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Numerical Solution for Two-Dimensional Flow Under Sluice Gates Using the Natural Element Method

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Numerical Solution for Two-Dimensional Flow Under Sluice Gates Using the Natural Element Method 1550 Numerical solution for two-dimensional flow under sluice gates using the natural element method Farhang Daneshmand, S.A. Samad Javanmard, Tahereh Liaghat, Mohammad Mohsen Moshksar, and Jan F. Adamowski Abstract: Fluid loads on a variety of hydraulic structures and the free surface profile of the flow are important for design purposes. This is a difficult task because the governing equations have nonlinear boundary conditions. The main objective of this paper is to develop a procedure based on the natural element method (NEM) for computation of free surface pro- files, velocity and pressure distributions, and flow rates for a two-dimensional gravity fluid flow under sluice gates. Natu- ral element method is a numerical technique in the field of computational mechanics and can be considered as a meshless method. In this analysis, the fluid was assumed to be inviscid and incompressible. The results obtained in the paper were confirmed via a hydraulic model test. Calculation results indicate a good agreement with previous flow solutions for the water surface profiles and pressure distributions throughout the flow domain and on the gate. Key words: free surface flow, meshless methods, meshfree methods, natural element method, sluice gate, hydraulic struc- tures. Re´sume´ : Les charges de fluides sur les structures hydrauliques et le profil de la surface libre de l’e´coulement sont impor- tants aux fins de conception. Cette conception est une taˆche difficile puisque les e´quations principales pre´sentent des conditions limites non line´aires. Cet article a comme objectif principal de de´velopper une proce´dure base´e sur la me´thode des e´le´ments naturels (NEM) pour calculer les profils de la surface libre, les distributions de vitesse et de pression ainsi que les de´bits pour un e´coulement gravitaire bidimensionnel sous les panneaux de vannes. La NEM est une technique nu- me´rique dans le domaine de la me´canique computationnelle et peut eˆtre conside´re´e comme e´tant une me´thode sans mail- lage. Dans la pre´sente analyse, le fluide e´tait pre´sume´ eˆtre exempt de viscosite´ et incompressible. Les re´sultats obtenus ont e´te´ ve´rifie´s par un essai du mode`le hydraulique. Les re´sultats des calculs montrent une bonne corre´lation entre les solu- tions d’e´coulement ante´rieures pour les profils de surface de l’eau et les distributions de pression dans tout l’e´coulement et sur la vanne. For personal use only. Mots-cle´s:e´coulement de surface libre, me´thodes sans maillage, me´thode par e´le´ments naturels, panneaux de vanne, struc- tures hydrauliques. [Traduit par la Re´daction] Introduction known part of the boundary, one boundary condition is The analysis of two-dimensional gravity affected flows in- specified while on the free surface of the flow two boundary volving a free surface is an important area of research in hy- conditions should be satisfied. The determination of the free draulic engineering. Examples include flows over hydraulic surface location as a part of the solution involves the solu- structures such as spillways, weirs and various types of tion of an intrinsically nonlinear problem. gates. Flow characteristics of interest may include the veloc- One of the major achievements in classical hydrodynam- ity and pressure distributions and the free surface profile. In ics was the use of complex analysis and conformal transfor- all these problems, when steady state prevails, the flow is mations in solving free surface flows as presented by Larock Can. J. Civ. Eng. Downloaded from www.nrcresearchpress.com by MCGILL UNIVERSITY on 04/19/15 governed by an elliptic partial differential equation. On the (1970). Such analytical solutions are limited in number be- Received 17 April 2009. Revision accepted 26 July 2010. Published on the NRC Research Press Web site at cjce.nrc.ca on 12 October 2010. F. Daneshmand1,2 and S.A.S. Javanmard. Faculty of Mechanical Engineering, Shiraz University, Shiraz, Iran. T. Liaghat. Dorj-Danesh Company, Shiraz 71348-51154, Iran. M.M. Moshksar. Department of Material Science and Engineering, School of Engineering, Shiraz University, Shiraz 71348-51154, Iran. J.F. Adamowski. Department of Bioresource Engineering, Faculty of Agricultural and Environmental Sciences, McGill University, 21111 Lakeshore Road, Ste. Anne de Bellevue, QC H9X 3V9, Canada. Written discussion of this article is welcomed and will be received by the Editor until 30 April 2011. 1Corresponding author (e-mail: [email protected]). 2Present address: Department of Mechanical Engineering, McGill University, Montreal, QC H3A 2K6, Canada. Can. J. Civ. Eng. 37: 1550–1559 (2010) doi:10.1139/L10-087 Published by NRC Research Press Daneshmand et al. 1551 cause of the difficulty in satisfying the nonlinear condition and far downstream, where the depths are d1 and d2, respec- of constant pressure along the free surface. tively, the flow is assumed to be uniform and it is taken nor- Among the numerical methods, the finite element method mal to the boundaries. In natural element analysis, these (FEM) has gained more popularity in solving the free sur- assumptions are used as boundary conditions in sections AF face problem. A lot of research in this area appears in the and DE. In the solved problems, satisfactory results were literature, including the work of Chan et al. (1973), Isaacs obtained with these sections located at x =±2B from the (1977), and Sankaranarayanan and Rao (1996). An impor- gate. The free surface profiles AS and CD must be found as tant contribution to the finite element analysis of potential part of the solution. Since the flow is assumed to be irrota- flows having free surfaces was provided by Daneshmand et tional, it is governed by the Laplace equation: al. (1999, 2000) and Daneshmand and Kazemzadeh Parsi @2j @2j (2004), all of whom conducted research to find the free sur- ½1 þ ¼ 0 face shapes of the flow under radial gates. A combination of @x2 @y2 a variable domain and a fixed domain finite element method where j is the stream function. Both the fixed boundaries was used by these authors and it was shown that the itera- and the free surface are streamlines; therefore j is taken to tion procedure converges rapidly. be a constant, i.e. Although the finite element method is robust and has been thoroughly developed, it necessitates regeneration of the ½2 j ¼ 0 on the lower boundary meshes in solving free surface problems. This is frequently j ¼ Q on the free surface and on the sluice gate done by the modeler, and is considered to be one of the most time-consuming tasks in finite element analysis. where Q is the flow rate per unit width. On the free surface To overcome the difficulty associated with remeshing, the the dynamic boundary condition requires past decade has seen a tremendous surge in the development v2 of a family of Galerkin and collocation-based numerical ½3 þ y ¼ B; p ¼ 0 methods known as meshless methods. For example, some of 2g the most widely used methods include the element-free Ga- lerkin method (Belytschko et al. 1994), the reproducing ker- where v is the velocity; g is the acceleration due to gravity; y nel particle method (Liu et al. 1995), and the natural is the free surface elevation measured from an arbitrary da- neighbor Galerkin methods, or natural element methods tum; B is the Bernoulli constant; and p is the pressure. Since (NEM) (Sukumar and Moran 1999). The essential character- @j ½4 v ¼ istic of the meshless methods is that there is no need for a @n highly structured mesh as required in the finite element methods. where n is the unit normal from the free surface; eq. [3] be- The natural element method (NEM) is a Galerkin-based comes For personal use only. method that is built upon the principle of natural neighbor 1 @j 2 interpolation. This interpolation scheme has several very ½5 þ y ¼ B on a free surface useful properties, such as its strictly interpolating character, 2g @n ability to exactly interpolate piece-wise linear boundary con- Q B ditions, and a well-defined and robust approximation with Either the flow rate, , or the Bernoulli constant, , are known; the other is required as part of the solution. no user-defined parameter on non-uniform grids. The aim of this paper is to present a numerical procedure based on For the purposes of the numerical solution the inflow and natural element discretization that treats the fluid flow outflow streams are cut at right angles to the primary veloc- through a sluice gate with a free surface. Despite some prog- ity. On the cut portions the boundary condition ress in solving gravity driven free surface flows through the @j ½6 ¼ 0 use of the various numerical methods, solving the problem @n by using NEM has not been investigated to date. In the present study, the free surface profile, velocity and pressure is applied, which means that there is no velocity normal to distributions and the flow rate per unit width Q are calcu- the main flow. lated for a known Bernoulli constant, B, using the natural el- For problems with a known flow rate Q, the free surface Can. J. Civ. Eng. Downloaded from www.nrcresearchpress.com by MCGILL UNIVERSITY on 04/19/15 ement method. Results for pressure distribution are profile is assumed and the problem is solved from eqs. [1], compared with measured values obtained by conducting a [2], and [6]. The constant, B, of eq. [3] is then calculated for hydraulic model test. the free surface nodes. If B is the same for all free surface points, the problem is solved.
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