux/influx bed discharge in the shallow water equations based on a wave propagation algorithm

In cities, flood waves may propagate over street surfaces below which lie complicated pipe networks used for storm drainage and sewage. The flood and pipe flows can interact at connections between the underground pipes and the street surface. The present paper examines this interaction, using the shallow water equations to model the flood wave hydrodynamics. Sources and sinks in the mass conservation equation are used to model the pipe inflow and outflow conditions at bed connections. We consider the problem reduced to one dimension. The shallow water equations are solved using a Godunov‐type wave propagation scheme. Wave speeds are modified in the wave propagation algorithm to enable flows to be simulated over nearly dry beds and dry states. First, the model is used to simulate vertical flows through finite gaps in the bed. Next, the interaction of the vertical flows with a dam break flow is considered for both dry and wet beds. An efflux number, En, is defined based on the vertical efflux velocity and the gap length. Comparisons are made with numerical predictions from STAR‐CD, a commercial Navier–Stokes solver that models the free‐surface motions, and a parameter study is undertaken to investigate the effect of the one‐dimensional approximation of the present model, for a range of non‐dimensional efflux numbers. It is found that the shallow flow model gives sensible predictions at all time provided En<0.5, and for long durations for En>0.5. Dam break flow over an underground connecting pipe is also considered. Copyright © 2010 John Wiley & Sons, Ltd.     

A new well‐balanced Hermite weighted essentially non‐oscillatory scheme for shallow water equations

Hermite weighted essentially non‐oscillatory (HWENO) methods were introduced in the literature, in the context of Euler equations for gas dynamics, to obtain high‐order accuracy schemes characterized by high compactness (e.g. Qiu and Shu, J. Comput. Phys. 2003; 193 :115). For example, classical fifth‐order weighted essentially non‐oscillatory (WENO) reconstructions are based on a five‐cell stencil whereas the corresponding HWENO reconstructions are based on a narrower three‐cell stencil. The compactness of the schemes allows easier treatment of the boundary conditions and of the internal interfaces. To obtain this compactness in HWENO schemes both the conservative variables and their first derivatives are evolved in time, whereas in the original WENO schemes only the conservative variables are evolved. In this work, an HWENO method is applied for the first time to the shallow water equations (SWEs), including the source term due to the bottom slope, to obtain a fourth‐order accurate well‐balanced compact scheme. Time integration is performed by a strong stability preserving the Runge–Kutta method, which is a five‐step and fourth‐order accurate method. Besides the classical SWE, the non‐homogeneous equations describing the time and space evolution of the conservative variable derivatives are considered here. An original, well‐balanced treatment of the source term involved in such equations is developed and tested. Several standard one‐dimensional test cases are used to verify the high‐order accuracy, the C‐property and the good resolution properties of the model. Copyright © 2010 John Wiley & Sons, Ltd.     

A well‐balanced explicit/semi‐implicit finite element scheme for shallow water equations in drying–wetting areas

Numerical solutions of the shallow water equations can be used to reproduce flow hydrodynamics occurring in a wide range of regions. In hydraulic engineering, the objectives include the prediction of dam break wave propagation, fluvial floods and other catastrophic flooding phenomena, the modeling of estuarine and coastal circulations, and the design and optimization of hydraulic structures. In this paper, a well‐balanced explicit and semi‐implicit finite element scheme for shallow water equations over complex domains involving wetting and drying is proposed. The governing equations are discretized by a fractional finite element method using a two‐step Taylor–Galerkin scheme. First, the intermediate increment of conserved variable is obtained explicitly neglecting the pressure gradient term. This is then corrected for the effects of pressure once the pressure increment has been obtained from the Poisson equation. In order to maintain the ‘well‐balanced’ property, the pressure gradient term and bed slope terms are incorporated into the Poisson equation. Moreover, a local bed slope modification technique is employed in drying–wetting interface treatments. The proposed model is well validated against several theoretical benchmark tests. Copyright © 2014 John Wiley & Sons, Ltd.     

Elxed Point Theorem of Composition g-Contraction Mapping and its Applications

ＩｎｔｒｏｄｕｃｔｉｏｎＭａｎｙｓｔｕｄｉｅｓｈａｖｅｂｅｅｎｄｏｎｅｏｎｃｏｎｔｒａｃｔｉｖｅｍａｐｐｉｎｇｓ,ｅ .ｇ .,Ｒｈｏａｄｅｓ[1]ｍａｄｅａｃｏｍｐａｒｉｓｏｎｏｆｖａｒｉｏｕｓｄｅｆｉｎｉｔｉｏｎｓ (ｍｏｒｅｔｈａｎ 1 0 0ｔｙｐｅｓｖａｒｉｅｄｆｒｏｍ 2 5ｂａｓｉｃｔｙｐｅｓ)ｏｆｃｏｎｔｒａｃｔｉｖｅｍａｐｐｉｎｇｏｎｃｏｍｐｌｅｔｅｍｅｔｒｉｃｓｐａｃｅｉｎ 1 977.Ａｎｄｕｐｔｏｎｏｗ ,ｓｕｃｈｓｔｕｄｙｉｓｓｔｉｌｌｇｏｉｎｇｏｎ ,ｅ .ｇ .,[2 ] .Ｉｎｔｈｉｓｐａｐｅｒ,ｔｈｅｓｅ…