ANZIAM J. 52 (CTAC2010) pp.C1102{C1125, 2012 C1102 Study of river flow based on finite element analysis and GPS-echo-sounder measurement Hashentuya1 Yoji Otani2 Kazuhiro Yamamoto3 Masaji Watanabe4 (Received 31 January 2011; revised 15 February 2012) Abstract Currents of a river are analyzed numerically by introducing data concerning the water depth into a finite element analysis of the partial differential equations of the flow. Techniques to simulate currents generated in the river are described. The region is divided in the vertical direction into layers, and a finite element discretization in the horizontal direction is applied. Results of the simulation of flow generated in the Yoshii River in Okayama Prefecture of Japan are shown. We also introduce our techniques to measure the river bed topography using a hardware system consisting of a global positioning system and an echo sounder. Data obtained by using the system are introduced into the analysis of flow and corresponding numerical results are generated. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/3966 gives this article, c Austral. Mathematical Soc. 2012. Published March 7, 2012. issn 1446-8735. (Print two pages per sheet of paper.) Copies of this article must not be made otherwise available on the internet; instead link directly to this url for this article. Contents C1103 Contents 1 Introduction C1103 2 Numerical solution by the finite element method C1105 2.1 Shallow water equations................... C1106 2.2 Initial condition........................ C1107 2.3 Application to the multi-layer finite element method.... C1109 2.4 Time-step approximation................... C1115 3 Flow simulation downstream of the Kamogoshi Dam C1118 4 Conclusion C1118 References C1123 1 Introduction Seto Inland Sea divides the main island and the Shikoku Island of Japan. The Yoshii River flows into the Kojima Bay which is connected to the Seto Inland Sea. It is one of three major rivers in Okayama Prefecture with 2, 110 km2 basin area, 133 km length, and 61.16 m3=s average flow. Kamogoshi Dam is approximately 7 km upstream from the Yoshii River mouth and the latter is called Kuban. Water from upstream has been washing the riverbed in the area below the Kamogoshi Dam, and a deep water area has developed. We analyze currents generated in the Yoshii River, downstream from the Kamogoshi Dam. The bottom topography is investigated by a global positioning system (gps) and an echo sounder. Figure1 shows the depth contours of the Yoshii River in the region near the Kamogoshii Dam. Note that the depth of the river ranges from 2 m to 9 m. The depth data are based on gps-echo-sounder measurements taken on March 14, 19, 22, 24, 29, and 30, 2010. Numerical results are obtained by introducing data concerning 1 Introduction C1104 DEPTH CONTOUR(Kamogoshi dam) -148200 -148400 -148600 -148800 -2 m -3 m -4 m -5 m -6 m -7 m -149000 -8 m -9 m -149200 -26200 -26000 -25800 -25600 -25400 -25200 Figure 1: The depth contours in the region near the Kamogoshi Dam in the Yoshii River. the depth of the river into a finite element analysis of equations that govern the dynamics of the currents. Experimental techniques to utilize the gps in analyses of currents are also introduced. Figure2 (based on the water level data of the river policy information of the Ministry of Land, Infrastructure, Transport and Tourism) shows the change of water levels of the the Yoshii River in the region near the Kamogoshii Dam and Kuban. The level rises and falls twice each day near the Kuban, because the Yoshii River is connected to the Seto Inland Sea. We study the lowest water levels from 13:00 to 16:00 jst, on July 23, 2010. Section2 shows how the system of partial differential equations is solved by the finite element method. Section3 applies this method to simulate the flow in the Yoshii River downstream from the 2 Numerical solution by the finite element method C1105 WATER LEVELS(July 23, 2010) 3.5 KUBAN KAMOGOSHI DAM 3 2.5 2 WATER LEVEL[m] 1.5 start end start end 1 SIM.1 SIM.2 0.5 00:00 04:00 08:00 12:00 16:00 20:00 00:00 TIME[hour] Figure 2: Change of water levels on July 23, 2010. Kamogoshi Dam. The difference between simulation results using three layers and five layers is obtained, and the results are discussed in the Conclusion. Numerical simulations such as these are useful for studying the water quality of a river, and for analyzing the effects of a red tide. 2 Numerical solution by the finite element method 2 Numerical solution by the finite element method C1106 2.1 Shallow water equations We apply a finite element method to Equations (1), (2), (3) and (4) to analyze the current generated in the Yoshii River. The shallow water equations consist of momentum equations and a continuity equation [1, 2, 3, 4], and model water in the river, @u @(uu) @(uv) @(uw) + + + - fv @t @x @y @z 1 @p 1 ∂τ ∂τ ∂τ + - xx + xy + xz = 0 , (1) ρ @x ρ @x @y @z @v @(vu) @(vv) @(vw) + + + + fu @t @x @y @z 1 @p 1 ∂τ ∂τ ∂τ + - yx + yy + yz = 0 , (2) ρ @y ρ @x @y @z @w @(wu) @(wv) @(ww) + + + @t @x @y @z 1 @p 1 ∂τ ∂τ ∂τ + - zx + zy + zz + g = 0 , (3) ρ @z ρ @x @y @z @u @v @w + + = 0. (4) @x @y @z Here, the components of the stress tensor are @u @v @w τ = -p + µ , τ = -p + µ , τ = -p + µ , (5) xx @x yy @y zz @z µ @u @v µ @u @w τ = τ = + , τ = τ = + , (6) xy yx 2 @y @x xz zx 2 @z @x µ @v @w τ = τ = + , (7) yz zy 2 @y @x and we use the following notation: 2 Numerical solution by the finite element method C1107 • x, y, z Cartesian coordinates positive eastward, northward, and upward, respectively; • u, v, w respective components of velocity; • t time; • f Coriolis parameter; • p pressure; • ρ = 1000 kg=m3 is the density of water; • µ viscosity coefficient of fluid; • g gravitational acceleration; • τxx, τxy, τyx, τyy, τxz, τyz components of the stress tensor. 2.2 Initial condition We also define the parameter γ2 = 0.0026 to be a bottom friction coefficient [5] and Ah to be a constant that represents eddy viscosity. We set Ah = 1.0 for the results obtained on July 23, 2010. The governing equations for the mass and momentum are integrated over the kth layer, where k = 1, 2, 3, ... , b. Let 1 k- 2 h ik = () dz (8) k+ 1 Z 2 1 where k ± 2 refers to z-levels of the interfaces between layers k and k ± 1. Water fluxes Mk and Nk are obtained by integrating the x-component and the y-component of the velocity over the depth, respectively: Mk = hkuk , Nk = hkvk . (9) Along the curve dx dy = u , = v , (10) dt k dt k 2 Numerical solution by the finite element method C1108 Equations (1) and (2) become dM b @M @N 1 M M k = l + l k-1 + k dt @x @y 2 hk-1 hk l=k X b @Ml @Nl 1 Mk Mk+1 ∂ζ - + + - ghk @x @y 2 hk hk+1 @x l=k+1 X 2 Mk Mk+1 2 Mk-1 Mk - γ (∆V)k - + γ (∆V)k-1 - hk hk+1 hk-1 hk A @2M A @2M + h k + h k , (11) ρ @x2 ρ @y2 and dN b @M @N 1 N N k = l + l k-1 + k dt @x @y 2 hk-1 hk l=k X b @Ml @Nl 1 Nk Nk+1 ∂ζ - + + - ghk @x @y 2 hk hk+1 @y l=k+1 X 2 Nk Nk+1 2 Nk-1 Nk - γ (∆V)k - + γ (∆V)k-1 - hk hk+1 hk-1 hk A @2N A @2N + h k + h k , (12) ρ @x2 ρ @y2 where p 2 2 (∆V)k-1 = (uk-1 - uk) + (vk-1 - vk) s M M 2 N N 2 = k-1 - k + k-1 - k . (13) hk-1 hk hk-1 hk At the rigid boundaries the values of Mk and Nk are set to zero. Initial conditions are given as Mk = 0 and Nk = 0 at all node points, where 2 Numerical solution by the finite element method C1109 k = 1, 2, 3, ... , b and b is the total number of layers. A finite element method is applied to the Equations (1), (2), (3), and (4) to simulate the flow generated downstream from the Kamogoshi Dam, with a mesh consisting of 1798 elements and 965 nodes. Figure3(a) shows the finite element mesh in the entire region of the Yoshii River Mouth. Figure3(b) shows the finite element mesh in the region near the Kamogoshi Dam. Figures4 shows a 3D view of this finite element mesh. 2.3 Application to the multi-layer finite element method After multiplying the equation (11) by δM, we integrate over a domain Ω and obtain [6,7,8] @M 1 b @M @N M M k δM dx dy = l + l k-1 + k δM dx dy @t 2 @x @y hk-1 hk Ω Ω l=k ZZ ZZ X 1 b @M @N M M - l + l k + k+1 δM dx dy 2 @x @y hk hk+1 Ω l=k+1 ZZ X ∂ζ - gh δM dx dy k @x ZZΩ M M - γ2(∆V) k - k+1 δM dx dy k h h ZZΩ k k+1 M M + γ2(∆V) k-1 - k δM dx dy k-1 h h ZZΩ k-1 k A @2M + h k δM dx dy ρ @x2 ZZΩ A @2M + h k δM dx dy.