ETH Zurich, Fall 2017 Numerical Hydraulics Assignment 2 Numerical solution of shallow water wave propagation (www.surfertoday.com)
1 Introduction 1.1 Equations Understanding the propagation of shallow water waves is important for a proper design of flood protection measures such as river dams in a populated area, the assessment of flooding risk or the study of morphological dynamics of a river. To assess the wave propagation through a channel, the 1D Saint Venant equations for open channel flow can be applied. The differences to the 1D pipe flow equations are the assumptions of hydrostatic pressure distribution, a small channel bottom slope (cos α 1 and sin α = - dz ), and a constant water density. The Saint Venant equations in 1D read: dx Continuity equation: Momentum equation: + v x + h v x t + vh t x v t = + q t x = 0 (non-conservative) (1a) = 0 (conservative) (1b) v + v + g g(i x x S I E ) = 0 (2) First, we consider possible simplifications of these equations. The first approximation is the steady state equation, which can be used to calculate water surface profiles. These profiles can be classified according to the bottom slope and the water depth in relation to the critical slope, the critical water depth and the uniform flow water depth (see slides). Steady state form: x = I S I E 1 Fr 2 (3) The second approximation is the kinematic wave where the momentum equation is simplified to: Kinematic wave: g(i S I E ) = 0 (4) Together with the continuity equation, this yields a non-linear wave equation. The advantage of the kinematic wave approximation is the reduction to a first order PDE, which is easier to solve and demands only one boundary condition. On the other hand, the wave steepens and no backwater effect can be implemented, as there is no downstream boundary condition. The third approximation is the diffusive wave where the momentum equation is reduced to: Diffusive wave: x = I S I E (4) The diffusive wave approximation leads to an advection-diffusion equation. The wave moves downstream and flattens out diffusively. A lower boundary condition is necessary due to the second derivative. Hence, this approximate solution allows for the implementation of a backwater effect. Often the diffusive wave approximation is sufficient to describe the wave propagation through a channel with a very simple and smooth geometry. For more complex cases, it is necessary to solve the full Saint-Venant equations which provide the solution of a dynamic wave. 1
1.2 Numerical solutions In this exercise, we will use the explicit finite difference method in an explicit form to solve the full Saint-Venant equations in 1D. From equation (1b) the water depth at the new time is obtained: h i = h j i Δt (h j j Δx i+1 v i+1 h j i v j i ) (5) Equation (2) provides the solution for the velocity at the new time: v i = v j i Δt ((vij ) 2 j (v 2Δx i 1 ) 2 ) g Δt (h j j Δx i h i 1 ) + gδt(i 0 I e ) (6) The energy slope Ie can be determined with the Strickler-formula: I e = j j v i vi k 2 4/3 (7) st r hy The hydraulic radius rhy is calculated as followed: r hy = j h i 1 b j 2 h i 1 +b (8) To solve the dynamic wave numerically two boundary conditions are required. The upstream boundary is usually implemented as an inflow hydrograph. This provides an equation for v 1 at the first cross-section; h 1 is obtained by formula (5). v 1 = Q in (9) b h i As downstream boundary condition a fixed water level can be used. This boundary condition provides an equation for h Nx+1 ; v Nx+1 can be determined with equation (6). h Nx+1 = h 0 (weir) (10) Other options for downstream boundary conditions are water level-flow rate relations or known slope of water level or energy. The explicit numerical solution method requires a stability condition named Courant-Friedrichs-Levy (CFL) criterion. The CFL criterion limits the timestep in relation to the spatial discretization and the flow and wave velocity to guarantee stability. Δt Δx (c+ v ) max (11) C is the relative wave velocity with respect to average flow depth: c = gh = g A(h) b(h) (12) Other possible solution schemes are the method of characteristics and the finite volume discretization. Compared to the pipe flow, the method of characteristics applied for open channel 2
flow needs an additional interpolation step, since the condition v + c v c c is not valid anymore (see lecture slides). In assignment 3 we will compare the numerical solutions obtained by finite difference and finite volume schemes. 2 Objective The goal of this assignment is to assess the propagation of a shallow water wave in a simple rectangular channel having constant slope and friction. In addition, the strengths and weaknesses of an explicit finite difference scheme shall be investigated. 3
3 Tasks and questions Complete the Matlab template NHY_Assignment_2_IncompleteMatlabCode.m. The channel properties are listed in table Tab. 1. The numerical parameters are shown in Tab. 2. Tab. 1: Channel properties Parameter Value Length 10 000 m Width 10 m Slope 2 Roughness (kst) 20 m 1/3 s -1 Tab. 2: Numerical parameters Parameter Time step Simulation time Number of spatial discretization intervals Value 10 s 14 000 s 40 The base flow Q0 can be set to 20 m 3 /s. The downstream boundary condition shall be implemented as a constant water depth of 2.2 m. The upstream boundary condition is given by an inflow hydrograph (Tab. 3). The discharge of the inflow hydrograph has to be added to the base flow. Tab. 3: Inflow hydrograph Time [h] 0.0 0.0 0.5 50.0 1.0 37.5 1.5 25.0 2.0 12.5 2.5 0.0 Discharge [m 3 /s] Tasks: 1. Determine the wave propagation (water surface profile, maximum water depth) for the rectangular channel. 2. Determine the outflow hydrograph and compare the result with the inflow hydrograph. 3. Try different values for the channel properties (slope, width, roughness) as well as for the spatial and temporal discretization. What differences do you observe? What is the cause of the observed differences? Additional Task: 1. Implement the CFL criterion (i.e. insert a condition that makes sure that the timestep is small enough) 4