Numerical Simulation of Water Flow Passing over the Steps using the Weakly Compressible Smoothed Particle Hydrodynamics Method with the Cubic Spline Kernel

The Eulerian framework is the most widely used method for modelling physical problems. This type of modelling requires the use of mesh-dependent numerical methods to solve differential equations. In addition to the problems of stability and expensive processing, these types of methods encounter serious problems in complex geometries and moving boundaries. On the other hand, the basic assumption of Lagrangian modelling is to divide the domain into particles. The smoothed particle hydrodynamics method is a common meshless method that has received considerable attention. This method replaces a fluid with a set of particles to achieve an approximate numerical solution. of fluid dynamics equations. Because of the capability of this method to simulate flow on a large and complex scale, in this study, the flood phenomenon was simulated on a smaller scale. Therefore, flood flow with less water volume and obstacles in the flood path is considered as a positive and negative step. According to the authors' information, such an application of the smoothed particle hydrodynamic method has not yet been simulated.

To simulate the water flow over the steps using the Weakly Compressible Smoothed Particle Hydrodynamics method, a two-dimensional model of the steps and water should be created first. Then, the governing equations of the fluid flow should be discretised, and the characteristics of the flow and modelling algorithm should be determined. In the next step, SPHYSICS code is used to define the boundaries. In this process, small particles are considered mass representatives of the water flow and walls. In this research, a comparison between the experimental results of a laboratory model of a dam-break flow and the numerical results was performed for verification. Considering that the particle distance and kernel type are very important in the Weakly Compressible Smoothed Particle Hydrodynamics method, the effects of these parameters on the results of the mentioned method are analysed and investigated. Subsequently, the modelling of water passing over the steps is performed and analysed in two scenarios: (I) dam break and water flow falling from two consecutive steps downwards, and (II) collision and passage of water flow over two fixed obstacles. The simulation results from both scenarios were compared with the results of the STAR-CD software, which is a mesh-based commercial software solving the Navier-Stokes equations based on the finite volume method.

The available experimental data of a dam-break problem were adopted to determine the validity of the Weakly Compressible Smoothed Particle hydrodynamics method. In addition, two kernel functions, the cubic spline and quintic spline, were considered to investigate their performance during this test. A comparison of the numerical model results with experimental data showed that the Weakly Compressible Smoothed Particle Hydrodynamics method produces acceptable outputs. Moreover, the cubic spline performed better than the other kernel functions for the dam-break problem. Then, the effect of the particle distance on the numerical results has been investigated and concluded that the use of the particle size of 0.01m compared to 0.0009m increases the error criteria. This analysis can be generalised to other particle distances. By examining the results of changing the particle distance, it is possible to understand the superiority of selecting smaller particle distances. However, it is worth noting that selecting smaller intervals increases the cost and computational time. The dam-break flow over two consecutive negative steps is simulated in the next step. The results were also compared with the STAR-CD software. In the second scenario, two obstacles in the form of steps were located, and a dam-break flow collided with these obstacles. Despite the complexity of the simulation process for these cases and the turbulent nature of the collision, the results obtained using the Weakly Compressible Smoothed Particle Hydrodynamics with cubic spline kernel function were in good agreement with the STAR-CD results. This proves the applicability of the numerical method.

This research showed that reducing the distance between particles causes more convergence of the results, although it increases the calculation time. In addition, the cubic spline kernel generates better results than the quintic spline kernel. A comparison was made between the results of the Weakly Compressible Smoothed Particle Hydrodynamics method and the outputs of STAR-CD. Moreover, the RMSE and Euclidean Error Norm criteria were calculated. Despite the differences between the methods, good agreement between the results was observed, which shows that the Smoothed Particle Hydrodynamics method can be used as a suitable method to simulate the flow over the steps.