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  • Int. J. Environ. Res., 5(2):381-394, Spring 2011ISSN: 1735-6865

    Received 10 March 2010; Revised 12 Aug. 2010; Accepted 25 Aug. 2010

    *Corresponding author E-mail: abedini@ut.ac.ir

    381

    3D Open Channel Flow Modeling by Applying 1D Adjustment

    Abedini, A. A.1*, Ghiassi, R.1 and Ardestani, M.2

    1 Faculty of Civil Engineering, University College of Engineering, University of Tehran,Tehran, Iran

    2 Department of Environmental Engineering, Graduate Faculty of Environment, Universityof Tehran,Tehran, Iran

    ABSTRACT: A three-dimensional (3D) finite volume model with a novel adjustment scheme was developedto solve shallow water equations in open channels. An explicit finite volume method was used to discretize thegoverning equations in a boundary-fitted structured and collocated grid system. Because a simple second-ordercentral scheme was used for spatial discretization and due to the occurrence of high Peclet numbers in openchannel flows, some treatments were needed to reduce oscillation. Thus, a special adjustment scheme designedto minimize differences in the averaged free surface elevation and flow discharge in a 3D model and 1D flowdata was applied to some cross-sections. The model was applied to simulate shallow water flow in a backward-facing step, a meandering channel with 90 bends and a 180 bend channel. A comparison of the model resultswith available experimental and numerical data illustrated that the proposed numerical procedure decreases thenumerical oscillations and increases the stability of the 3D numerical model in open channel flow modeling.

    Key words: Finite volume, Open channel flow, 3D model, Adjustment scheme, Central scheme

    INTRODUCTIONThe prediction of flow patterns and characteristics

    of fluid flows in rivers and open channels have beenthe subjects of research for many years. A remarkablenumber of different models have been used insimulating water bodies regarding both hydraulics andcontaminants fate and transport (Rowshan et al., 2007;Nakane and Haidary, 2010; Monazzam, 2009; Ardestaniand Sabahi, 2009; Etemad-shahidi et al., 2009;Rajasimman et al., 2009; Sadashiva Murthy et al., 2009;Praveena et al., 2010; Etemad-shahidi et al., 2010; NabiBidhendi et al., 2010). Because it is expensive and time-consuming to conduct physical model tests and fieldmeasurements, many numerical models have beendeveloped and applied to simulate river and openchannel flow problems. Most of the models are two-dimensional and depth- or width-averaged. Depth-averaged models provide satisfactory results for manypractical purposes. However, they give no informationabout the depth variation of longitudinal velocity orsecondary flow; therefore, 3D models are necessaryfor computing all velocity components in threedirections. Lane et al. (1999) compared the predictiveability and accuracy of both 3D and 2D models by using

    high-quality field data of a gravel-bed river confluence.They showed that the 3D model had a higherpredictive ability, particularly when the 2D model wasnot corrected for the effects of secondary circulationon flow structure.

    The finite difference method has been usedextensively to solve the basic governing equations in3D hydrodynamic models. Shankar et al. (2001)investigated flow characteristics in a channelconstriction using a 3D finite difference multi-levelmodel. Chen (2003) proposed a 3D, hydrodynamicmodel for free surface flow using a finite differencescheme that involved two predictorcorrector steps.Due to the rectangular grid, the classic finite differencemethod (FDM) requires a rather fine mesh size to givea satisfactory representation for complicated andcurved boundary conditions (Abbott, 1979; Chaudhry,1979), which leads to a high computational effort.Although the finite element method (FEM) does notalways require a regular mesh, it leads to dense matricesand, therefore, involves lengthy matrix inversionprocedures that cause difficulties in recursivecalculations of unsteady flow problems (Bauer andSchmidt, 1983). The finite volume method (FVM),

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    Abedini, A. A. et al.

    which has the merits of both the FDM and FEM, wasfirst introduced into the field of numerical fluiddynamics (independently by McDonald (1971) andMacCormack and Paullay (1972)) for the solution oftwo-dimensional, time-dependent Euler equations(Hirsch, 1988). Recently, Ye et al. (1998) developed athree-dimensional hydrodynamic model for free surfaceturbulent flow using an implicit finite volume methodwith a standard k- turbulence model in a collocatedgrid. Chiavassa et al. (2003) investigated the effects ofdiscretization schemes on a 3D finite volume model ofthe Rhne River plume. Zarrati and Jin (2004)developed a three-dimensional finite volume multi-level model to simulate free surface flows.

    In general, the width and length of a river are muchlarger than the depth, and the vertical velocity of waterflow is much smaller than the horizontal velocities. Thisassumption reduces the momentum equation in thevertical direction to the hydrostatic pressure law, andthe 3D Navier-Stokes equations are simplified to thetwo momentum equations in the x-y plane. Theresulting equations are referred to as the 3D shallowwater equations. In this study, an explicit 3D finitevolume model was developed to simulate shallow waterflow in rivers and open channels. The central differencescheme was used to discretize the governing equationsin a collocated and structured grid.

    In numerical models, when convection is strongerthan diffusion (i.e., Peclet number > 2), a central schememay produce numerical oscillations and physicallyunrealistic results. An upwind scheme does not exhibitoscillation, but it has first-order accuracy and mayoverestimate diffusion at large Peclet numbers(Patankar, 1980). To decrease the numerical oscillation,there are several high-order upwind-type schemes suchas flux-vector splitting, flux-difference splitting, totalvariation diminishing (TVD) and fluctuation splittingas well as central-type schemes such as the Nessyahu-Tadmor method for the discretization of convectivefluxes (Blazek, 2001). These high-order schemes on astaggered (or collocated) grid usually reduce thenumerical oscillation appropriately. In addition, forcollocated grids, the well-known checkerboardinstability may appear (Rhie and Chow, 1983).Therefore, for non-staggered grids, the Rhie and Chowinterpolation should be applied.

    In this study, for spatial discretization, a simplesecond-order central difference scheme on a collocatedgrid without the Rhie and Chow interpolation was used.Thus, the model may confront numerical oscillationand checkerboard instability. To solve these problems,a special adjustment scheme based on adjusting a 3Dmodel with 1D flow data was applied.

    The presented model was verified and validated withthree different case studies of flow in open channels:(1) flow over a backward-facing step, (2) flow in ameandering channel with 90 bends and (3) flow in achannel with a 180 bend. The numerical results arecompared to available experimental and numerical datain terms of free surface profiles and velocities.

    Governing equationsFree surface flow can be studied through mass

    and momentum conservation equations (Streeter andWylie, 1958; Shames, 1962). Under the assumption of aconstant density and hydrostatic pressure distributionand neglecting wind and Coriolis forces, the resulting3D shallow water equations can be expressed in integralforms as follows (Ghiassi, 1995):

    where v is the velocity vector; u and v are the velocitycomponents in the horizontal x- and y-directions,respectively; t is time; is the water surface elevationmeasured from the undisturbed water surface; g is

    gravitational acceleration; and x and y are thecomponents of the shear stress tensor in the x- and y-directions, respectively.Because the main idea of the research was to establisha 3D-1D adjustment scheme for shallow water flowmodeling, a basic algorithm for shear stress evaluationwas applied. Thus, the components of the shear stresstensor were calculated based on the Boussinesqapproximation, using Prandtls mixing length model todetermine the eddy viscosity (Cea et al., 2007). Theresults show that this model, which is categorized inzero-equation turbulence models, is generallysufficiently accurate for shallow water flows. However,for investigating a local full 3D flow, more appropriateturbulence models such as two-equation k- or k- oranisotropic turbulence models (Obi and Peric, 1994;Wallin and Johansson, 2000) can be applied.

    Integrating the continuity equation over the watercolumn together while using the kinematic conditionat the free surface leads to the following free surfaceequation (Casulli, 1999):

    0. =s

    sdv (1)

    +=+

    sx

    sx

    ssdsdgsdvuud

    t i .1.).(

    (2)

    +=+

    sy

    sy

    ssdsdgsdvvvd

    t i .1.).(

    (3)

  • Int. J. Environ. Res., 5(2):381-394, Spring 2011

    383

    where =

    h udzU ; =

    h vdzV ; and h is thewater depth from the bed to the undisturbed freesurface. This means that H(x, y, t) = h(x, y) + (x, y, t)is the total depth of the water column.

    Numerical MethodTo solve the governing 3D equations, we used a

    cell-centered finite volume method on a boundary-fittedstructured and collocated grid. The boundary-fittedstructured grid has the following advantages: it isrelatively straightforward to impose proper boundaryconditions, the solution is sufficiently accurate nearthe boundary, and the conservation property isobserved. However, depending on the shape of theboundary, the quality of the structured boundary-fittedgrid can be a concern (Blazek, 2001). The computationaldomains were divided into a number of hexahedral cells,and all geometrical characteristics of the cells,