#### Transcript 2d_fem

Environmental Hydrodynamics Lab. Yonsei University, KOREA RCEM 2005 2D finite element modeling of bed elevation change in a curved channel S.-U. Choi, T.B. Kim, & K.D. Min Yonsei University Seoul, KOREA Introduction • Most natural streams are sinuous and meandering. • In a curved channel, the centrifugal force makes the flow structure and sediment transport mechanism extremely complicated. • To simulate the flow and morphological change in a curved channel, the secondary currents and the gravity effect due to morphological change should be properly considered (Kassem & Chaudhry, 2002). Environmental Hydrodynamics Lab. Yonsei University, KOREA RCEM 2005 Why 2D Model? • 1D Model – Impossible to account for sediment transport in the transverse direction • 3D Model – Still Expensive – Not readily applicable to many engineering problems • Turbulence Closure • Sediment Transport Model • Boundary Conditions Environmental Hydrodynamics Lab. Yonsei University, KOREA RCEM 2005 Previous Study • Only applicable to the steady flow condition, constant channel width and constant radius curvature – Koch & Flokstra (1981), Struiksma et al. (1985), Shimizu & Itakura (1989), Yen & Ho (1990) and so on. • The coordinate transformed, unsteady FDM & FVM – Kassem & Chaudhry (2002), Duc et al. (2004), Wu (2004) • The finite element model for bed elevation change in a curved channel has never been proposed!! – FEM provides greater flexibility in handling spatial domain. Environmental Hydrodynamics Lab. Yonsei University, KOREA RCEM 2005 Purpose • To Develop a 2D FEM model – capable of predicting time-dependent morphological change in a curved channel. • For flow analysis, the shallow water equations are solved by the SU/PG scheme. • To assess the be elevation change, Exner’s equation is solved by BG scheme. • For validation, we applied the model to two laboratory experiments. Environmental Hydrodynamics Lab. Yonsei University, KOREA RCEM 2005 Limitations • Decoupled modeling approach • Flow equations and Exner’s equations are solved separately. • Uniform sediment • Neglecting armoring or grain sorting effects. Environmental Hydrodynamics Lab. Yonsei University, KOREA RCEM 2005 Flow Equations • 2D shallow water equations with the effective stress terms h p q 0 t x y 1/ 2 zb gn 2 p p 2 gh 2 pq p p q 2 2 2 gh p p q 0 t t t x h 2 y h x x y y x x h 7 / 3 1/ 2 zb gn 2 q pq q 2 gh 2 p q q 2 2 2 gh q p q 0 t t t x h y h 2 x y x y y y h7 / 3 • Eddy viscosity model t 6 U *h Environmental Hydrodynamics Lab. Yonsei University, KOREA RCEM 2005 Bed Sediment Conservation • Exner’s equation 1 p qtx cos qt zb qtx qty 0 t x y qty sin qt • Total Sediment Load 1/ 2 d qt 0.05 sV 2 g s / 1 Environmental Hydrodynamics Lab. Yonsei University, KOREA 0 s d 3/ 2 Engelund & Hansen’s formula RCEM 2005 Finite Element Method (1) • Flow Equations U U U Dx Dy A B F 0 t x y x y • Weighted Residual Equations U Ni Ni U U Dx Dy N x W y W A B F d 0 x y i x y t x y x y • 2D SU/PG Method (Ghanem, 1995) Ni N Wx y i Wy x y A B Wx , Wy A2 B2 A2 B2 Ni* Ni x Environmental Hydrodynamics Lab. Yonsei University, KOREA RCEM 2005 Finite Element Method (2) • Exner’s Equation 1 p zb qtx qty 0 t x y • Weighted Residual Equation zb qtx qty * N 1 p ' d 0 i t x y • BG Method Ni* Ni Environmental Hydrodynamics Lab. Yonsei University, KOREA RCEM 2005 Boundary Conditions • Upstream & Downstream BCs • Sidewall BC N k y dA tan N k x dA Environmental Hydrodynamics Lab. Yonsei University, KOREA (Akanbi, 1986) RCEM 2005 Flow Characteristics of a Curved Channel (a) Under a flat (& fixed) bed condition - The centrifugal force makes higher flow depth, but lower mean velocity, at the outer bank. This generates the secondary flows satisfying the continuity. - Observed in Experiments and Numerical simulations. (b) Under a mobile bed condition - Secondary flows induces sediment erosion & deposition at the outer & inner banks, respectively. - The flow depth and mean velocity at the inner bank is lower. - Observed in natural meandering rivers and Experiment by Yen (1967 & 70) Environmental Hydrodynamics Lab. Yonsei University, KOREA RCEM 2005 Direction of sediment transport • Gravity effect on a slope 1 zb f s* y tan 1 zb cos f s* x sin (Struiksma et al., 1985) • Angle of bed shear stress due to the secondary flow effect v tan 1 u F 2 n g tan 1 h , F 2 1 1/ 6 h Rs (Rozovskii, 1957) 1 1 v v u u 3 u 2 uv uv v 2 Rs V x y x y Environmental Hydrodynamics Lab. Yonsei University, KOREA RCEM 2005 Applications 1. 180º Curved Channel Experiment Lab. of Fluid Mech. (LFM) in Delft Univ. of Tech. (Sutmuller & Glerum, 1980) 2. 140º Curved Channel Experiment Delft Hydraulics Lab. (DHL) (Struiksma, 1983) Environmental Hydrodynamics Lab. Yonsei University, KOREA RCEM 2005 LFM 180º Curved Channel • Experimental Conditions Q (m3/s) B (m) h (m) u (m/s) S0 (×10-3) C (m1/2/s) d50 (mm) Rs (m) L (m) 0.17 1.7 0.2 0.5 1.8 26.4 0.78 4.25 13.35 • 1400 elements, 1551 nodes Flow • Porosity = 0.4 10 y (m) • 10 times extension of width • Fr = 0.36 5 • Fixed bed B.C. for upstream & downstream boundaries 0 0 5 10 15 20 x (m) Environmental Hydrodynamics Lab. Yonsei University, KOREA RCEM 2005 Direction of Sediment Transport (LFM) 5 10 y (m) y (m) 10 10 min. 0 16 5 150 min. 0 18 20 x (m) 22 16 18 20 22 x (m) • At the initial stage, the particles are heading for the inner bank. This induces sediment deposition & erosion at the inner and outer banks. • After for a while, the gravity effect due to changed bed reduces the secondary flow effect. Environmental Hydrodynamics Lab. Yonsei University, KOREA RCEM 2005 Flow Depth (LFM) 0.22 0.21 0.2 0.19 0.18 0.17 0.16 0.15 0.14 5 10 min. 0 16 10 y (m) y (m) 10 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 5 150 min. 0 18 20 x (m) 22 16 18 20 22 x (m) • At the initial stage, the flow depth near the outer bank is higher than that near inner bank. • A similar pattern at 150 min. But, considering deposition & erosion, the water surface elevation across the width is nearly uniform. Environmental Hydrodynamics Lab. Yonsei University, KOREA RCEM 2005 Velocity Distribution (LFM) 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 5 10 min. 0 16 10 y (m) y (m) 10 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 5 150 min. 0 18 20 22 16 18 x (m) 20 22 x (m) • At the initial stage, the mean velocity near the inner bank is slightly higher. • Later, we have an opposite situation after bed deformation. Environmental Hydrodynamics Lab. Yonsei University, KOREA RCEM 2005 Evolution of Depth-Averaged Velocity Environmental Hydrodynamics Lab. Yonsei University, KOREA RCEM 2005 Bed Elevation Change (LFM) 0.5 0.25 0 -0.25 -0.5 -0.75 -1 Measured data by Sutmuller & Glerum (1980) Simulated Result • In the numerical simulation, the bed elevation change became negligible after 150 min. • A good agreement. But the location of max deposition is slightly different. This may be … Environmental Hydrodynamics Lab. Yonsei University, KOREA RCEM 2005 Longitudinal Bed Profile (LFM) • Overall trend is the same. • Near the inner bank, the amount of sediment deposition is over-predicted. •Near the outer bank, the amount of sediment erosion is under-predicted. Environmental Hydrodynamics Lab. Yonsei University, KOREA RCEM 2005 DHL 140º Curved Channel • Experimental Conditions (Struiksma, 1983) B (m) h (m) u (m/s) S0 (×10-3) 0.062 1.5 0.1 0.41 2.03 C (m1/2/s) d50 (mm) Rs (m) L (m) 28.8 0.45 12.0 29.35 • 945 elements, 804 nodes • Porosity = 0.4 • 10 & 15 times extension of width for US & DS, respectively • Fixed bed B.C. for both US & DS 30 y (m) Q (m3/s) 20 10 Flow 0 0 5 10 15 20 25 30 x (m) Environmental Hydrodynamics Lab. Yonsei University, KOREA RCEM 2005 Longitudinal Bed Profile (DHL) • Simulated result is after 10 hr. • Overall trend is the same. • Especially good agreement in max deposition & erosion. • The simulated results fluctuate with distance while... Environmental Hydrodynamics Lab. Yonsei University, KOREA RCEM 2005 Variation with Time (DHL) • Spatial fluctuation increases with time. • This is due to BG scheme applied to Exner’s eq. Environmental Hydrodynamics Lab. Yonsei University, KOREA RCEM 2005 Conclusions • Development of 2D FEM model for bed elevation change – SU/PG method for shallow water eqs. – BG method for Exner’s eq. – Secondary flow effect and gravity effect on sloping bed • Applications to 2 curved channel experiments – The model predicts the flow and bed morphology well. • Specially, the time-evolution of changing bed morphology from the flat bed. • Sediment deposition & erosion at the inner & outer banks. • Necessity of introducing the upwind scheme to Exner’s eq. – Spatial fluctuations in the simulated bed profiles increase with time. – Weighting is required in the upwind direction along the trajectory of sediment particles. Environmental Hydrodynamics Lab. Yonsei University, KOREA RCEM 2005