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 ͑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 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 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- 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. ␯ ficient. For deep channels, in FESWMS-2DH code, t values in the The turbulent normal and shear stresses are formulated accord- range of 2.4–14.4 m2 sϪ1 are suggested, increasing with element ͑ ␶ ץ ץ ␶ ϭ ␳␯ ing to the Boussinesq’s assumption as xx 2 t( u/ x), yx dimensions and the dynamic nature of the problem King and ͒ ץ ץ ␶ ϭ ␳␯ ץ ץϩ ץ ץ ϭ␶ ϭ␳␯ yx t( u/ y v/ x), and yy 2 t( u/ y) where Norton 1978 . These values are larger than the ones that can be ␯ ϭ t depth averaged eddy viscosity. computed on physical bases or measured in regular channels. In ␯ ϭ␣ ϭ The computational domain, as presented in Fig. 1, is dis- fact, for regular channels, t u*h, where u* shear velocity cretized adopting six node ͑vertexes and mid-side points͒ triangle and ␣ϭdimensionless coefficient equal to 0.07 ͑e.g., Shiono and

238 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / MARCH 2004 ͚ ␩ Ϫ␩ ␯ 2 ͓ meas ͑ t͒comp͔ ␰͑␯ ͒ϭ (2) t ͚͓␩ Ϫ␩͑␯ ͒ ͔2 meas t* comp ␩ ␩ ␯ ϭ where meas , ( t)comp water surface elevations, respectively, measured and reproduced by the numerical model, at all locations ͑ ͒ ␰ ␯ ϭ ␯ Fig. 1 . The minimum value of ( t) 1 corresponds to a t ϭ␯ t* value for which the mean square error is at its minimum. From Fig. 2, one can note that, at increasing upstream dis- ␯ charge, a larger value of t* is detected, as suggested in literature ͑e.g., King and Norton 1978͒, while for larger floods, a relatively ␯ ␯ constant t* is obtained: varying t in the range of 12.5–15 m2 sϪ1 for a 10,000–12,000 m3 sϪ1 upstream discharge, the dif- ference in discharge along the minor branch is less than 3% of the solution as obtained by the physical model ͑Fig. 2͒, that is within the measurement and simulation rounding errors. In Fig. 2 the discharge in the Po di Goro branch in prototype ␯ condition as a function of t is shown, starting from its minimum ␯ value needed for computational stability. The influence of t is ␯ relevant: in fact, with the minimum t , the percentage discharge increase in the Po di Goro branch attains 20%, while, increasing ␯ t , a stable solution is obtained, being characterized by a progres- ␯ Fig. 2. Relationship between Po di Goro discharge and t : solid sive decrease of the diverted discharge. lines indicate Po di Goro discharge, depending on labeled upstream With reference to the water level, an increase in eddy viscosity ␰ ␯ discharge; diamonds indicate mean square error ratio ( t); and hori- increases the energy dissipation due to the Reynolds stress ͑1͒ and zontal dotted lines indicate the Po di Goro discharge as measured in vice versa. This effect must be considered in a global sense, hav- the physical model ␯ ing assumed a constant t over the computational domain. One ␯ must observe that small variations in t might cause locally rather large fluctuations of the water surface where high velocity gradi- ͑ ͒ Knight 1991͒. In the case of natural channels with variable depth, ents occur e.g., flow separation, localized phenomena, etc. so these values can increase, especially on the floodplain, depending that gauging points must be properly chosen to represent the over- on water depth differences; in such conditions, the Shiono and all free water surface. If this condition is fulfilled, higher or lower Knight ͑1991͒ experiments ascertain locally ␣ coefficient values water levels computed at gauging points, with respect to the ones 2 ␯ Ͼ␯* ␯ up to 10 . As a consequence ␣ is a geometry-dependent param- measured in the physical model, are expected if t t or t Ͻ␯ eter ͑Rodi 1993͒ and larger values than the ones detected in regu- t* , respectively. lar channels are expected in the river bifurcation analysis. With reference to the discharge separation, this is determined ␯ by momentum and force balance as applied to the bifurcation To avoid uncertainty in the choice of a t value, one could apply depth averaged physically based turbulence closures, e.g., a node depending on river geometry, water level, and discharge depth averaged k-␧ model ͑Rastogi and Rodi 1978͒. This ap- separation. One must note that, according to Boussinesq’s as- ␯ proach, when applied to river engineering problems as well as to sumption, t plays a different role in global energy losses in the large physical models, requires a considerably larger computa- two branches depending on flow dynamics. By considering con- tional effort, not only because additional equations of k and ␧ tinuity in the node, a change in eddy viscosity can produce the must be taken into account in the turbulence model in order to same absolute change in discharge rates along the branches, ob- ␯ viously with a different sign. In relative terms, this change affects evaluate the spatial distribution of t , but mainly because it re- quires so fine a grid as to obtain grid-independent solutions. Fur- more effectively the smaller branch ͑Po di Goro͒ than the larger ther complications may arise from the analysis of cases with a one ͑Po di Venezia͒ where 85–90% of the total discharge is ex- variable boundary due to drying-wetting areas or the presence of pected to flow. In the first instance, one can consider the momen- hydraulic control structures. As a consequence these models are tum and external forces conditions in the major branch practically not widely used in practical applications. Thus the choice of a unaffected by the change in eddy viscosity, thus focusing on the ␯ ␯ suitable t value, that takes into account the subgrid scale effects, minor branch. Here, a decrease in t will produce lesser dissipa- becomes not only a turbulence-model problem but also a matter tions at fixed discharge rates from the boundary section to the of numerical calibration and for this reason it is often assumed as node, so that discharge must increase in order to maintain mo- a constant ͑Hardy et al. 2000͒. A more refined calculation method mentum and force equilibrium at the node. ␯ could be inappropriate and also difficult to calibrate successfully For these same reasons, a change in the t parameter is not in numerical 2D models as applied to natural or large crucial for water levels, compared with the freeboard ͑1m͒ physical models that present a 3D behavior of the flow field, and water depth ͑10 m͒: in fact, variations less than 0.1 to 0.2 m complex geometry, heterogeneous nature of the boundary rough- were detected. ness, etc. ͑Shiono and Knight 1991͒. The mean mesh side varies in a range of 25–50 m in the main In Fig. 2, the solutions of the numerical model, as represented channel and 50–200 m on the floodplain. A mesh refinement was by the discharge in the Po di Goro branch, are reported as a adopted where larger gradients in flow field and water elevation ␯ function of t , in comparison with the observed values in the were expected, particularly in the surroundings of the branch ␯ physical model as scaled to the prototype size. The influence of inlet, as shown in Fig. 1. The sensitivity of t* to a mesh refine- ␯ the t on water surface elevation is computed according to ment was checked comparing the results obtained with two dif-

JOURNAL OF HYDRAULIC ENGINEERING © ASCE / MARCH 2004 / 239 ␯ Table 1. Change in t* with Mesh Dimensions Number of Number of Upstream discharge ͑m3 sϪ1͒ 6,000 8,000 10,000 12,000 elements nodes ␯ ͑ ͒ t* coarse mesh Fig. 1 3.0 11.5 15.0 13.0 2,088 4,318 ␯ t* fine mesh 1.6 10.0 13.0 12.0 4,761 9,492

ferent meshes: a coarse mesh with 4,318 nodes and 2,088 ele- 2D problems is questionable since they take into account in an ments ͑Fig. 1͒, and a fine mesh with 9,492 nodes and 4,761 overall manner channel cross section irregularities, , and ␶ ͑ ͒ elements, close to the maximum of 5,000 elements allowed by the meandering, while in 2D models, in the b evaluation 1 , the code. bottom roughness is mostly accounted for; for this reason, a lesser The results are reported in Table 1: the eddy viscosity values value in n should be expected. decrease, appreciably for the inflow discharge of 6,000 m3 sϪ1 This reduction should be more relevant for the main channel ͑approximately scaled with grid size͒, and only slightly with than for floodplains, where velocity is moderate and where the larger discharges. From the physical point of view, the eddy vis- equivalent roughness largely depends on vegetation ͑its height, cosity is proportional to a velocity scale Vˆ and a length scale L density, distribution, and type͒. The Manning’s coefficient was Ϫ characterizing the ͑large scale͒ turbulent motion depending on the assumed equal to 0.05 s m 1/3 for floodplains, as suggested in Reynolds number ͑Rodi 1993͒. Indicating with l a grid character- previous field studies ͑SIMPO 1980͒, while for the main channels istic length, a decrease in grid size as long as LϽl cannot produce some considerations are needed. ␯ The river bed geometry is described starting from surveyed appreciable changes in t* because the same subgrid effects are taken into account. Conversely, if LϾl, as long as the grid size cross sections at an average distance of about 200 m. This dis- decreases, smaller and smaller scales of turbulence are solved and tance prevents the recognition of bed forms, considering also that ␯ such surveys are generally made in drought conditions, so that the t* decreases too. These considerations can explain the results in Table 1. More- bed is described in terms of average elevation. As a consequence, over, these results are in agreement with the ones presented by the bed form contribution in the shear stress must be taken into King and Norton ͑1978͒ that show eddy viscosity coefficients account in the choice of the Manning coefficient for the channel. scaled approximately to the element size in analyzing tidal effects The van Rijn method ͑1984͒ gives bed form dimensions and the in the San Francisco Bay Delta System: that is a phenomenon effective hydraulic roughness of the bed forms from grain size ϭ characterized by a scale L larger than the grid dimension l. distribution. In our case, we have D50 0.2 mm and D90 ϭ Table 1, together with Fig. 2, points out a relative stability of 0.6 mm. Relatively to D90 grain size, the van Rijn method pro- ␯ vides nϭ0.016 s mϪ1/3 with a plane bed; in further steps, the t* with higher discharges; this allows the calibration of the nu- merical model using level data relative to nonextreme flood con- length and height are calculated and a correction of the plane bed ditions, and the study of the flow field in extreme design condi- roughness coefficient is derived. In our case, this correction is less tions, for which case data are generally unavailable. than 10% for the Po di Venezia branch, where the Froude number In order to define the influence of the Reynolds number on the during major floods is in the range of 0.25–0.3, close to a transi- eddy viscosity coefficient scaling the model according to the tional stage with washed-out . In the Po di Goro branch, Froude similarity, some numerical experiments were made using with a Froude number less than 0.2, the correction is of the order the same mesh with a geometrical scale ␭ varying between the prototype ͑␭ϭ1͒ and the model ͑␭ϭ100͒ scale. For each experi- ␯ ment the t*,model value was selected as the one that minimizes the mean square error between the levels, as measured in the physical model and scaled to ␭, and the computed levels, according to the criterion previously defined by Eq. ͑2͒. ␯ ␯ In Fig. 3 the ratio t*,model/ t*,prototype is shown versus ϭ␭Ϫ3/2 Rmodel /Rprototype according to the Froude similarity. From ␯ the figure one can observe a linear relationship of t* with the Reynolds number, so that the numerical model seems not to suffer from scale effects.

Roughness Coefficient Effects

For the evaluation of bed shear stresses, the Manning’s coefficient n must be assigned. This one does not depend on the mesh di- mensions and can be estimated based on physical considerations, within a limited range of variability. Values for the Manning’s coefficients are suggested in technical literature for 1D problems, based on surface roughness, vegetation effects, and water depth as well as on the basis of channel geometry ͑e.g., Chow 1959͒. Studies regarding the Po River in the study area ͑SIMPO Ϫ Fig. 3. ␯* 1980͒ give typical estimates of n of 0.05 s m 1/3 for the floodplain Relationship between the t and the Reynolds number ac- Ϫ cording to Froude similarity with ␭ equal to 1, 2, 10, 20, 40, 80, 100 and 0.025 s m 1/3 for the channel. The use of these coefficients in

240 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / MARCH 2004 Table 2. Influence of the Manning’s Coefficient in Diverted Discharge and Water Levels Upstream discharge ͑m3 sϪ1͒ 6,000 8,000 10,000 12,000 Main channel Po di Goro discharge m3 sϪ1 1,003 1,266 1,581 1,858 Ϫ1/3 ␰ ␯ nϭ0.017 s m ( t) 1.0 1.0 1.0 1.0 ͑maximum deviation͒ ͑0.03 m͒ ͑0.05 m͒ ͑0.06 m͒ ͑0.08 m͒

Main channel Po di Goro discharge m3 sϪ1 1,017 1,285 1,632 1,937 nϭ0.025 s mϪ1/3 ͑percentage error͒ ͑1.3%͒ ͑1.5%͒ ͑3.2%͒ ͑4.2%͒ ␰ ␯ ( t) 7.4 7.7 2.8 2.8 ͑maximum deviation͒ ͑0.08 m͒ ͑0.10 m͒ ͑0.18 m͒ ͑0.12 m͒

of 30–40%. As a consequence, the Manning coefficient for chan- depend on the Reynolds number: in the main channel, where Ϫ nels ranges between 0.016 and 0.022 s m 1/3. In the physical water depth and mean flow velocity are in the range of 10–14 m model, using the Colebrook White formula, a Manning coefficient and 1.3–2 m sϪ1, respectively, the Reynolds number lies in the Ϫ equal to 0.013 s m 1/3 could be estimated. range of 13ϫ106 –28ϫ106. In a 1:100 model, the Reynolds num- In the comparison between the results of the physical and the ber scales in the range of 13ϫ103 –28ϫ103. numerical models at the same scale, the value of n that minimizes These differences in Reynolds number can produce scale ef- the mean square error according to Eq. ͑2͒ was found to be 0.017 fects in the application of the physical model results. Field sur- Ϫ1/3 sm , within the range of physical acceptance. One must recog- veys ͑Desiderio 1984͒, as compared to the physical model repro- nize that, according to the Froude similarity, the Manning’s coef- ducing the actual river conditions, show a difference in diverted 1/6 Ϫ1/3 ficient is scaled with ␭ ; so that nϷ0.013 s m in the physical discharge of the order of 10%. This discrepancy can give an idea ͑ ␭ϭ ͒ Ϸ Ϫ1/3 model with 100 corresponds to n 0.025 s m in the pro- of the physical model approximations, considering also the errors totype. in field measurement procedures when applied to large flood con- For these reasons, sensitivity of the numerical model, both at ditions. Conversely, the numerical model does not suffer from the prototype scale and at the physical model scale, to a change scale effects as shown in Fig. 3, where the eddy viscosity coeffi- in the Manning’s coefficient was tested in the range of n cient scales with ␭Ϫ3/2 according to the Froude condition. This ϭ Ϫ1/3 0.017– 0.025 s m , covering uncertainty in roughness related result is a reconfirmation of the traditional estimation of the con- to the prototype and the physical model, taking into account the stant eddy viscosity coefficient as proportional to the product of a roughness scale law. The same results, scaling with the Froude characteristic flow velocity and hydraulic depth. The problem is similarity, were obtained, as expected with reference to the results the local variability of the proportionality coefficient ␣, particu- reported in Fig. 3. larly difficult for cases of complex geometry. For example, even For the two extreme values of the Manning’s coefficient, with in the Shiono and Knight ͑1991͒ experiment regarding a relatively reference to the prototype, Table 2 reports: ͑1͒ the discharge rate ͑ ͒ simple two stage straight channel, the dimensionless eddy viscos- in the Po di Goro branch; 2 the dimensionless mean square error ␣ in water levels according to Eq. ͑2͒; and ͑3͒ the absolute deviation ity parameter increases exponentially on the side-slope domain, between simulated and measured water levels. The analysis of the in the presence of secondary flow, and attains constant but larger ␣ results shows that, for practical purposes, the Manning’s coeffi- values on the flood plain than in the main channel where is cient has a negligible effect on discharge partition and water lev- close to 0.07. Therefore in application to river reaches where els; the change in discharge rate is of the order of some percent complex geometry, heterogeneous nature of boundary roughness, while the maximum difference in water level is of the order of 0.1 and uncertainty in assigning values to the boundary roughness m. This limited effect on water levels can be expected as a con- coefficients are expected, and averaging of parameters over a sequence of the limited longitudinal extension of the reach, where large area in a grid representation of the flow physics is needed, ␯ head losses are mainly due to localized and geometry-dependent the use of the eddy-viscosity models with constant t might be a phenomena taken into account by turbulent stresses. convenient alternative, especially when the main purpose is to evaluate global quantities ͑e.g., the discharge bifurcation rate͒. Furthermore, in this assumption, the calibration procedure is Discussion and Conclusion greatly simplified. As long as the behavior of the eddy viscosity coefficient is known for a particular case, the scale ␭Ϫ3/2 could be The above reported analysis illustrates some of the problems that used to produce an estimate of the eddy viscosity coefficient in affect physical and numerical models in reproducing river bifur- application to cases almost similar in geometry. cations. The principal problems related to physical models regard The absence of scale effects in the eddy viscosity coefficient roughness similarity and Reynolds number. As related to the main due to a change in Reynolds number cannot be extended a priori channels, where discharge is mostly concentrated, values in the to other types of closure models and principally to the physical physical model are of the same order of magnitude of the corre- models, where the flow is 3D and all the turbulence scales are sponding roughness coefficients as estimated in the prototype. In implicitly solved. The physical model and the numerical model, this condition, a moderate change in roughness seems to have a though affected by different, independent, sources of approxima- limited impact on water levels, caused by the limited length of the tions, seem to produce consistent results; this fact allows us to study reach, as well as on discharge separation, as pointed out in conclude that, reasonably, their results should have the same order Table 2. In effect, for the analysis of localized phenomena ͑e.g., of accuracy. river junction, bifurcation, etc.͒ the bed shear stresses are less The constant eddy viscosity commercial numerical models, as important than the turbulent stresses ͑1͒. The turbulent stresses FESWMS-2DH, give acceptable results for engineering purposes,

JOURNAL OF HYDRAULIC ENGINEERING © ASCE / MARCH 2004 / 241 ␯ ␯ ͑ ͒ but need calibration, with reference to t . The influence of t in Khan, A. A., Cadavid, R., and Wang, S. S. Y. 2000 . ‘‘Simulation of discharge separation is highlighted in Fig. 2. This fact could have channel confluence and bifurcation using the CCHE2D model.’’ Proc. a relevant influence in flood risk evaluation along the downstream Inst. Civ. Eng., Waters. Maritime Energ., 142, 97–101. ͑ ͒ branches. A proper choice of this parameter can be performed: ͑1͒ King, I. P., and Norton, W. R. 1978 . ‘‘Recent application of RMA’s using known results for similar conditions ͑this is the case of the finite element models for two dimensional hydrodynamics and water quality.’’ Finite elements in water resources II, Pentech, London, present application, when extended to other Po Delta’s branches͒; ͑ ͒ ͑ ͒ 2.81–2.99. and 2 using river stage data according to Eq. 2 as shown in Ramamurthy, A. S., Carballada, L. B., and Tran, D. M. ͑1988͒. ‘‘Com- Fig. 2. The use of literature parameters, without any other infor- bining open channel flow at right angled junctions.’’ J. Hydraul. Eng., mation, might bring to light important inaccuracies in the results 114͑12͒, 1449–1460. because of their large range of variability ͑King and Norton Ramamurthy, A. S., and Satish, M. G. ͑1988͒. ‘‘Division of flow in short 1978͒. Once properly calibrated, commercial 2D numerical mod- open channel branches.’’ J. Hydraul. Eng., 114͑4͒, 428–438. els might prove to be competitive with the physical ones in the Ramamurthy, A. S., Tran, D. M., and Carballada, L. B. ͑1990͒. ‘‘Dividing analysis of river systems. flow in open channels.’’ J. Hydraul. Eng., 116͑3͒, 449–455. Rastogi, A. K., and Rodi, W. ͑1978͒. ‘‘Predictions of heat and mass trans- fer in open channels.’’ J. Hydraul. Div., Am. Soc. Civ. Eng., 104͑HY3͒, 397–420. References Rodi, W. ͑1993͒. Turbulence models and their application in hydraulics—A state of the art review, International Association for Chow, V. T. ͑1959͒. Open channel hydraulics, McGraw-Hill, New York. Hydraulic Research. Desiderio, A. ͑1984͒. ‘‘Po di Goro: a useful Po Delta branch.’’ Proc. II Shettar, A. S., and Murthy, K. K. ͑1996͒. ‘‘A numerical study of division Convegno di idraulica padana, Parma, 113–136 ͑in Italian͒. of flow in open channels.’’ J. Hydraul. Res., 34͑5͒, 651–675. FESWMS-2DH, version 2. ͑1988͒. Turner-Fairbank Highway Research Shiono, K., and Knight, D. W. ͑1991͒. ‘‘Turbulent open-channel flows Center, McLean, Va. with variable depth across the channel.’’ J. Fluid Mech., 222, 617– Gurram, S. K., Karki, K. S., and Hager, W. H. ͑1997͒. ‘‘Subcritical junc- 646. tion flow.’’ J. Hydraul. Eng., 123͑5͒, 447–455. SIMPO. ͑1980͒. ‘‘Study and preliminary design of the hydraulic regula- Hardy, R. J., Bates, P. D., and Anderson, M. G. ͑2000͒. ‘‘Development of tion of the River Po main channel for civil and environmental pur- a reach scale two dimensional finite element model for floodplain poses.’’ Magistrato per il Po—SIMPO Spa. ͑in Italian͒. sediment .’’ Proc. Inst. Civ. Eng., Waters. Maritime Energ., van Rijn, L. C. ͑1984͒. ‘‘, part III: Bed forms and 142, 141–156. alluvial roughness.’’ J. Hydraul. Eng., 110͑2͒, 1733–1754.

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