The University of Edinburgh^{1}

**Introduction**

This study presents two unique curvilinear methods to produce grids generation for any curvature boundary. Partial differential equations with different kinds of boundaries, i.e., circular, cylindrical, elliptical or any other arbitrary boundaries could not be solved easily. One of the most applicable methods in this respect is curvilinear system that solves the partial differential equation with any arbitrary boundary. This study investigates the curvilinear system modelling of shallow water equation as governing PDE in open channels. The present study involves the following stages: development of an iterative solver of the curvilinear grid generation including Successive over relaxation (SOR) and Multi-grid (MG) solvers for producing grid generation ,evaluation of Grids produced by SOR and MG against each other , derivation and numerical solution of the curvilinear shallow water equations, simulation of two dimensional shallow water equations for non-distorted grid generation and distorted curvilinear grid generation and last but not least the verification and validation of the solver against benchmark test cases. In this research, steady uniform flow in a sloping rectangular channel is used to verify the shallow flow solvers. The findings are directly applicable to river engineering and river basin management. In this paper two powerful iterative methods to solve Shallow water equations are compared against each other for the first time.

**Methods/Materials**

Successive over relaxation (SOR) is the first method to develop curvilinear systems. SOR was introduced by Thompson (1977) to produce grids generation for any arbitrary two dimensional bodies. Thompson maintains that any curvature boundary transforms to grids lines and solves grids based on a numerical method such as finite difference method (FDM). In this method, Poisson-type elliptic equations are commonly used to transform the physical plane onto a rectangle. This method is applied Jacobi or Gauss-Seidel iteration schemes to create grids. Multi-grid (MG) is the second grid producer to develop curvilinear systems. The method was first introduced by Fedoranko in (1964). Multi-grid method is an efficient iterative method For solving boundary-value elliptic problems such as Laplace, Poisson, and Helmholtz equations. This iterative method suggests linear equation system with high order. In this paper Multi-grids are applied to solve the Poisson equation that is discretised by finite difference method. The features of Multi-grid's method are as follows: an iterative method with high speed of convergence which is replaced for pervious iterative methods with slow speed of convergence such as Jacobi. This new solver may optimize and decrease the number of iteration, by applying relaxation method to speed up convergence in iteration numbers. After comparing the results of Grid generation for the two grid generation producers, the hydraulic part of project is carried out and different kinds of Curvilinear methods are applied to solve Shallow water equation in two dimensions. The Shallow Water Equations (SWE) applies to almost horizontal flows in wide open channels, rivers, and lakes. These equations can be derived from fundamental control volume analysis of three-dimensional elements and depth-averaging, control volume analysis of elements that extend through the depth with velocity assumed uniform in the vertical, and from potential theory. The SWEs comprises of fundamental fluid dynamic equations related to mass conservation (continuity) and force/momentum balance. In order to solve the shallow water equations on a curvilinear grid, the equations must themselves be transformed into the mapped system, by means of the transformation Jacobian.

**Results and Discussion**

The benchmark case of steady uniform flow in a sloping rectangular channel is used to verify that the shallow flow solvers provide the correct balance between flow gradient and bed resistance terms. The local and advective acceleration terms are also tested as the flow builds up from initially still conditions. In this case, the open channel has 1000 m length and 240 m width. Its lateral walls are frictionless. The initial water depth is 5 m throughout the channel, bed slop is 1:1000, and the initial mass flux is set to zero everywhere. To investigate the problem in the context of curvilinear grid generation, the MG and SOR programs are used to create rectangular channel.

To determine the velocity in the open channel, shallow water equation is used and the velocity vectors are plotted for the rectangular channel. Uniform flow in an open channel is simulated using two time integration methods (Adams-Bashforth and Runge-Kutta). To validate numerical schemes against analytical solution for predicting velocity in open channel, Chezy friction law is employed. Numerical schemes converged to equilibrium the same as the analytical solution. In this study, Converged time is 5000 seconds and time step is 7.0 for Runge-Kutta method and 1.0 for Adams-Bashforth method.

**Conclusion**

This research compares two powerful iterative methods to solve one of the most important equations in Hydraulic engineering. The comparison of these iterative methods against each other and their validation for arbitrary curvature physical boundary proves the curvilinear system's ability to address and simulate any fluid mechanic problem. The simulation of 2-D Shallow Water Equations (SWE) reveals the agreement between Cartesian SWE and Transformed SWE. Both Cartesian SWE and Transformed SWE have same velocity profile for the rectangular channel. For further studies, the application of curvilinear system is recommended to solve other Hydrodynamic equations, such as Euler equations and Navier-Stokes equations. It is to be mentioned that this study is directly applicable to river engineering and open channel simulation. *1. Borthwick, A. G. L. and Barber, R. W. (1992). River and reservoir flow modelling using the transformed shallow water equations. international journal for numerical methods in fluids. 14,1193-1217. *

*2. Fedorenko (1964). The speed of convergence of an iterative process, USSR Comput. Math, and Math. Phys. 4(3), 227-235. *

*3. Lee, Wei-Koon, Borthwick ,A. G. L. and Taylor P.H. (2011). A fast adaptive quadtree scheme for a two-layer shallow water model. Journal of Computational Physics. 230, 4848Â–4870. *

*4. Liang,Qiuhua and Borthwick,A. G. L.(2008). Simple treatment of non-aligned boundaries in a Cartesian grid shallow flow model. International Journal for Numerical Methods in Fluids. 56, 2091Â–2110. *

*5. Scott A. Yost, Prasada Rao (2001). A multiple grid approach for open channel flows with strong shocks. Journal of Applied Mathematics and Computation. 124, 381-395. *

*6.Thompson, J. F. , Thames, F. C. and Martin, C. W. (1974). Automatic numerical generation of body fitted curvilinear coordinate systems on fileds containing any number of arbitrary two dimensional bodies. Journal of Computational Physics. 15, 299Â–319. *

*7. Triki (2014). Multiple-grid finite element solution of the shallow water equations: water hammer phenomenon. Journal of Computers & Fluids. 90, 65Â–71. *

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