Simulating 2D open-channel flows through an SPH model
Introduction
Smoothed Particle Hydrodynamics (SPH) is a meshless Lagrangian method which proves to be well suited for studying complex fluid dynamics. First applied to astrophysics [1], this method has been successfully used to model free-surface flows (see e.g. [2]) especially when strong free-surface deformations take place, such as impact flows (see e.g. [3]), sloshing phenomena (see e.g. [4]) and breaking bores (see e.g. [5]).
Despite this, the SPH scheme has not been widely used to model open-channel flows. In the present work, attention is focused on this kind of problem which represents a key topic in the field of hydraulic river engineering.
In Eulerian models the imposition of inflow and outflow boundary conditions is relatively simple because each cell of the mesh describes a part of the domain and ghost cells can be used to impose boundary conditions. Conversely, the implementation of suitable upstream and downstream boundary conditions in the SPH model is not straightforward because of the Lagrangian nature of this scheme. Indeed, SPH particles move during the simulation and, consequently, they have to be conveniently inserted and removed from the domain. Furthermore, the interpolation procedure which is the basis of the SPH scheme makes the implementation of this kind of boundary condition rather difficult. Kajtar and Monaghan [6] simulated a confined flow past a tethered cylinder adopting free outflow conditions. However, they did not provide details of their inflow/outflow conditions as their work mainly focused on the coupling between fluid and rigid bodies. Later, Lastiwka et al. [7] proposed a model for the imposition of permeable boundary conditions in the context of gas dynamics. Unfortunately, this method cannot be straightforwardly applied to hydrodynamic problems because of the presence of the free surface.
As a result, a more general algorithm for in/out-flow boundary conditions is proposed here and validated in order to deal with free-surface channel applications. Specifically, this permits the study of uniform, non-uniform and unsteady free-surface flows. Through the use of suitable inflow and outflow buffer particles, the proposed algorithm allows the imposition of open-boundary conditions in continuous currents as well as the enforcement of different upstream and downstream conditions.
First, the proposed model is applied to viscous free-surface channel flows at low Reynolds numbers. The suitability of the in/out-flow algorithm is shown comparing the obtained velocity field with the analytical Poiseuille solution for uniform flow (see e.g. [8]). Then the capabilities of the algorithm are tested in non-uniform conditions through different upstream and downstream conditions. A typical phenomenon is the hydraulic jump, that is characterized by sharp discontinuities of the water level and strong dissipative effects. Varying the Froude number, different types of jumps are obtained: undular, breaking undular and weak jump. The results are validated with the classical hydraulic jump theory (see e.g. [9]) based on the balance of the upstream and downstream pressure forces. A comparison of the flow field with experimental data by Hornung et al. [10] is also provided. Finally, the proposed SPH model is used to model an unsteady flow case regarding a flash flood hitting a fixed bridge. This simulation mainly focuses on the evolution of the free surface and vorticity field generated by the impact and the time history of the global loads acting on the bridge.
Section snippets
Field equations
For a viscous, weakly compressible and barotropic fluid, the governing equations are: where and are, respectively, the position of a generic material point, its velocity, pressure and density, represents the mass force acting on the fluid, the initial density at the free surface, the initial sound speed and the viscous stress tensor.
For computational reasons, it is common practice in the weakly-compressible SPH solvers not to use
Free surface
The free-surface boundary conditions can be easily handled by the SPH method. Indeed, due to the Lagrangian character of the solver, the kinematic condition is intrinsically satisfied. As concerns the dynamic boundary condition, the enforcement of null-pressure along the free-surface is satisfied implicitly by the discrete Eq. (5) since these can be derived through a Lagrangian Variational Principle where the work done by the external pressure field on the free-surface is forced to zero even at
Test cases
Three different hydraulic test cases were simulated:
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The first one refers to a viscous free-surface channel flow in laminar regime. For this case only the upstream boundary conditions were imposed (specifically, a continuous intake of time-constant values of surface elevation, velocity and pressure) while the downstream boundary of the fluid domain was modelled as an open-boundary.
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In the second test case a hydraulic jump was considered. Here, both upstream and downstream boundary conditions
Conclusions
A new model to enforce in/out-flow boundary conditions in the SPH framework has been presented. This allows the assignment of different upstream and downstream conditions in open-channel flows. The proposed numerical scheme permits the investigation of uniform, non-uniform and unsteady flows.
Different sets of particles are defined in order to model the fluid flow, the channel bottom, the inflow and the outflow. The inflow and outflow particles affect the fluid particles but not vice versa, and
Acknowledgements
This research has received funding from the European Community’s Seventh Framework Programme (FP7/2007–2013) under grant agreement no. 225967 “NextMuSE”. This work was also partially supported by the Centre of Excellence for Ship and Ocean Structures of NTNU Trondheim (Norway) within the “Violent Water–Vessel Interactions and Related structural loads”.
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