Fukuoka, Shoji

Takemura, Yoshiharu

English

Hydraulic Engineering Repository

Conference Paper, Published VersionTakemura, Yoshiharu; Fukuoka, ShojiEvaluation Method of Positive and Negative RetardingStorage Volumes for Unsteady Two Dimensional Flows andPropagation Mechanisms of Peak Discharge and PeakWater LevelZur Verfügung gestellt in Kooperation mit/Provided in Cooperation with:Kuratorium für Forschung im Küsteningenieurwesen (KFKI)Verfügbar unter/Available at: https://hdl.handle.net/20.500.11970/99445Vorgeschlagene Zitierweise/Suggested citation:Takemura, Yoshiharu; Fukuoka, Shoji (2014): Evaluation Method of Positive and NegativeRetarding Storage Volumes for Unsteady Two Dimensional Flows and PropagationMechanisms of Peak Discharge and Peak Water Level. In: Lehfeldt, Rainer; Kopmann,Rebekka (Hg.): ICHE 2014. Proceedings of the 11th International Conference onHydroscience & Engineering. Karlsruhe: Bundesanstalt für Wasserbau. S. 303-312.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. 1 INTRODUCTION  When floods occur in river reaches, channel areas are temporally filled with flood water and they are gradually released to downstream. Discharge hydrographs of flood flows passing through river channels change in shapes through this process and it causes attenuation of peak discharge, delay in propagation and difference between arrival times of the peak discharge and the peak water level at river cross sections (Takahashi (1971) and Menedez et.al. (1982)). They are known as the storage effects of river channels and largely affected by unsteadiness of flood flows and channel geometries such as cross sectional shapes or its longitudinal changes (Henderson (1966) & Fukuoka (2003)).  Understanding of the storage effects of river channels is a great interest of river engineers or research-ers and many fundamental studies are proposed (e.g. Lighthill and Whitham (1955), Ponce and Simons (1977)). However, applications of these studies are limited to the specific problems in rivers because the most of studies are proposed under the ideal conditions (e.g. uniform and quasi-steady flow condition or linear stability analysis). Now, flood propagations in river channels can be analyzed with high accuracy for the development of the observation technique and numerical analysis method (e.g. Fukuoka and Watanabe (2004)). How to evaluate the storage effects of river channels during flood events by using the numerical analysis results becomes an important problem. Mishra et.al. (1997) adopts the area of the loop of the rating curve (discharge - water level curve) in non-dimensional form as the parameter to explain the propagation characteristics of flood flows passing through river reaches in their study. However, the rat-ing curve cannot represent the amount of the deformation of discharge hydrographs in the river channels which is the dominant factor to evaluate the storage effects of the river channels quantitatively because it represents only the discharge - water level relationship at each river cross section during flood events. Figure 1 shows the discharge hydrographs at upstream and downstream ends of a river reach. The dis-charge hydrograph deforms from the black broken line to the gray solid line in the reach. Areas surround-Evaluation Method of Positive and Negative Retarding Storage Volumes for Unsteady Two Dimensional Flows and Propagation Mechanisms of Peak Discharge and Peak Water Level Y. Takemura & S. Fukuoka Research and development initiative, Chuo University, Japan ABSTRACT: Understanding of the storage effects of river channels, such as attenuation of peak dis-charge, delay in propagation and difference between arrival times of peak discharge and peak water level at river cross sections during flood events, has been important for river management and improvement. This paper defines the parts of flood water stored in river channels during the rising period and flood wa-ter released from river channels during the descending period that cause deformation of discharge hydro-graphs as positive retarding storage and negative retarding storage of flood water, respectively. The eval-uation method of the positive and negative retarding storage volumes in river channels is proposed by deriving the advective equation for flood discharge from the unsteady two dimensional flow equations. In addition, the mechanism causing the difference between arrival times of peak discharge and peak water level at river cross sections is discussed with relation to the positive retarding storage around the flood peak time. Finally, the evaluation method is applied to a large flood in the valley of the upper Kitakami River to clarify the storage effects of the valley.  Keywords: Flood propagation, Storage, Retarding storage, River geometries, Discharge hydrograph, Water level hydrograph, Valley ICHE 2014, Hamburg - Lehfeldt & Kopmann (eds) - © 2014 Bundesanstalt für Wasserbau ISBN 978-3-939230-32-8303( ) ( )− ∂∂ ∂ ∂ ∂ = + = + − − ∂ ∂ ∂ ∂ ∂  ∫rlyyu U uhDQ Q Q u uU h vh u U dyDt t x t y x  ( ) ( )2 1= =∂∂− − + −∂ ∂lry y y yyyu U uh u U uhx x  (3) where , ,     = = =∫ ∫r rl ly yy yQQ uhdy  A hdy  UA  (4) Q = discharge, A = cross sectional area, U= mean velocity, yl and yr = positions of water edges at rive cross sections. As the result, second term of the right side of Equation (2) is canceled out and third - fifth terms are appeared on the right side of Equation (3). These three terms represent the effects of velocity distributions in river cross sections on the rate of change in Q. Equation (3) can be written in the more simplified form      ∂ ∂ ∂ = + = ∂ ∂ ∂ DQ Q Q UU ADt t x t  (5) The rate of change in Q can be represented by only the product of partial derivative ∂U/∂t and flow cross sectional area A when we follow the mean motion of a fluid. This indicates that deformation of discharge hydrographs in river channels is controlled by temporal changes in mean velocity at river cross sections.  2.2 Evaluation method of the positive and negative retarding storage volumes Rate of storage in a river reach is represented by the difference between inflow and outflow discharge of the reach or L∂= ∂∫dS A dxdt t  (6) where S = storage volume in the river reach, L = longitudinal distance of the river reach. Differentiating discharge Q (=UA) respect to time, ∂A/∂t on the right side of Equation (6) can be written 1∂ ∂ ∂= −∂ ∂ ∂A Q A U  t U t U t  (7) It can be easily understood with relation to Equation (5) that the second term of the right side of Equation (7) (A/U)∂U/∂t represents the rate of storage per unit distance due to the deformation of discharge hydro-graphs. To assign a meaning of the first term (1/U)∂Q/∂t, we think the case that mean velocity does not temporally change throughout river reaches; i.e., discharge hydrographs are unchanged in shape as it moves downstream since the right side of Equation (5) A∂U/∂t is 0. In this case, the difference of arrival time of the discharge hydrograph between two river cross sections is only the factor causing storage in river channels. Then rate of storage per unit distance ∂A/∂t is equal to (1/U)∂Q/∂t since the second term of Equation (7) (A/U)∂U/∂t is 0. It allows us that the first term of the right side of Equation (7) (1/U)∂Q/∂t can be taken as the rate of storage per unit distance due to the difference of the arrival time of the dis-charge hydrograph.  From the above discussions, it can be understood that the positive and negative storage volumes de-fined as the parts of the storage and release volumes in the river reaches that cause the deformation of the discharge hydrographs (hatched areas in Figure 1) are produced by the second term of the right side of Equation (7). The rate of the positive and negative retarding storage volumes per unit distance are written as 1 11 0,  δ   at the rising period0 0 ∂ ≤  ∂∂ ∂   = − = ∂ ∂ ∂  >  ∂ srU  tr A Uδt U t U t  (8) 305Qx-dx/2:Discharge hydrograph at x-dx/2 section, Qx+dx/2:Discharge hydrograph at x+dx/2 section, Q’x+dx/2:Discharge hydrograph at x+dx/2 sec-tion (unchanged in the shape), TpQ: Arrival time of the peak discharge, Tph: Arrival time of the peak water level  Figure 2. Discharge hydrographs around the flood peak time at river cross sections infinitesimal space dx away from x section in upstream and downstream directions. 2 21 0,δ   at the descending period0 0 ∂ ≥  ∂∂ ∂   = − = ∂ ∂ ∂  <  ∂ sdU  tr A Uδt U t U t  (9) where rsr = positive retarding storage volume per unit distance, rsd = negative retarding storage per unit distance, δ1 and δ2 = response functions.  Figure 2 shows the discharge hydrographs around the flood peak time at river cross sections infinites-imal space dx away from x section in upstream and downstream directions. The times TpQ and Tph are the arrival times of peak discharge and peak water level at x section, respectively. The rate of storage and the rate of positive retarding storage in the space dx are represented by ∂A/∂tdx, ∂rsr/∂tdx, respectively. Con-sidering the right side of Equation (7), the first term is less than 0 (i.e., (1/U)∂Q/∂t < 0) and second term is equal to the rate of positive retarding storage (i.e., -(A/U)∂U/∂t = ∂rsr/∂t) from the time TpQ to the time Tph. Therefore, the rate of positive retarding storage is larger than the rate of storage (i.e., ∂rsr/∂tdx > ∂A/∂tdx) in this time as shown in Figure 2. This indicates that the volume of the positive retarding storage obtained by integrating Equation (8) respect to time during the rising period overestimates the volume surrounded by the points a, b and c in Figure 2. The overestimated volume can be eliminated by using Equation (10) instead of Equation (8). 1 11 0min , ,  δ0 0 ∂ ≤  ∂∂ ∂ ∂    = − =  ∂ ∂ ∂ ∂    >  ∂ srU  tr A U Aδt U t t U t  (10) Generally, it can be neglected because the overestimated volume would be considered sufficiently smaller than the total volume of the positive retarding storage in river reaches. However, Equation (10) should be used to discuss the positive retarding storage around the flood peak time. Integrating Equation (8) (or Equation (10)) and Equation (9) respect to the length of river reaches L, we obtain the rate of the positive retarding storage and negative retarding storage in the river reaches Ld∂= ∂∫sr srdR r xdt t   (11) Ld∂= ∂∫sd sddR r xdt t   (12) where Rsr = positive retarding storage volume in river reaches, Rsd = negative retarding storage volume in river reaches. Time0=∂∂tQ 0=∂∂=∂∂xQtAbTpQ TphcaDifference between arrival times of the peak discharge and the peak water level at the x sectiondxtrsr∂∂ dxtA∂∂dxtQUQQ dxxdxx ∂∂−=′ −+ 12/2/dxtUUAdxtQUQQ dxxdxx ∂∂+∂∂−= −+ 12/2/2/dxxQ −DischargeRising period Descending period :Release volume of flood water  :Storage volume of flood water + + :Positive retarding storage volume of flood water :Negative retarding storage volume of flood water 306Lighthill, M. J. and Whitham, G. B. (1955): On kinematic waves. I. Flood movement in long rivers”. Proceedings of the Royal Society A. pp. 281-316. Menedez, A. N., and Norscini, R. (1982): Spectrum of shallow water waves: An analysis, J. Hydraul. Div. Am. Soc. Civ. Eng, 108(HY1), pp. 75-94. Mishra, S. K. and Seth, S. M. (1997): Use of hysteresis for defining the nature of flood wave propagation in natural channels, Hydrological. Sci., 41:2, pp. 153-170. Ponce, V. M. and Simons, D. B. (1977): Shallow wave propagation in open channel flow, J. Hydraul. Div. Am. Soc. Civ. Eng, 103(HY12), pp. 1461-1478. Takahashi, T. (1971): Study on propagation characteristics of flood flows in river channels, thesis, Kyoto University, Kyoto, Japan. Takemura Y. and Fukuoka S. (2012): Effects of river morphology on propagation and transformation of discharge and water level hydrographs of flood flows through river valley, Journal of Japan Society of Civil Engineers, Ser. B1 (Hydraulic En-gineering) Vol. 68, No. 1, pp. 35-54. Takemura Y. and Fukuoka S. (2014): Effects of channel shape on propagation characteristics of flood flows through a valley, Journal of Flood Risk Management, Volume 7, Issue2, pp. 152-158. 311

Evaluation Method of Positive and Negative Retarding Storage Volumes for Unsteady Two Dimensional Flows and Propagation Mechanisms of Peak Discharge and Peak Water Level

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Evaluation Method of Positive and Negative Retarding Storage Volumes for Unsteady Two Dimensional Flows and Propagation Mechanisms of Peak Discharge and Peak Water Level

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