Smoothed particle hydrodynamics applications in hydraulic engineering

Computational fluid dynamics is an important tool to solve many practical engineering problems. It offers insights to the fluid motion and the physics behind, and it plays an essential role in interpretations of the fluid phenomena. To solve the fluid motion equations, two main numerical techniques are adopted: Euleraian grid-based methods and Lagrangian particles methods. This research studies the smoothed particle hydrodynamics (SPH) method and its capability to solve problems in hydraulic engineering. SPH is a mesh-free Lagrangian method in which the fluid is discretized into non-connected particles with no grids. The particles are interpolation points carry all the fluid properties. The particle properties change with time based on the surrounding particles within its influence domain where the spatial dependencies are determined using a smoothing function. The non-connectivity nature of SPH allows the methods to easily handle complex dynamic problems. The SPH formulations are carried out for full Navier-Stokes equations (SPH-NS) and Saint Venant equations (SPH-SWE). The main difference between the two is that the particle in Saint Venant represents a water column and the assumption of hydrostatic pressure is hold. In SPH-NS, the weakly compressible (WCSPH) approach is used where it allows slight variations in the density field to calculate the density time derivate. The equation of state is utilized to determine the pressure-density relation. Incompressible flow (ISPH) approach is proposed in literature where the ideal density is used and the Poisson implicit equation is solved, however, ISPH is not cover in this report. Artificial viscosity is used as a viscous and stabilization term. Boundaries are simulated using boundary particles with repulsive force method and fluid boundary particles method. In SPH-SWEs the notion of two-dimensional density is introduced. A variable smoothing length is applied for stabilization purpose. Smoothing length is given as a function of the 2D density and therefore the density interpolation equation becomes implicit. The implicit set is solved using Newton-Raphson iteration. Lax Friedrichs flux is adopted as stabilization term. Virtual particles method and ghost particles method are implemented as close boundary methods. To simulate open boundaries, two rows of particles are placed just outside of the domain and parallel to the open boundary and their positions, velocities, and water depth are updated every time step. The simulations were carried out for both SPH-NS and SPH-SWEs using an open SPH source code called SPHysics. For SPH-NS, simple test cases were performed comprise of static, steady flow, and water column impact. In static case the exact hydrostatic pressure was not achieved due to the density variation of 1% in WCSPH approach. The boundary with a repulsive force prevented the fluid particles penetration but unphysical gab between the boundary and the fluid particles was observed. The fluid boundary particles method did not guarantee fully block and some fluid particles escaped outside of the domain. Both closed boundary methods do not provide a full support for the particles close to the boundary which affects the method consistency. The steady flow case was conducted to investigate the open boundaries implementation. The open boundaries were simulated by introducing a column of particles in the inflow boundary and removing the particles column that leaves the outflow boundary. To have a complete support the two columns of particles near each boundary were mirrored outside of the domain. This case showed that implementation of open boundaries in SPH-NS is a difficult task as it can be associated with great disturbance near the boundary. The use of WCSPH resulted in oscillations and noises in the pressure field which was exercised in the water column impact case. Shepard and Moving Least Squares density filters were used in this case and they provided better pressure field with less noise. Most of the hydraulic problems come with open boundaries and large domains and therefore most of the cases in this report were simulated using SPH-SWEs. SPH-SWEs was used in more practical applications with large scale. The simulation test cases start with simple problems of omputing water profile in non-uniform flow and predicting the flood propagation behaviour in a rectangular channel. The results of water profiles were satisfactory with negligible errors compared with the results obtained analytically using direct step method. The flood propagation behaviour in a mild channel and steep channel was typically achieved. To examine the method capability to conserve momentum a case of inflow water flowing over a hump was simulated and further compared to the results of other 2D modelling packages. The results showed that the method is momentum conserved and it also highlights the importance of determining the correct velocity and water depth at the inflow open boundary. Particle splitting procedure was adopted as one of the scenarios. A low momentum flooding simulation case in a floodplain with depressions was carried out to assess the method ability to predict the final inundation extent and water levels. In comparison to other 2D software results and with the use of small particle resolution and appropriate open boundary parameters, the method provided satisfactory results for the water levels and final inundation extent with acceptable agreement. Finally, the ghost boundary particle is implemented and it showed better outcomes than the virtual boundary particles. The ghost boundary particles SPH-SWEs was coded and added to the SPHysics solver as a subroutine. In order to achieve stable solution, a very limited time step is required for both SPH-NS and SPH-SWEs which makes the method computationally expensive. Efforts have been made to parallelized the code and run on GPUs which reduce the computational time considerably, however, this comparison is not covered in this research. The test cases in this research show that SPH has a great potential to be used in different problems in hydraulic engineering. Although, the method is still not fully matured numerically and algorithmically, it provides satisfactory results, and with some improvements it will become a powerful and robust technique