Robust boundary treatment for open-channel flows in divergence-free incompressible SPH

Highlights

• Exact specification of inflow/outflow boundary for open channel flow.

• Hybrid free-surface boundary treatment to minimize error in velocity divergence.

• The ISPH model can simulate laminar and turbulent flow alike.

• Provides smoothed and structured pressure variation.

Abstract

A robust Incompressible Smoothed Particle Hydrodynamics (ISPH) framework is developed to simulate specified inflow and outflow boundary conditions for open-channel flow. Being purely divergence-free, the framework offers smoothed and structured pressure distribution. An implicit treatment of Pressure Poison Equation and Dirichlet boundary condition is applied on free-surface to minimize error in velocity-divergence. Beyond inflow and outflow threshold, multiple layers of dummy particles are created according to specified boundary condition. Inflow boundary acts as a soluble wave-maker. Fluid particles beyond outflow threshold are removed and replaced with dummy particles with specified boundary velocity. The framework is validated against different cases of open channel flow with different boundary conditions. The model can efficiently capture flow evolution and vortex generation for random geometry and variable boundary conditions.

Introduction

Open channel flows are difficult to deal with in Lagrangian framework due to the presence of free-surface and varying inflow/outflow boundary conditions. Moreover, it is challenging to maintain divergence-free nature of the flow throughout the simulation period. In recent years, Smoothed Particle Hydrodynamics (SPH) has emerged as one of the best methods for treating incompressible flows in a Lagrangian framework. Being a purely meshfree particle method, advection is treated exactly in SPH. It can precisely handle unsteady flow fluctuation inside a closed domain. However, open-channel flow modelling becomes difficult in SPH because of varying boundary conditions. Most of the previous unsteady free-surface cases take comparatively less computational overhead due to closed domain. Open channel flow simulations will consume much larger computational burden because of time required for flow stabilization and continuous injection/removal of particles at the boundary. The present work proposes a robust divergence-free framework for handling free-surface and inflow/outflow boundary conditions for open-channel flow simulation.

SPH, invented for solving astrophysical problems (Gingold and Monaghan, 1977, Lucy, 1977), was first utilized in free-surface flow simulation by Monaghan (1994). However, weakly compressible SPH models (Liu and Liu, 2003, Gomez-Gesteira et al., 2010, Violeau and Rogers, 2016) require an artificial equation of state (Liu and Liu, 2010) and consequently very small time-step (Lee et al., 2008). Incompressibility in SPH can be handled with two different approaches (a) density-invariant (Shao and Lo, 2003) (b) divergence-free (Cummins and Rudman, 1999). Density invariance approach solves Pressure Poisson Equation (PPE) in terms of particle number density. However, numerical noise in pressure variation (Xu et al., 2009) may occur in the process. The divergence-free approach considers the density of fluid particles remains unchanged throughout the simulation (Pahar and Dhar, 2016b). Obtained pressure profile also shows smoother transition compared to density-invariant approach. In SPH, a periodic condition is frequently used to simulate inflow/outflow boundary. The particles going out from the fluid domain is injected into the inlet (Lee et al., 2008). Inlet/outlet of the fluid domain is presumed to be continuous. On both sides, particle appears in each other’s support domain. The principle of a periodic boundary can be analogous to the experimental idea of recirculating flume. However, periodic boundary condition does not offer specific inlet/outlet boundary as the flow is allowed to develop based only on initial condition. Lastiwka et al. (2009) used analytical expression between pressure, density, and velocity for modelling open channel flow. Shakibaeinia and Jin (2010) proposed a weakly compressible moving particle semi-implicit method for modelling different inflow/outflow boundaries. Later their strategy of recycling outgoing particles as inflow boundary was used for incompressible models (Shakibaeinia and Jin, 2011). Vacondio et al. (2012) modelled explicit shallow water equations using buffer particles at boundary thresholds. Federico et al. (2012) modelled inflow/outflow boundary using weakly compressible SPH model. Multiple layers of particles are positioned just outside fluid domain to take care of specified boundary conditions. Chang et al. (2014) solved one-dimensional non-hydrostatic shallow water equation with open boundary treatment. A concept of cylindrical particles was utilized for horizontal flood inundation in SPH with shallow water equations by Kao and Chang (2012). However, shallow water equations consider a depth-averaged approach (Chang et al., 2011, Chang et al., 2016) leading to loss of information regarding vertical velocity component (Pahar and Dhar, 2014). Almost every weakly compressible SPH model requires a direct/indirect artificial viscosity term. They also do not necessarily conserve solenoidal velocity criteria. Weakly compressible SPH has an advantage of solving pressure directly from the intermediate density of the fluid particles unlike Incompressible SPH (ISPH). ISPH with inflow/outflow was addressed by Khorasanizade and Sousa, 2016, Tan et al., 2015 for flow in closed and open domain respectively. However, the pressure obtained from Pressure Poisson Equation with density related source/sink term may contain numerical noises (Xu et al., 2009). Numerical viscosity in the form of XSPH may be required to ensure proper particle movement. Khorasanizade and Sousa (2015) simulated flow around a cylinder in a domain enclosed at top and bottom with pure divergence-free ISPH with modified particle shifting algorithm. Beyond boundary threshold, particles progress with specified boundary condition. However, they did not consider the effect of free-surface. Tan et al. (2015) considered a density invariant approach for laminar and turbulent open-channel flow over a smooth bed. They also investigated the effect of artificial drag force resulting from boundary treatment. XSPH variation was used by them to achieve steady open channel flow. Hirschler et al. (2016) developed a multiphase incompressible flow model with open boundary. They have utilized muti-phase ISPH-DFDI approach by Hu and Adams (2007) for flow through a closed domain. It has been mentioned that ISPH-DFDI cannot be extended for free-surface flow. However, purely divergence-free incompressible SPH model with particle shifting (Lind et al., 2012) is capable of precise open channel flow modelling. Density-invariant ISPH produces larger error in velocity divergence (Gui et al., 2015) compared to the divergence-free formulation.

It is evident that majority of existing models did not concentrate on purely divergence-free open channel flow simulation in SPH. Moreover, the boundary conditions are not specified directly to handle complex flow situations. Most of the studies in the literature require renormalization algorithm and artificial viscosity to reduce the effect of spurious oscillations during simulations. This study proposes a robust divergence-free incompressible SPH model for simulating specified inflow/outflow boundary conditions. Inflow/outflow boundary conditions are satisfied by providing multiple layers of particles and specifying velocity on them beyond domain thresholds. Being a purely divergence-free model, the framework provides smooth and structured pressure variation. Implicit shifting at the end of every time level ensures proper distribution of fluid particles throughout the domain. Proposed framework is validated against existing analytical/experimental flow systems ranging from steady laminar to unsteady turbulent. The first scenario considers a laminar uniform flow in an open channel. The second scenario simulates the evolution of weak hydraulic jump. In the third scenario, time-averaged flow through a trapezoidal trench is considered and compared with experimental data. Free-surface flow over a rigid block is taken in the fourth scenario.