Evaluation of well-balanced form of Weighted Average Flux scheme for simulation of flow in open channels

Document Type : Research Article


Department of Civil and Environmental Engineering, Shiraz University


Water is the most strategic liquid in the world. The life of all humankind and animals and plants are relying on the water. The water should supply to the location of demands. One of the most common water transmission ways is open channels. If a sudden change occurs in the channel section, it can affect the whole water flow in the channel. These changes can happen naturally, like aggregation of sediments in a section of the channel. Moreover, the changes may build by humans, like sharp and broad-crested weirs. Thus, it is necessary to simulate open-channel flows to predict possible changes in water surface profile and velocity.
Basically, researchers follow three approaches to simulate water flows: the analytic, the experimental, and the numerical approaches. Analytical approach for solving the flow equations is not sufficient due to the complexity and nonlinearity of the equations so there are several restrictions in the modeling. On the other hand, experimental approach is time consuming and expensive. Since the high-performance computers have been developed, researchers attracted to numerical approaches. There are different numerical solutions which are used to solve the water flow equations such as Finite Difference Method, Finite Element Method, Finite Volume Method, etc.
The Finite Volume Method is one of the most applicable methods in several computational aspects of engineering, such as computational fluid dynamics and heat transfer problems. In this method, it is necessary to have a strong approximation of numerical flux term for solving flow equations. The Riemann solver provides a reliable approximation for the numerical flux term. The Riemann problem for a set of PDEs is an initial value problem for such PDEs in which the initial condition has a special form. In order to apply numerical solutions, one can use the exact Riemann solver or approximate Riemann solver. The exact Riemann solver uses Newton-Raphson method that takes noticeable cost in time and money and the results rely on the first guess of Newton-Raphson. Therefore, researchers prefer the approximate Riemann solvers such as Harten Lax van Leer (HLL) scheme, Harten Lax van Leer Constant (HLLC) scheme and Weighted Average Flux (WAF) scheme that have acceptable results and running time.
WAF scheme can be categorized as a branch of Finite volume method. The scheme was first applied to the Euler equation. This scheme is one of the approximation solution (besides HLL and HLLC methods) of the Riemann problem. Then, Toro used the WAF scheme to simulate two-dimensional shallow water equations. Subsequently, WAF has been utilized to simulate flow over different kinds of open channels. Although the scheme shows reasonable results, it is noticeable that the numerical scheme is not well-balanced essentially. Thus, a well-balanced WAF scheme should be developed to simulate flow in open channels accurately without non-physical fluctuations in flow surface.
The aim of this research is using the ability of the WAF scheme to simulate shallow water and applying some consideration on the scheme to prevent non-physical fluctuations in water surfaces.
In this paper, a well-balanced form of WAF which is combined with HLL for estimating flux has been employed to simulate one-dimensional flow open channels. MINMOD as an effective slope limiter has been used in order to prevent non-physical oscillations. Moreover, Runge-Kutta has been employed as the time integration method to renew depths and velocities. Several different cases have been used to show that the scheme has an excellent shock-capturing ability and can handle the wet and dry condition of channel bed. Importantly, the linear reconstruction for the scheme has been applied to have second-order accuracy and to prevent the negative depth effect on computations. The scheme is shown to be well-balanced by evaluating stationary solutions at steady state conditions. Besides, the capability and accuracy of the scheme are verified by the comparison of scheme numerical results with literature analytical and experimental results. The numerical results have shown that the scheme can satisfy the continuity equation and prevent negative depth. For real applications of the scheme, the simulations of flow over sharp changes and dam-break show that RMSEs are in acceptable ranges and there is no non-physical fluctuations on the water surface profile. Simulating dam break on wet and dry beds, show that the scheme is capable in shock capturing as well as solving wetting-drying problems. In addition, flow with wide range of Froude number over different forms of broad crested weirs, have been employed to verify the robustness, accuracy and stability of the scheme. Hence all of these results prove that the presented well-balanced scheme is able to simulate different cases of shallow water equation examples accurately.


Asi, P. and Amiri, S.M. (2016). Numerical evaluation of flow variables in the presence of sudden changes in level of channel bottom applying weighted average flux scheme, MSc Thesis, Shiraz University of Technology, Shiraz. (In Persian)
Audusse, E., Bouchut, F., Bristeau, M.O. and Klein, R. (2004). Perthame, A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows, SIAM Journal on Scientific Computing, 25(6), 2050–2065.
Nieto, E.D. and Reina, G. (2008). Extension of WAF type methods to non-homogeneous Shallow Water Equations with pollutant. Journal of Scientific Computing, 37, 193-217.
Mahdavi, A. and Rakhshandehroo, G.R. (2012). Numerical Simulation of Unsteady Dam Break Flow Using Weighted Average Flux Scheme. Journal of Iran-Water Resources Research, 8, 64-80.
Pongsanguansin, T., Maleewong, M. and Mekchal, K. (2016). Shallow-water simulations by a well-balanced WAF finite volume method: a case study to the great flood in 2011, Thailand. Compute Geosci, 20, 1269–1285.
Toro, E.F. (2009). Riemann Solvers and Numerical Methods for Fluid Dynamic A Practical Introduction. 3rd ed. Springer, Verlag.
Toro, E.F. (2001). Shock-Capturing Methods for Free-Surface Shallow Flows. Wiley and Sons, LTD.
Toro, E.F., Ata, R., Pavan, S. and Khelladi, S. (2013). A Weighted Average Flux (WAF) scheme applied to shallow water equations for real-life applications. Advances in Water Resources, 62, 155-172.
Zhou, j.G., Causon, D.M., Ingram, D.M. and Mingham, C.G. (2002). Numerical solutions of the shallow water equations with discontinuous bed topography. International Journal of Numerical Methods in Fluids, 38, 769–788.