European Journal of Mechanics B/Fluids 34 (2012) 35–46 Contents lists available at SciVerse ScienceDirect European Journal of Mechanics B/Fluids journal homepage: www.elsevier.com/locate/ejmflu Simulating 2D open-channel flows through an SPH model I. Federico a , S. Marrone b,c , A. Colagrossi c,d,∗ , F. Aristodemo a , M. Antuono c a Department of Soil Conservation, University of Calabria, Arcavacata di Rende (CS), Italy b Department of Mechanics and Aeronautics, University of Rome ‘‘Sapienza’’, Rome, Italy c CNR-INSEAN, The Italian Ship Model Basin, Rome, Italy d CESOS: Centre of Excellence for Ship and Ocean Structures, NTNU, Trondheim, Norway article info Article history: Received 24 June 2011 Received in revised form 14 December 2011 Accepted 14 February 2012 Available online 26 February 2012 Keywords: Smoothed Particle Hydrodynamics Inflow/outflow boundary conditions Open-channel flows Laminar flow Hydraulic jump Flash flood abstract The present work deals with the development and application of a 2D Smoothed Particle Hydrodynamics (SPH) model to simulate a broad range of open-channel flows. Although in the last decades the SPH modelling has been widely used to simulate free-surface flows, few applications have been performed for free-surface channels. For this reason, an appropriate algorithm is developed to enforce different upstream and downstream flow conditions and simulate uniform, non-uniform and unsteady flows. First, the proposed algorithm is validated for a viscous laminar flow in open channel characterized by Reynolds numbers of order O(10 2 ). The second test case deals with a hydraulic jump for which different upstream and downstream conditions are needed. Varying the Froude number, several types of jumps are investigated with specific focus on the velocity field, pressure forces, water depths and location of the jump. Comparisons between numerical results, theory and experimental data are provided. Finally, the interaction between a flash flood generated by an unsteady inflow condition and a bridge is shown as an example of an engineering application. © 2012 Elsevier Masson SAS. All rights reserved. 0. Introduction Smoothed Particle Hydrodynamics (SPH) is a meshless La- grangian method which proves to be well suited for studying com- plex 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 bound- ary 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 na- ture of this scheme. Indeed, SPH particles move during the simula- tion and, consequently, they have to be conveniently inserted and ∗ Correspondence to: CNR-INSEAN, Via di Vallerano 139, Rome 00128, Italy. Tel.: +39 06 50 299 343; fax: +39 06 50 70 619. E-mail address: a.colagrossi@insean.it (A. Colagrossi). removed from the domain. Furthermore, the interpolation proce- dure which is the basis of the SPH scheme makes the implementa- tion 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 fo- cused on the coupling between fluid and rigid bodies. Later, Lasti- wka 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 0997-7546/$ – see front matter © 2012 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.euromechflu.2012.02.002