Source: ICHE Conference Archive - https://mdi-de.baw.de/icheArchiv

Iwakoke, Yasuhiro

Kawahara, Yoshihisa

Morishita, Yu

Uchida, Tatsuhiko

English

Hydraulic Engineering Repository

Conference Paper, Published VersionKawahara, Yoshihisa; Morishita, Yu; Uchida, Tatsuhiko; Iwakoke,YasuhiroAccurate Estimation of Discharge Hydrograph in aCompound Meandering Channel Using a Two-DimensionalNumerical ModelZur Verfügung gestellt in Kooperation mit/Provided in Cooperation with:Kuratorium für Forschung im Küsteningenieurwesen (KFKI)Verfügbar unter/Available at: https://hdl.handle.net/20.500.11970/110244Vorgeschlagene Zitierweise/Suggested citation:Kawahara, Yoshihisa; Morishita, Yu; Uchida, Tatsuhiko; Iwakoke, Yasuhiro (2008): AccurateEstimation of Discharge Hydrograph in a Compound Meandering Channel Using a Two-Dimensional Numerical Model. In: Wang, Sam S. Y. (Hg.): ICHE 2008. Proceedings of the8th International Conference on Hydro-Science and Engineering, September 9-12, 2008,Nagoya, Japan. Nagoya: Nagoya Hydraulic Research Institute for River Basin Management.Standardnutzungsbedingungen/Terms of Use:Die Dokumente in HENRY stehen unter der Creative Commons Lizenz CC BY 4.0, sofern keine abweichendenNutzungsbedingungen getroffen wurden. Damit ist sowohl die kommerzielle Nutzung als auch das Teilen, dieWeiterbearbeitung und Speicherung erlaubt. Das Verwenden und das Bearbeiten stehen unter der Bedingung derNamensnennung. Im Einzelfall kann eine restriktivere Lizenz gelten; dann gelten abweichend von den obigenNutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte.Documents in HENRY are made available under the Creative Commons License CC BY 4.0, if no other license isapplicable. Under CC BY 4.0 commercial use and sharing, remixing, transforming, and building upon the materialof the work is permitted. In some cases a different, more restrictive license may apply; if applicable the terms ofthe restrictive license will be binding.ACCURATE ESTIMATION OF DISCHARGE HYDROGRAPHIN A COMPOUND MEANDERING CHANNELUSING A TWO-DIMENSIONAL NUMERICAL MODELYoshihisa Kawahara1, Yu Morishita2, Tatsuhiko Uchida3and Yasuhiro Iwakoke41Professor, Department of Civil and Environmental Engineering, Hiroshima UniversityKagamiyama, Higashi-Hiroshima, 739-8527, Japan, e-mail: kawahr@hiroshima-u.ac.jp2Engineer, River Engineering Division, Pacific Consultants Co., LTDAzuchimachi, Chuo-ku, Osaka, 541-0052, Japan, e-mail: yuu.morishita@os.pacific.co.jp3Associate Professor, Research and Development Initiative, Chuo University1-13-27 Kasuga, Bunkyo-ku, Tokyo, Japan, email; utida@tamacc.chuo-u.ac.jp4Graduate Student, School of Engineering, Hiroshima UniversityKagamiyama, Higashi-Hiroshima, 739-8527, Japan, e-mail: m082392@hiroshima-u.ac.jpABSTRACTAn accurate estimation of flood discharge hydrographs has been one of the mostimportant issues in River Engineering. This paper explores a numerical estimation method ofdischarge hydrograph at arbitrary cross sections by using 2-d shallow water equations andwater level hydrographs measured at gauging stations. The numerical method adopts CIP-CLS2 scheme for convective terms in the shallow water equations. It is applied to anexperiment on unsteady flow in a compound meandering channel. The comparison betweenthe experiment and the numerical simulation reveals that the propagation speed of flood peakcan be used to estimate the flow resistance in the channel and thereby the present method canreproduce the discharge hydrograph with high accuracy.Keywords: flood discharge hydrograph, water level hydrograph, 2-d numerical simulation,compound meandering channel, CIP-CLS2 scheme1. INTRODUCTIONA compound channel, consisting of main channel and floodplains, is a representativeshape of large rivers in Japan. It is known that the attenuation of peak discharge and thedeformation of discharge hydrograph in meandering compound channels are significantlyaffected by river channel shape and floodplain roughness, which is necessary to be clarifiedfor river improvement plans.Fukuoka et al. (2004) paid special attention to the accuracy of and accessibility towater level hydrographs and proposed a methodology to estimate flood discharge hydrographwith a 2-D numerical method coupled with water level hydrographs along the river reach andat the upstream and downstream boundaries. They demonstrated that their approach can givebetter estimation of discharge hydrograph at an arbitrary cross section when the floodplainroughness and the permeability of vegetation are well predicted, which is realized in theirmethod to reproduce the discharge hydrograph observed at a station.Recently Uchida and Kawahara (2006) developed a 2-D numerical model for unsteadyshallow flows on the Cartesian coordinate system with the CIP-CLS2 scheme originallyproposed by Nakamura et al. (2001). Uchida and Kawahara demonstrated that their numericalmodel works very well for complex flows even with coarse grids.In this study, following Fukuoka et al. (2004), we develop an estimation method fordischarge hydrograph using water level hydrographs at gauging stations. Our method,however, differs from that of Fukuoka et al. in two points. First, our method does not use anyobserved discharge hydrograph but uses the propagation speed of flood peak to estimate theflow resistance in the river reach. Second, it also uses the 2-D numerical model by Uchida-Kawahara (2006). This paper consists of three parts; 1) the description of the experiment onunsteady flow in a compound meandering channel (Morishita et al., 2006), which is used tovalidate our method, 2) the explanation of our numerical model and 3) the application of ourmodel to the experiment to show that the floodplain roughness can be well estimated from theobserved propagation speed of the flood peak and that thereby discharge hydrograph can becalculated accurately.2. EXPERIMENTAL SETUP AND CONDITIONSFigure 1 shows the compound meandering channel used in this study. The channel has22.5 m in length, and its slope is fixed at 1/1,000. It has a smooth main channel and adjacentroughened floodplains. The height of the floodplains is 5.2cm including the artificial turf onthe floodplains. The roughness coefficients of the main channel and the floodplains arenmc=0.010 and nfp=0.021, respectively. It has five waves of meandering reaches with asinuosity of 1.10.We repeatedly generate the discharge hydrograph of an isosceles triangle to measurethe time variation in water level at nine cross-sections shown in Figure 1. Three velocitycomponents are measured by two-component electromagnetic current meters in the reachbetween sections 6 and 7. The gate at the downstream end is automatically controlled tomaintain a constant water surface slope in the measuring reach throughout the flood.3. NUMERICAL MODELUchida and Kawahara (2006) developed a 2-D numerical simulation model tosimulate inundation flows in urban areas which have complex structures and boundaryconditions. They demonstrated that the model can give more accurate results thanconventional methods even with relatively coarse numerical grids on Cartesian coordinatesystem.The governing equations of two-dimensional unsteady flows in a compound(c) Discharge hydrographat upstream end6810121416180 500 1000 1500 2000Time (s)Discharge(l/s)Figure 1 Experimental setup and condition.(b) Cross-sectionArtificial turf (0.7 cm)5.2cm(a) Plan viewAutomatic controlgate22.5m2.2mControl section of water surface profile41 2 3 5 6 7 8 9Flow 0.5m widthmeandering channel are described by the following shallow water equations0jjxfhutfh(1)jijiijijixhhhxgxhfuuthfufh   01 (2)where subscripts i,j follow the summation convention, indicating 1 = x and 2 = y, respectively; h =water depth; ui = velocity in the xi direction; f= occupancy ratio of fluid; g = acceleration due togravity;  = water surface elevation (h+z); z = ground elevation; 0i=bed shear stress ; and  ij=horizontal shear stress tensor due to diffusion and dispersion. In this study, bed shear stress isexpressed by the Manning equation. The Reynolds stresses are represented by an eddy-viscosity model.Figure 2 shows a computational grid consisting of permeable and impermeable areastogether with the arrangement of main variables in the grid. All the variables in the governingequations are set at the same location. Each control volume ij has three kinds of variables, i.e.,the value at the intersection of the grid (Point Value (P), denoted by lower-case characters),the averaged value along the side of the grid (Line-averaged Value (Lxa, Lya), upper-casedwith subscript of x or y) and the averaged value over the grid (Area-averaged Value (Aa),shown with capital letters with subscript of xy). The staggered grid systems usually requirethe interpolation of the variables in many numerical schemes including the original CIPscheme whereas in the present scheme no interpolation is necessary. Many types of boundaryconditions can be taken into account by using the three kinds of valuables. In the original CIPscheme for the shallow water equations, the advection terms are transformed fromconservative form into non-conservative form to eliminate water depth h in the advectionterms with the help of the Continuity equation (1). In the present scheme, time variation of thedepth-integrated momentums and water depths are computed directly by CIP-CSL2(Nakamura et al., 2001) without the transformation of the advection terms.The grid size used in the study is 10 cm×5 cm and the time step is 0.005 sec. Thewater levels are changed in the imaginary ponds set at the upstream and downstream ends insuch a way that the observed water levels are reproduced at the upstream and downstreamends of the calculation domain.4. NUMERICAL RESULTS AND DISCUSSIONSxxFyij yF jxi 1yImpermeable wallyxFxyij yFxij yijUjyiU 1xijV1ijij ji 1ji 1ijxijy1ijijyijyijyijyxFHVUaL ,,,:xijyijyijyijy FVUHaL ,,,:xyijxyijxyijijFUHVAa,,:ijijijij fhuP ,,,: Figure 2 Computational grids with impermeable walls and the arrangement ofmain variables on the volume ij.Comparison with experimental resultsFigure 3 shows the experimental andcomputational results of the longitudinal water levelprofiles in rising and falling stages with thefloodplain roughness of 0.021. We divide the floodhydrograph into three parts, which are Stage 1,Stage 2, and Stage 3, based on the relative depth Dr(defined as the ratio of depth over the flood plainsto that in the main channel). Main emphasis isplaced on the numerical results in Stage 2 where theeffects of the flow interaction between mainchannel and floodplains are strong and the effect ofthe artificial turf over the floodplains is significant.Figure 4 also compares the calculated resultsand the observed ones of discharge and water level hydrographs in cross-sections 1, 5, and 9.The computational results agree fairly well with the experimental ones, particularly in Stage 2.Figure 5 depicts the depth averaged velocity distributions at t=880 sec near the peak ofthe flood. The calculated velocity vectors show good agreement with the measured onesexcept near the interface between main channel and floodplains where the flow field becomehighly three-dimensional.Waterlevel(cm)45678910110 500 1000 1500 2000ExperimentCalculationHeight of flood plainDistance from downstream end (cm)Waterlevel(cm)45678910110 500 1000 1500 2000Height of flood plainExperimentCalculationDistance from downstream end (cm)Figure 3 Time change in longitudinal profile of water level.(Left : Rising stage, Right : Falling stage)45678910110 500 1000 1500 2000Stage 1 (Dr<0.3) Stage 2 (Dr>0.3) Stage 3 (Dr<0.3)Times (s)Waterlevel(cm)Figure 4 Discharge and water level hydrographs.(Left: Discharge, Right: Water level)6810121416180 500 1000 1500 2000Times (s)Stage 1 (Dr<0.3) Stage 2 (Dr>0.3) Stage 3 (Dr<0.3)Discharge(l/s)ExperimentCalculationCross section : 5 991Cross section : 1 5Experiment CalculationFigure 5 Comparison of depth-averaged velocity.20(cm/s)Sec6 Sec7Estimation of floodplain roughnessWe carry out numerical calculations fornine cases to estimate the effects of the floodplain roughness on the longitudinal water levelprofiles, the water level and the dischargehydrographs. Table 1 summarizes the ninecases where the floodplain roughness nfp areconsistently changed around 0.021 in Run 5,because it leads to the best agreement with themeasured data.Figure 6 shows the longitudinal profilesof water levels in rising and falling stages fortwo values of nfp. In Figure 6 it is confirmedthat the longitudinal water level profiles are notmuch affected by the different value ofnfp since the water levels at boundariesare the same in all cases.Figures 7 demonstrate thedischarge and the water levelhydrographs in cross section 5 for severalvalues of nfp. It is clear that the dischargein cross-section 5 changes significantlyaccording to the value of nfp, particularlyin Stage 2 near the flood peak.Meanwhile, the water level hardlychanges irrespective of the change infloodplain roughness.We scrutinize the sensitivity ofthe discharge and the water level to nfp.Figure 8 shows the relations between therelative errors in discharge and waterlevel and the relative error of nfp. Thecomputational result in Run 5 is treatedas the reference and the deviations from itin the other cases are regarded as errors.Figure 6 Time change in longitudinal waterlevel profiles.Waterlevel(cm)Waterlevel(cm)45678910110 500 1000 1500 200045678910110 500 1000 1500 2000nfp=0.017 Rising stage nfp=0.017 Falling stageExperimentCalculation45678910110 500 1000 1500 200045678910110 500 1000 1500 2000nfp=0.025 Rising stage nfp=0.025 Falling stageExperimentCalculationExperimentCalculationExperimentCalculationTable 1 Computational conditionsFloodplainroughness nfpChange comparedwith Run 5 (%)Run 1 0.010 -52.4Run 2 0.017 -19.0Run 3 0.019 -9.5Run 4 0.020 -4.8Run 5 0.021 0.0Run 6 0.022 4.8Run 7 0.023 9.5Run 8 0.025 19.0Run 9 0.030 42.9Figure 7 Effects of floodplain roughness on discharge and water level hydrographs.(a) Discharge hydrographDr > 0.3 Dr > 0.3(b) Water level hydrograph681012141618200 500 1000 1500 2000nfp=0.017nfp=0.020nfp=0.021nfp=0.022nfp=0.025Time (s)Discharge(×10-3m3/s)Waterlevel(cm)Time (s)56789100 500 1000 1500 2000nfp=0.017nfp=0.020nfp=0.021nfp=0.022nfp=0.025Waterlevel(cm)Waterlevel(cm)The relative error of discharge increases linearly from -10% to 20% in Stage 2 when the errorof nfp varies from 20% to -20% in Stage 2. Contrarily the sensitivity of the water level to nfp ismuch smaller than that of discharge. This means that even if the calculated water level showsgood agreement with the measured one with accuracy of less than 1%, the accuracy ofdischarge is not high. For this reason, the information on observed discharge at a cross-sectionis important to estimate the discharge hydrograph with high degree of accuracy, which is thebasic concept of Fukuoka et al. (2004).Figure 9 shows the relationship between the occurrence time of peak water level andthe location, where the distance and the time are non-dimensionalized by the distance betweenSections 1 and 9 and by the time difference of the peak water level at Sections 1 and 9,respectively. We can obtain this relationship from water level hydrographs at multiple pointsalong the channel and we do not need measured discharge data. Hence it can be said that if wecan determine the floodplain roughness through the tuning of the occurrence time of waterlevel, we can calculate the discharge hydrograph with enough accuracy. Naturally the flooddischarge data taken at some water depth, not at its peak, are useful to estimate the magnitudeof floodplain roughness.5. CONCLUSIONSWe have developed a numericalestimation method of dischargehydrograph at arbitrary cross sectionsby using 2-d shallow water equationsand water level hydrographs measuredat multiple locations. The numericalmethod adopting CIP-CLS2 scheme isapplied to an unsteady flow in acompound meandering channel. Thecomparison between the calculatedresults and observed ones reveals thatthe propagation speed of flood peak-1.0-0.50.00.51.0-60 -40 -20 0 20 40 60-0.3-0.2-0.10.00.10.20.3-60 -40 -20 0 20 40 60-6-4-20246-60 -40 -20 0 20 40 60-0.6-0.4-0.20.00.20.40.6-60 -40 -20 0 20 40 60-60-40-200204060-60 -40 -20 0 20 40 60-10-50510-60 -40 -20 0 20 40 60Relative error of nfp (%) Relative error of nfp (%) Relative error of nfp (%)(a) Stage 1 (b) Stage 2 (c) Stage 3Figure 8 Relation between the error in discharge/water level andthe error in nfp.(a) Stage 1 (b) Stage 2 (c) Stage 3Relative error of nfp (%) Relative error of nfp (%) Relative error of nfp (%)Relativeerroraverageofdischarge(%)Relativeerroraverageofdischarge(%)Relativeerroraverageofdischarge(%)Relativeerroraverageofwaterdepth(%)Relativeerroraverageofwaterdepth(%)Relativeerroraverageofwaterdepth(%)00.20.40.60.810 0.5 1 1.5 2nfp=0.021nfp=0.017nfp=0.025Figure 9 Relation between the time of waterlevel peak and the floodplain roughness.Section 1Section 9Non-dimensionalized timeNon-dimensionalizeddistancefromthedownstreamendcan be used to estimate the floodplain roughness and thereby the present method canreproduce the discharge hydrograph with high accuracy. The observed data of discharge areuseful to determine the floodplain roughness but are not always necessary with the presentmethod.REFERENCESFukuoka, S., Watanabe, A., Hara, T., Akiyama, M. (2004), Highly accurate estimation ofhydrograph of flood discharge and water storage in rivers by using an unsteady2ddimensional flow analysis based on temporal change in observed water surface profiles,Journal of Hydraulics, Coastal and Environmental Engineering, JSCE, No. 761/II-67, pp.45-56, in Japanese.Uchida, T., Kawahara, Y. (2006) Development of an explicit conservative CIP scheme forshallow water flows, Journal of Applied Mechanics, JSCE, Vol. 9, pp. 917-924, inJapanese.Nakamura, T., Tanaka, R., Yabe, T., and Takizawa, K. (2001), Exactly conservative semi-Lagrangian scheme for multi-dimensional hyperbolic equations with directional splittingtechnique, Journal of Computational Physics, Vol.174, pp. 171–207.Morishita, Y., Watanabe, A., Uchida, T., Kawahara, Y. (2006) Flood flow structure duringrising and falling stages in a compound meandering channel, Journal of Applied Mechanics,JSCE, Vol.9, pp.977-985, in Japanese.

Accurate Estimation of Discharge Hydrograph in a Compound Meandering Channel Using a Two-Dimensional Numerical Model

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Accurate Estimation of Discharge Hydrograph in a Compound Meandering Channel Using a Two-Dimensional Numerical Model

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