In this paper, we propose a simplified approach of open boundary conditions for particle-based fluid simulations using the weakly compressible smoothed-particle hydrodynamics (SPH) method. In this scheme, the values of the inflow/outflow particles are calculated as fluid particles or imposed desired values to ensure the appropriate evolution of the flow field instead of using a renormalization process involving the fluid particles. We concentrate on handling the generation of new inflow particles using several simple approaches that contribute to the flow field stability. The advantages of the

Open boundary conditions pose an interesting challenge that has attracted a number of investigations by different authors. The simplest and fastest technique, periodic boundary conditions [

The rapid development of variants of the SPH model, specifically the

To simplify the treatment of the open boundary condition, in the present study, we propose a fast and simple approach based on the

In the SPH method, the governing equations for weakly compressible SPH in their Lagrangian form are:

Recently, the

We adopted the leapfrog method for the integration time. For stability, the time step

The PST was derived by Lind et al. [

here, the symbols

After the time integration, the pticle position is shifted slightly by an amount

Open boundary conditions pose an interesting challenge in numerical simulations. The inflow and outflow zones can be defined as being upstream and downstream, respectively, of the fluid domain [

In general, particle-based simulion with open boundary conditions requires four sets of particles that correspond to fluid, wall, inflow and outflow particles as shown in

In order to avoid the extrapolation process and perform the computation with a simple procedure, we modify the algorithm so that the SPH can be applied not only at the fluid zone but also at inflow/outflow zones. Similar to other inflow/outflow algorithms, we assume that the inflow zone is placed in front of the fluid region so that the attached zone covers a region as wide as the radius of the kernel support

The inflow particles move according to their velocity until they cross the border between the inflow and fluid zones.

The inflow particles that cross the border turn into fluid particles and behave as fluid particles.

At the same time, the particles that cross the border produce new inflow particles periodically at the front end of the inflow zone and we copy all the information of the particle to the new inflow particle.

Assuming

In terms of the particles at the outflow zone, we apply the SPH algorithm in the same way as the fluid zone and we simply eliminate particles that go out of the outflow zone as shown in

In some cases of the extrapolation-based open boundary conditions, prescribed particle quantities given a priori are applied to some of the inflow particles during the entire simulation [

To demonstrate the effectiveness and applicability of the proposed technique, in this section, we perform several numerical test cases. Specifically, we simulate:

Viscous open-channel flow;

Flow past a circular cylinder at Re = 200; and

Flow over a backward-facing step with prescribed and non-prescribed inflow boundary conditions.

A test case of viscous open-channel flow in the laminar regime was conducted to illustrate the stability of the flow over the entire computational domain. The objective of the simulation was to authenticate the steadiness of the velocity field after a sufficiently long period of time and to compare it to the analytical solution shown in

The initial setup is shown in

The Reynolds number is calculated such that

The sound speed was selected to be equal to

The initial boundary condition was imposed as follows:

To verify the stability, three different

To check the convergence of the velocity field between the proposed technique and the analytical solution, the mean square error percent (MSEP) was calculated using

The second test case consistedf flow past a circular cylinder, where the circular cylinder is represented by an implicit function:

No-slip boundary conditions were enforced on the body surfaces, and the information concerning these solid particles was extrapolated from the fluid particles following the method in [

For the upper and lower walls, symmetry boundary conditions for the velocity were imposed as described in [

^{3}, a constant x-velocity of ^{2}/s.

To prevent an impulsive start of the velocity field, a constant acceleration

The drag and lift force coefficients on the body are calculated such that:

Validation for this case was performed by comparing the time histories of the drag and lift coefficients using the technique proposed in Tafuni et al. [

A 2D backward-facing step problem was simulated at

The step height was set to

The reattachment length was determined to be

The velocities at the four different marked positions, P1–P4, were considered over the channel height and compared to the profiles of the reference results from [

In this case, a 2D backward-facing step problem was simulated at

In this simulation, the density was

The fluid flow was driven by a constant body force

Assessing the quality of simulation, the axial velocities at four different marked positions, P1–P4, were considered over the channel height and compared to the reference profiles from [

In this paper, we developed a simplified approach of open boundary conditions. Taking advantage of the

The numerical tests demonstrate that the proposed technique obtains good results with a high agreement with the reference solutions. The first test case demonstrated the stability of the flow field over a sufficiently long time with the MSEP value of the velocity field being approximately 0.1%. Given this stability, we compressed the computational domain to a lower resolution in a second test case to demonstrate the high accuracy of the simulation. The third test case examined the flexibility of the inflow boundary conditions to prescribed or non-prescribed values. This case developed results well suited to the wall boundary and the evolution of the flow field. The performance of the results demonstrated the versatility of the proposed technique. Overall, the proposed technique has addressed the information on inflow/outflow particles without extrapolation from the fluid zone.

One of the limitations is that particles in the fluid zone can move across the borders of inflow zones in case of flow reversion. This issue is pointed out also in [

This research was supported by the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan.

^{+}-sPH model: Simple procedures for a further improvement of the SPH scheme