River Bifurcation Analysis by Physical and Numerical Modeling Gianluca Zanichelli1; Elpidio Caroni2; and Virgilio Fiorotto3 Abstract: In the framework of a river regulation design of the Po River Delta ͑Northern Italy͒, a study on a large physical model of the bifurcation Po di Goro-Po di Venezia was conducted with the main objective of determining the discharge subdivision rate at the river node, in order to assess the inflow conditions in the Po di Goro River for flood risk analysis. In this context, a two-dimensional depth averaged numerical model was tested against measured values, with reference to the prototype. In this paper a comprehensive analysis and discussion of the results is reported in order to highlight the applicability of numerical models in comparison with physical ones in river engineering applications. DOI: 10.1061/͑ASCE͒0733-9429͑2004͒130:3͑237͒ CE Database subject headings: Two-dimensional models; Eddy viscosity; Calibration; Bifurcations; Numerical models. Introduction which produces uncertainty in its determination as well as in its influence on the results ͑e.g., King and Norton 1978͒. The study of channel junctions is an important topic in river en- Despite these uncertainties, the interest in numerical modeling gineering problems. The design and evaluation of a channel bi- of river engineering problems depends largely on its cheapness furcation requires an accurate analysis of the flow field, e.g., and flexibility. In addition, it can be useful to obtain a compre- water surface elevation and discharge subdivision in the branches. hensive flow field from measurements taken on laboratory mod- Laboratory flume tests on channel junctions have been conducted els. Nevertheless, physical modeling in Froude similarity is gen- by Ramamurthy and Satish ͑1988͒; Ramamurthy et al. ͑1988͒, erally preferred, as it is supposed to be calibration free. It must be ͑1990͒; and Gurram et al. ͑1997͒. In addition to these physical emphasized that some problems arise in roughness similarity; in experiments, Shettar and Murthy ͑1996͒ conducted numerical addition, the Reynolds number in the model is two or three orders analyses using a two-dimensional ͑2D͒ depth averaged equation of magnitude lower than in the prototype. These effects give un- with a k-␧ closure scheme ͑e.g., Rastogi and Rodi 1978͒. Further certainties in transposing the model results to the prototype. numerical simulations of the cited laboratory tests were recently The aim of this paper is to compare results obtained on the performed by Khan et al. ͑2000͒ in order to test the capability of physical model with the ones produced by a well-known numeri- ͑ the CCHE2D model ͑a 2D depth averaged model developed at the cal model FESWMS-2DH, developed by the U.S. Federal High- ͒ National Center for Computational Hydroscience and Engineer- way Administration as applied to both model and prototype di- ing, Univ. of Mississippi͒ based on zero-equations turbulence clo- mensions and to discuss problems and limits of the two different sure schemes ͑Rodi 1993͒. These closures provide a good repro- approaches. This is presented in the perspective of applying nu- duction of laboratory results and prove simpler than the two- merical modeling to other similar situations in the Po River Delta. equation closure, thus providing a more efficient tool, in view of larger scale river engineering applications. In their work, Khan Po River Study Case et al. ͑2000͒ used mesh element size of order 1 cm, which is not feasible in application to river engineering problems. If a numeri- A river bifurcation project, Po di Venezia-Po di Goro in the Po cal model needs to add numerical viscosity to the zero equation River Delta, was studied by a 1:100 physical model according to closure in the cases of large mesh sizes, the depth averaged eddy the Froude similarity in the Hydraulic Laboratory of the Univer- viscosity parameter loses its physical meaning. In these cases, the sity of Trieste, in the framework of a contract with the Magistrato depth averaged eddy viscosity depends on the mesh element size, per il Po, Parma. Fig. 1 shows the river reach bed geometry and on the current speed, and on the dynamic nature of the problem, the element meshes used for numerical simulations ͑2,088 grid elements and 4,096 nodes͒. The figure shows also the locations 1Senior Engineer, Magistrato per il Po, Parma I-43100, Italy. where, in the physical model, water levels were measured, with 2Associate Professor, Dept. of Civil Engineering, Univ. of Trieste, stilling wells allowing an accuracy of 0.1 mm, i.e., with an esti- Trieste I-34100, Italy. mated prototype error less than 0.01 m. Inflow discharge, at the 3 Professor, Dept. of Civil Engineering, Univ. of Trieste, Trieste upstream section, was measured by an induction flow-meter, I-34100, Italy. while the outflow discharge in the Po di Goro River was mea- Note. Discussion open until August 1, 2004. Separate discussions sured by means of a triangular weir; the error in discharge evalu- must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASCE Managing ation is less than 1%. The water levels in the downstream section Editor. The manuscript for this paper was submitted for review and pos- of the Po di Venezia and Po di Goro branches were controlled sible publication on June 5, 2001; approved on September 3, 2003. This with sluice gates; these levels must accomplish given stage- paper is part of the Journal of Hydraulic Engineering, Vol. 130, No. 3, discharge relationships, which were obtained by steady state 1D March 1, 2004. ©ASCE, ISSN 0733-9429/2004/3-237–242/$18.00. numerical modeling and some direct measurements of water level JOURNAL OF HYDRAULIC ENGINEERING © ASCE / MARCH 2004 / 237 Fig. 1. River Po: Po di Goro-Po di Venezia bifurcation area, in one of the proposed regulation designs tested in the physical model. Bed elevation, mesh discretization ͑2,088 elements͒, and gauging locations are shown. and discharge. The model study area was extended both upstream elements, where a linear function is used to interpolate depth, and and downstream so that the flow in the study reach is not influ- a quadratic function is used to interpolate depth averaged velocity enced by the boundary: ͑1͒ inflow from a reservoir at the up- components ͑FESWMS-2DH͒. The boundary conditions are dis- stream end; and ͑2͒ sliding gates at the two downstream ends. The charge at the upstream section and water level at the downstream downstream control sections, with given stage discharge relation- sections according to the given stage-discharge relationships, the ships, are located about 20 times the maximum depth upstream same as have been used for the physical model boundary condi- from the model outlet to avoid disturbances in the measured lev- tions. These conditions are known with reference to their overall els caused by the gate. Runs were performed with upstream dis- value along the boundary sections, as obtained with 1D modeling, charges in the range of 6,000–12,000 m3 sϪ1, the latter corre- and they are very difficult to apply properly in a 2D scheme that sponding to the maximum design flood. requires the knowledge of their values at every point. For this reason, the stage discharge relationship is not assigned at the boundary, rather it is checked at proper control sections, Numerical Modeling which are located about 20 times the maximum depth upstream from the outlets of the model, to let the flow conditions adjust The 2D, depth averaged, mass and momentum conservation equa- freely. This was done in order to have solutions compatible with tions are the ones measured in the physical model and free from errors due to the imperfect knowledge of the boundary conditions. The clo- hvץ huץ hץ ϩ ϩ ϭ0 sure of the numerical model ͑1͒ needs a choice for the depth y ␯ץ xץ tץ averaged eddy viscosity t and the Manning’s coefficient n. h␶ ␶ץ h␶ 1ץ ␩ 1ץ uץ uץ uץ ϩ ϩ ϩ ϭ xx ϩ xyϪ bx (␳ (1 ץ ␳ ץ ␳ ץ g ץ v ץ u ץ t x y x h x h x h Eddy Viscosity Effects ␶ ␶ ץ ␶ ץ ␩ץ ץ ץ ץ v v v 1 h yx 1 h yy by ϩu ϩv ϩg ϭ ϩ Ϫ In natural river simulations, where elements are generally larger x ␳hץ x ␳hץ y ␳hץ yץ xץ tץ than those used to simulate small laboratory experiments when where hϭdepth of flow; u and vϭvelocity components in the finite element analysis is applied, the depth averaged eddy viscos- ␯ horizontal x and y coordinate directions; t represents time; ity coefficient t represents the influence of turbulent energy gϭgravitational acceleration; ␩ϭwater surface elevation; losses at the subgrid scale. Moreover, it includes ‘‘numerical vis- ␳ϭ ␶ ␶ ϭ water density; xx and yy normal turbulent stresses in the x cosity’’ which is required for stability, whether it is implicit in the ␶ ␶ ϭ and y directions; xy and yx lateral turbulent shear stresses; and numerical scheme or is added in part as is the case of FESWMS- ␶ ␶ ϭ bx and by bed shear stresses in the x and y directions, respec- 2DH: thus the choice of an appropriate eddy viscosity coefficient tively. cannot be entirely based on physical considerations. Increasing The bed shear stresses are computed by the following formu- the eddy viscosity coefficient from the minimum value required ␶ ϭ␳ ͉ ͉ ␶ ϭ␳ ͉ ͉ ͉ ͉ϭ las: bx c f u V and by c f v V where V modulus of the for computational stability, one obtains a stable solution, but if it ϭ 2 1/3 ϭ velocity vector, c f gn /h and n Manning’s roughness coef- is too high, the flow field is not properly evaluated.