Solution of Saint Venant's Equation to Study Flood in Rivers, through Numerical Methods.

Transcription

1 Hydrology Days 005 Solution of Saint Venant's Equation to Study Flood in Rivers, through Nuerical Methods. Chagas, Patrícia Departent of Environental and Hydraulics Engineering, Federal University of Ceará Souza, Raiundo Departent of Environental and Hydraulics Engineering, Federal University of Ceará Abstract. The probles of flood wave propagation, in bodies of waters, caused by intense rains or breaking of control structures, represent a great challenge in the atheatical odeling processes. The solution of this proble passes, invariably, for the developent of atheatical odels that, in their forulations, describe, in the closest it way, the real scenery of the process. On the other hand, the atheatical anipulation of those odels iplies in the solution of nonlinear differential equations, as it is the case of Saint Venant's Equation, of great application in the studies of rivers, for gradually varied flow. This research uses a discretization, for the equations that governs the propagation of a flood wave, in natural rivers, with the objective of a better understanding of this propagation process. The results have shown that the hydraulic paraeters play iportant gae in the propagation of a flood wave. Keywords: Hydrodynaics Models, Flood Control, Flow in Natural Rivers.. Introduction The study of the flow in open channels is an iportant topic of the hydraulics and the hydrology subjects. As Brazil is a privileged country in ters of water resources, the necessity of the rational and the sustained use of the rivers becoes an indispensable task. Being like this, the technical and scientific counity linked to the thee is seeking, ore and ore, to supply subsidies inherent to probles related with water resources anageent. In this work it intends to study the propagation of a flood wave natural river, originating fro torrential rains or of the breaking of a control structure. A wave is defined as a teporary and space variation in the properties and hydraulic characteristics of the flow, just as the change in the flow or in the height of the surface of water (Porto, 003). This is a physical process of high coplexity and it needs for its solution the knowledge of all hydrodynaics of the hydraulic syste. The hydrodynaic odel, which is coposed by the differential equations of Saint-Venant, allows, in their ain analysis, that the study of the hydraulic and hydrologic behavior of this body of water could be ade. In that context, through those equations, it is possible to develop a ethodology capable to study such variables as flow, velocity and depth in function of the space and Departent of Environental and Hydraulics Engineering, Federal University of Ceará Capus do Pici Bloco 73 P. O. Box 608, CEP , Fortaleza CE Brasil. Tel: (55) (85) ; e-ail: pfchagas@yahoo.co & rsouza@ufc.br Hydrology Days 05

2 Chagas and Souza teporal coordinates and, finally, to deterine every flow structure of the river. The results obtained in this work allow evaluating how it happens the change in the depth of the surface of water, with the entrance of a wave in the channel, for different roughness coefficient and different bed slope of the channel.. Methodology The basic equations that describe the propagation of a wave in an open channel are the Saint Venant's equations. Those waves can be classified as: dynaic wave, gravitational wave, diffuse wave and cineatic wave, according to the nuber of eleents considered in the odel. The dynaic wave, for instance, considers all the ters of the oentu equation. The gravitational wave neglects the bed slope effects, and the friction effect that develops between the water and the walls of the river. In this case, the neglected ter is g(s 0 -S f ). The other ters of the oentu equation are considered in the odeling. The cineatic wave just considers the effects of the bed slope and the effects of the friction that develops between the water and the walls. In this work it will be used the dynaic wave. For the developent of the atheatical odel, soe siplifications were ade: the flow will be considered one diensional; the distribution of the pressure in the vertical is hydrostatic one; the water will be considered as incopressible and hoogeneous, and the density will be considered constant in the tie and in the space. Thus, the odel equation could be defined by, Continuity Equation A + = 0 x t () Moentu Equation + t ( Q / A) y + ga( S ) + = 0 x x 0 gas f () Where x is the longitudinal distance along the channel (), t is the tie (s), A is the cross section area of the flow ( ), y is the surface level of the water in the channel (), S 0 is the slope of botto of the channel, S f is the slope of energy grade line, B is the width of the channel (), and g is the acceleration of the gravity (.s - ). In order to calculate S f, the Manning forulation will be used. Thus, / 3 / V = R S f (3) n 06

3 Solution of Saint Venant Equations Where V is the ean velocity (/s), R is the hydraulic radius () e n is the roughness coefficient. Operating algebraically (), () and (3), Keskin (997), one can find, Where, and + + β = 0 t x (4) ga Q Q = + B A (5) A Q 5 4R A 3 3B β = ga( S S 0) (6) f In this hydrodynaic odel it will certain two dependent variables. The first refers to the cross section area A(x,t), along the channel, for each interval of tie. The second one refers to the flow field Q(x,t) along the channel, for the sae previous conditions. As the investigation deands the knowledge of two dependent variables, there is the necessity of two differential equations: the equation () and the equation (4) will copose the odel. Initial Conditions: Q(x,0)=Q 0 (7) A(x,0)=A 0 (8) Where Q 0 is the steady state flow of the channel, and the A 0 is the cross section area for the steady state conditions. Boundary Conditions: πt Q( 0, t) = Q0 ( + a.sin( )), for 0<t<t b (9) T Q ( 0, t) = h( t), for t>t b (0) Where a represents the wave aplitude; and T represents the flood wave period; and t b is the axiu tie required for the entrance of the wave into the channel. 07

4 Chagas and Souza 3. Nueric Solution of the Hydrodynaic Model With respect the nueric solution of the differential equations of the dynaic wave, the explicit discretization forulation will be used and defined through the relationship: Q Q t j Q Qi + ( ) + β = 0 () x Making an arrangeent of the equation () it is possible to find: j t Q + Qi β t Q x = () t + x Where and β are defined, respectively: And j j i = (3) + + j j β i β β = (4) + + Finally with the Q calculate for the next tie step, it is possible calculate the A(x,t), through: j Dt Dx j = Ai + ( Q Qi ) + ( q qi ) (5) Dx A Results For this work it was developed a hydrodynaic odel capable to realize siulations to analyze the behavior of the depth of water, when a flood wave coes into the channel, as function of tie. Soe considerations were ade for the nueric siulation, such as, length of the channel eters, width of the channel 50 eters, and sinusoidal function to represent the entrance of a wave in the channel. The discretization in the x direction was ade in 50 intervals of.000 eters each, totaling a length of 50 k. Regarding discretization of the tie, it becae divided in 360 intervals, with 50 seconds, totaling a tie of 5 hours. The Figure shows the coparison aong the values of the depth of the water in the channel, in eters, after the entrance of a wave, for different ties and with a sae value of bed slope. It was considered the bed slope of the channel equal to /. Thus, for n=0.0 and a tie of approxiately 3 08

5 Solution of Saint Venant Equations hours, the depth is around.8 eters, while for n=0.03 and a tie of 4 hours, the pick of the height is close to 3.0 eters. 4 Water Depth () 3 n=0.0 n= Tie (h) Figure Coparison aong values of water depth for different roughness. As it can be observed, as bigger is the roughness, the will be the values of the depth of the water in the channel. This happens because the roughness coefficient, n, represents the ost iportant factor in the resistance process concerning with friction effect (Silva et al., 003). The Figure shows the coparison aong the values of the depth of the water in the channel, in eters, after the entrance of a wave, for different ties and with a sae value of the roughness n=0.0. As it can be observed, as bigger is the bed slope, the less will be the values of the depth of water in the channel. Thus, for S 0 =0.000 and a tie of approxiately :30 hours, the depth is around.8 eters, while for S 0 = and a tie of :40 hours, the pick of the height is close to.4 eters. 3 Water Depth () S0=0.000 S0= Tie (h) Figure - Coparison aong values of water depth for different bed slope. 5. Conclusions It is iportant to reind that the objective of the present work is the developent of a hydrodynaic odel to evaluate the behavior of the water 09

6 Chagas and Souza depth of the channel, with the entrance of a wave, such that it could be able to deterine the region with ore susceptibility of the occurrence of inundations. A relevant aspect of this research is in the ipleentation of a ethodology capable to accoplish soe siulations, through a coputational progra, with base in the ethod of the finite differences, where quite satisfactory results were obtained. In that way, it was iportant to verify the reach of the depth of the channel under the influence of a dynaic wave, for different bed slope and roughness coefficient, being verified that the propagation of the flood wave suffers great influence of these paraeters. In the reality, this study allows that subsidies could be obtained, in order to apply in the developent of progras of water resources anageent, especially, in the prevention of inundations. However, it is iportant to point out that there are still any aspects to research and to ipleent in the odel. References Bajracharya, K. Barry, D.A. (999). Accuracy Criteria for Linearised Diffusion Wave Flood Routing, Journal of Hydrology, v. 95, p. 00-7, Elsevier. Barry, D.A.; Bajracharya, K. (995). On the Muskingu-Cunge Flood Routing Method. Environental International, v., n. 5, p , Elsevier. Chalfen, M.; Nieiec, A. (996). Analytical and Nuerical Solution of Sain-Venant Equations. Journal of Hydrology, v. 86, p. 3. Chow, V. T., 988. Applied Hydrology, New York: McGraw-Hill. 57p. Keskin, M. E.; Agiralioglu, N. A., 997. Siplified Dynaic Model for Flood Routing in Rectangular Channels. Journal of Hydrology, v. 0, p , Elsevier. Porto, R. de M. P., 003. Hidráulica Básica. Projeto Reenge. São Carlos: EESC USP, ª ed. 540p Moussa, R.; Bocquillon, C. (996). Criteria for the choice of Flood-Routing Methods in Natural Channels. Journal of Hydrology, 86, p. -30, Elsevier. Silva, R. C. V., Mascarenhas, F. C. B., Miguez, M.G., 003. Hidráulica Fluvial, v. I, Rio de Janeiro: COPPE/UFRJ, 304p. 0