MODELING OF 2D FLOOD FLOW ANALYSIS BY FINITE ELEMENT METHOD

4th International Symposium on Flood Defence: Managing Flood Risk, Reliability and Vulnerability Toronto, Ontario, Canada, May 6-8, 2008 MODELING OF 2D FLOOD FLOW ANALYSIS BY FINITE ELEMENT METHOD Kun-Yeun Han1 and Sang-Ho Kim2 and Seung-Yong Choi3 Professor of Civil Engineering, Kyungpook National University, Daegu, Korea Professor of Civil Engineering, Sangji University, Wonju, Korea Researcher of Civil Engineering, Kyungpook National University, Daegu, Korea ABSTRACT: The understanding and prediction of the behavior of flood flow in open channels are important to resolve a wide variety of practical problems including levee-break and dam-break flow Recently, the frequency of flood has increased, therefore, the effective flood control and management of river flows is essential The challenging problem facing two-dimensional model is the treatment of wet and dry areas This situation is encountered in most practical river and coastal engineering problems To solve the dry/wet problems, deforming grid method, transition element method and hybrid method are adopted The model is verified by applying to an U-shaped laboratory channel The simulation results agree with observed data for various flow conditions.Trapezoidal channel with partly dry side slopes and straight channel with various drying conditions are examined for flow model validations RAM2 model shows reasonable flow distribution compared with existing model in dry area simulation in Milyang river and Keum river RAM2 model can be used for constructing a hydrodynamic input for water quality and sediment transport model as well as for analyzing dam-break flow levee-break flow and rapidly varied flow Key Words: finite element method, RAM2 model, SU/PG scheme, dry/wet problem INTRODUCTION Two dimensional hydrodynamic model can provide good estimates compared to one dimensional model in the application of complex features in the flow around island/obstructions, flow at confluences and flow in braided channel The challenging problem today facing two dimensional finite element model is the treatment of wet/dry areas This situation is encountered in most practical river such as flood propagation, dam-break/levee-break analysis and tidal processes The main objective of this study is to develop an accurate and robust two-dimensional finite element model (RAM2) with wet/dry simulation in complex natural rivers The developed model by SU/PG scheme can successfully simulate river hydrodynamics without significant parameter variations by taking turbulence effects into consideration This success of RAM2 is partially due to its consideration of the foundational role of characteristics in quantifying the upwind weighting of SU/PG scheme The model output can be used as basic hydraulic data for the water quality and sediment transport models The suggested model adopts the two dimensional shallow water equations with wetting/drying problems using moving boundaries It treats the effect of the bed friction by using Manning’s equation The model employs the Newton-Raphson method to solve the nonlinear equations A system of linear equations is analyzed by the frontal method and the modified frontal method Either a water level or a discharge condition can be specified as an initial condition This model handles upstream and downstream boundary conditions flexibly in various forms such as the surface elevation, discharge, and a relation of surface elevation and discharge RAM2 model is able to analyze a rapidly varied unsteady flow such as an analysis of dam break and levee break problems, which contains transcritical flow regime The model is also able to carry out the two-dimensional analysis of chutes and pools for ecological modeling 2.1 RAM2 MODEL Governing Equations The governing equations for two-dimensional shallow free surface flows can be obtained by the integration of three-dimensional Reynolds equations with respect to the depth, or applying laws of conservation mass and momentum for a uniform water column When the vertical variation of velocity can be ignored, governing equations of two-dimensional continuity and momentum equation describes the flow phenomenon The basic assumption is that the pressure distribution is hydrostatic Under these assumptions conservation laws can be written as follows [1] U F G    D 0 t x y where, U , F , G , D matrices are defined as       p  h      p gh    [2] U  u  , F  , G     h v  q2   pq      h    h     q i   2 1/ pq   z p( p  q )  gn , D  gh  h x h7 /   2 1/ gh  gh z  gn q ( p  q )     y h7 /  Applying the chain rule, equation [1] can be written as follows [3] U F U G U U U U    D 0 or A B  D 0 t U x U y t x y            F  p   gh where, A  U  h   pq  h 2 2.2 2p h q h  0  0  p h   0 G  pq q B   U  h h q   gh  h 2  1 p  h 2q  h SU/PG(Streamline Upwind Petrov Galerkin) Method The Streamline Upwind Petrov Galerkin (SU/PG) Finite Element method is applied to the system of equation [4] The weight function [4] B B Bˆ i  Bi  xW x i  yW y i x y   2 [5] Ax  c  U    UV 2.3 Bˆ i for the two-dimensional case has the following form 0 2U  , V U    Ay    UV V c  V 1 U  2V  Constitution of Jacobian Matrix In order to apply the SU/PG finite element method for the continuity equation and momentum equations, arranged as follows [6] x , y direction E1 , E , E3 equations are differentiated with respect to time and can be p q E1 hˆ n 1     x y 2 z   p gh    pq  n 1 p( p  q ) [7] E p ˆ         gh  gn   x  h  y  h  x h7 / [8] E3 qˆ n 1    2 2 z  q gh    pq  q( p  q )        gh  gn y  h  x  h  y h7 / Equation represents rearranged equation, which can be solved by Newton-Raphson method to solve nonlinear simultaneous equations  Bi  [9]   0 Bi 0   E1   Bi        E   xW x   x  Bi   E  Bi 0   E1   Bi        E   y W y y     Bi   E    Bi 0   E1      E   0 Bi   E  Computation flow chart for the analysis of two dimensional shallow water equations in natural rivers is presented in Figure Figure 1: Computation flow chart of RAM2 model 3.1 APPLICATIONS Flow in U-shaped channel Unsteady flow simulation in U-shaped channel was conducted to validate the developed model Results were compared to hydraulic flume results reported in Bell, Elliot, and Chaudhry(1992) A plan view of flume facility is shown in Figure The flume simulated a dam break through a horseshoe bend Because this was a 2-D problem and model results were being compared to hydraulic flume results Initially, the reservoir has an elevation of 0.1898m relative to the channel bed; the channel is located at a depth (and elevation) of 0.0762 m The velocity was zero and then the dam was removed The surge location and height were recorded at several stations and compared to the model at two of these, at stations 4, Station was 6.0m from the dam along the channel center line of the bend and Station was 7.62m from the dam near the conclusion of the bend The test results for stations and are shown in Figures 3~4 It turned out that the depths match fairly closely between flume and numerical model at stations and Figure 2: Details of the dam break test flume and computational mesh Figure 3: Flume and developed model depth histories (station 4) Figure 4: Flume and developed model depth histories (station 6) 3.2 Flow with drying/wetting problem This test solves for a steady state flow in a hypothetical trapezoidal channel, 20m bed-width and to side slopes The channel has a total length of 300m, with 100m transition lengths from rectangular to trapezoidal at the upstream end, and from trapezoidal to rectangular at the downstream end The longitudinal bed slope is 0.0013 A subcritical inflow boundary condition with a total inflow discharge of 80m3/sec is specified Initial conditions set the water depth at 5.0m throughout the channel The downstream boundary condition specifies a depth of 4.0 m The finite element mesh is composed of square linear quadrilateral elements Figure shows a cross section at x = 100 m and Figure shows depth contours for the steady state solution Figure shows the computed steady state velocity vectors Figure 5: Cross section at X=100m, Showing Wet and Dry Areas Figure 6: Depth Contours Figure 7: Velocity Vectors 3.3 Natural river applications The applications to several natural rivers with wetting and drying area were performed for the purpose of verification of the developed model Results using RAM2 model show reasonable flow distribution compared with those of existing model in dry area simulation in Milyang river and Keum river The suggested model revealed that it can successfully simulate the wet/dry condition for the flood and low flow respectively The simulation results agree with observed data in terms of velocity and flow depth Accordingly, RAM2 model will contribute to river-basin analysis for modeling the real world applications including various flow conditions Figure 8: Velocity Vectors and Depth Contours (Milyang river) Figure 9: Velocity Vectors (Keum river) Figure 10: Depth Contours (Keum river) CONCLUSIONS The suggested RAM2 model was developed to simulate two dimensional flood flow analysis by SU/PG scheme Several tests were performed with different kinds of element(4-Node, 6-Node, 8-Node elements) to validate the applicabilities of the model RAM2 model is able to analyze a rapidly varied unsteady flow such as an analysis of dam break and levee break problems, which contains transcritical flow regime RAM2 model was verified with analysis of dam break flow, U-shaped channel flow, and natural river flow Simulation results were compared with those of observed in laboratory experiments and the field measured data Also, results showed more reasonable velocity distribution compare to those of the existing model in meandering domain for application of natural river flow Accordingly, the developed finite element general-use model is feasible and produces reliable results for simulation of two dimensional natural river flow One contribution of this study is to present that results can lead to significant gain in analyzing the accurate flow behavior including dry/wet problem with large floodplain and flow around hydraulic structure Moreover, the model can be used as a basis for extending to sediment transport model as well as be used for analyzing and predicting of two dimensional features in river flow ACKNOWLEDGEMENTS This research was supported by a grant (2-2-3) from Sustainable Water Resources Research Center of 21st Century Frontier Research Program, Korea REFERENCES Bell, S.W., Elliot, R.C and Chaudhry, M.H 1992 Experimental results of two-dimensional dam-break flows Journal of Hydraulics Research 30(2), pp 225-252 Berger, R.C and Howington, S.E 2005 Discrete Fluxes and Mass Balances in Finite Elements Journal of Hydraulics Engineering, Vol 128, pp 97-92 Han, K.Y et al 2006 Development of Hydrodynamic Analysis System(RAM2) in River Ministry of Science and Technology, Korean Government Samuels, P.G 1985 Modeling of river and floodplain flow using the finite element method Research Report No SR 61, Hydraulics Research Ltd Wallingford, UK, pp 1-198 ... Development of Hydrodynamic Analysis System(RAM2) in River Ministry of Science and Technology, Korean Government Samuels, P.G 1985 Modeling of river and floodplain flow using the finite element method. .. unsteady flow such as an analysis of dam break and levee break problems, which contains transcritical flow regime RAM2 model was verified with analysis of dam break flow, U-shaped channel flow, ... two dimensional flood flow analysis by SU/PG scheme Several tests were performed with different kinds of element( 4-Node, 6-Node, 8-Node elements) to validate the applicabilities of the model RAM2