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When investigating water flow in spillways and energy dissipation, it is important to know the behavior of the free surfaces. To capture the real dynamic behavior of the free surfaces is therefore crucial when performing simulations. Today, there is a lack in the possibility to model such phenomenon with traditional methods. Hence, this work focuses on a parameter study for one alternative simulation tool available, namely the meshfree, Lagrangian particle method Smoothed Particle Hydrodynamics (SPH). The parameter study includes the choice of equation-of-state (EOS), the artificial viscosity constants, using a dynamic versus a static smoothing length, SPH particle spatial resolution and the finite element method (FEM) mesh scaling of the boundaries. The two dimensional SPHERIC Benchmark test case of dam break evolution over a wet bed was used for comparison and validation. The numerical results generally showed a tendency of the wave front to be ahead of the experimental results,
*i.e.* to have a greater wave front velocity. The choice of EOS, FEM mesh scaling as well as using a dynamic or a static smoothing length showed little or no significant effect on the outcome, though the SPH particle resolution and the choice of artificial viscosity constants had a major impact. A high particle resolution increased the number of flow features resolved for both choices of artificial viscosity constants, but at the expense of increasing the mean error. Furthermore, setting the artificial viscosity constants equal to unity for the coarser cases resulted in a highly viscous and unphysical solution, and thus the relation between the artificial viscosity constants and the particle resolution and its impact on the behavior of the fluid needed to be further investigated.

In Sweden, hydropower plays a significant role in the supply of energy and it generates roughly 45% (66.0 TWh in 2011) of the total electricity power production [

The superscripts α and β are used to denote coordinate directions and the subscripts i and j denotes particle indices. The continuity and momentum equations can be written in the SPH formalism according to,

where _{j} is the particle mass, g is the acceleration due to gravity and x_{i} and

In Equation (3), dim is the number of space dimensions and

and h is the smoothing length. The commonly used cubic B-spline kernel is obtained by choosing

where C is a constant of normalization defined for one-, two- and three-dimensional spaces as: C_{1D} = 3/2, C_{2D} = 45/14π and C_{3D} = 9/4π. In Equations (3)-(5), R is the normalized distance between two particles, i.e.

The NULL material model will be applied that defines the deviatoric viscous stress as [

where

and for expanded materials as,

where S_{1}, S_{2} and S_{3} are coefficients of the slope of the u_{s}-u_{p} curve, u_{s} and u_{p} are the chock and particle velocity respectively. The variable c_{0} is the intercept of the curve corresponding to the adiabatic speed of sound, γ_{0} is the Gruneisen gamma, _{0}, E_{0} is the initial internal energy and

References [

where c_{0} in Equation (11) is the adibatic speed of sound and

which can be obtained by setting the constants

where

q_{1} and q_{2} are constants and _{1}in the artificial viscosity expression produces both bulk and shear viscosity. The second quadratic term is incorporated to handle high Mach number shocks and is intended to suppress particle penetrations [_{1} and q_{2} are according to [_{2} to zero and q_{1} between 0.01 and 0.1 [_{1} and q_{2} are usually set around unity. The impact of the choice of the constants will be another parameter to be studied in the present work. With the reduction of the total stress tensor, the inclusion of the artificial viscosity term together with the symmetric properties of the

A first order time integration scheme is applied and the time step is determined according to [

S_{1} | S_{2} | S_{3} | c_{0} [m/s] | γ_{0} | α | E_{0} [J/kg] | 𝜌_{0} [kg/m^{3}] |
---|---|---|---|---|---|---|---|

1979 | 0 | 0 | 1484 | 0.11 | 3.0 | 3.072 × 10^{5 } | 1000 |

where C_{CFL} is the Courant number usually set to unity. The smoothing length h is either static or dynamic. For the dynamic case h is obtained from,

For numerical reasons, a minimum and maximum value is imposed on the smoothing length, i.e. H_{min}h_{0} < h < H_{max}h_{0}. In the present study, H_{min} and H_{max} were set equal to 0.5 and 1.5 respectively while h_{0} is the initial smoothing length given by_{0} is the initial distance between particles. The impact on the results from using a static or dynamic smoothing length will also be studied in the current study.

Solid wall boundaries are modelled as rigid shell finite elements and the coupling between the boundaries and the SPH-particles are governed by a penalty based “node to surface” contact-algorithm [

To validate the SPH model experiment, the SPHERIC Benchmark test case of dam break evolution over a wet bed is used. Numerous authors have adopted this test case for validation proposes, e.g. [

In the experiment by [_{TD} = 0.15 m and channel depth L_{CD} = 0.018 m. The tank length L_{TL} = 0.38 m was set as in the experiment and the channel length L_{CL} = 1.62 m was smaller than in the experiment (L_{CL} = 9.55 m) in order to reduce the overall number of particle and hence computational time. According to [

the result section. The gate velocity was constant and measured to 1.5 m/s. According to the Benchmark test case instruction, it is vital to include the gate movement in simulations.

As stated above, the three EOSs including different values of the artificial viscosity constants q_{1} and q_{2} are to be tested and compared with experimental results. Thus, six cases is set up, see _{CD}, here defined as the characteristic length scale of the problem, by 4, 5, 6 and 10. All SPH particles were initially placed on a structured grid with zero initial velocity. Secondly, the scaling of the FEM boundaries was tested where the side of a FEM element was set equal to 2x, 1x and 0.5x the initial interparticle spacing. Finally, the dynamic (D) and static (S) smoothing length was compared. All parameters are summarized in

The initial density ^{3} and all simulations were stopped at 0.488 s due to the time shift proposed by [

The post processing method used in this work incorporates the detection of the free surface as well as interpolation of data. The method is divided into two steps. Firstly, all nodes representing the boundary must be detected and secondly a Delaunay triangulation and Barycentric interpolation is performed, for a detailed description of Delaunay triangulation and Barycentric interpolation see [

where b_{x} is the barycentric coordinate and f_{x} are the function values such as velocity at the triangle vertices. As

EOS | Artificial viscosity constants |
---|---|

Gruneisen | (q_{1} = 0.1; q_{2} = 0) and (q_{1} = 1; q_{2} = 1) |

Linear polynomial | (q_{1} = 0.1; q_{2} = 0) and (q_{1} = 1; q_{2} = 1) |

Morris | (q_{1} = 0.1; q_{2} = 0) and (q_{1} = 1; q_{2} = 1) |

SPH resolution L_{CD}/4 | SPH resolution L_{CD}/5 | SPH resolution L_{CD}/6 | SPH resolution L_{CD}/10 | |
---|---|---|---|---|

FEM scaling: 2x | S1 and D1 | S2 and D2 | S3 and D3 | S4 and D4 |

FEM scaling: 1x | S5 and D5 | S6 and D6 | S7 and D7 | S8 and D8 |

FEM scaling: 0.5x | S9 and D9 | S10 and D10 | S11 and D11 | S12 and D12 |

data is mapped to a structured grid, traditional techniques for data visualization such as streamline plotting can then be used. Apart from data visualization, the post processing method can be used to statistically quantify the difference between numerical and experimental results. By imposing both the numerical and experimental results on the same grid the non-overlapped grid points can be identified. The error is here defined as number of non-overlapped grid points divided by the total number of points in the experiment for each time step. The mean error is then the average of all time step errors. Non-overlapped grid points can be seen as an area and hence when the non-overlapped area goes to zero the mean error goes to zero as well.

The post processing method described in the section above was used to visualize the results. To the left in _{1} and q_{2} set equal to unity. One of the most significant discrepancies when comparing the experimental and numerical results is that the wave front obtained from the SPH simulation is ahead of the experimental wave front, see _{CD}/10 and to some extent in the L_{CD}/6 cases only. One possible explanation that the SPH wave front is ahead of the experiments is due to the frictionless contact between the SPH particles and the FEM mesh, i.e. the free-slip boundary condition. There is a possibility to include a friction force in the contact formulation. However, the proposed friction formulation available is not applicable for these types of problems [

Having these zones in mind, Figures 5-8 show the effects of particle resolution on the outcome for the three EOSs and artificial constants q_{1} and q_{2}. Dummy nodes are used to visualize the free surface and the

Absolute Velocity Field [m/s] and Streamlines―Gruneisen q_{1} = 1 q_{2} = 1 D12

shaded grey area is the experimental results. Furthermore, the choice of setting q_{1} = 1 and q_{2} = 1 is shown in blue and q_{1} = 0.1 and q_{2} = 0 in red respectively. Few flow features are visible for the case with lowest resolution (L_{CD}/4), see _{1} and q_{2}. When increasing the resolution to L_{CD}/5 (_{CD}/6 the mushroom jet is more pronounced than in the cases with lower resolution, see _{1} = 0.1 and q_{2} = 0. For the cases with finest resolution (L_{CD}/10) most of the flow features shown in the experiment are now visible in the numerical results for both choices of the artificial viscosity constants, see _{CD}/10 and to some extent in the L_{CD}/6 cases, see

As shown above there is a qualitative agreement between simulations and experiments if the set-up is fine enough. Using the method proposed in the post-processing section a quantitative comparison can also be carried out in the form of the mean error that is summarized for all cases in _{1} and q_{2} not the EOS. Selecting q_{1} and q_{2} equal to unity seems to produce the better result. However, as can be seen in _{1} and q_{2} equal to unity is due to

the highly viscous solution which reduces the wave front velocity and hence lowering the mean error. This makes it cumbersome to find a good relation of spatial resolution and artificial viscosity constants to obtain a trustworthy solution. Secondly, the results connected to the L_{CD}/10 cases or the case with the highest resolution seems to produce the worst results. This is in sharp contrast to the expected outcome, as generally more nodes in any numerical method should produce a better result and also the qualitative investigation indicated this. One plausible explanation to this behavior is that more features are resolved using more particles such as the breaking wave and zone four. Also, the height of the secondary splash produced mainly in the L_{CD}/10 cases are generally over predicted thus increasing the number of none overlapping grid points, i.e. increasing the mean error, as can be seen in _{0} which was set equal to the physical speed of sound of 1484 m/s. Usually, c_{0} is set equal to at least 10 times the expected maximum velocity in the flow [_{ref} is the maximum water height in the tank (H_{ref} = 0.15 m) as according to [

In the zeroth order Shepard density filter, the simplest and quickest type of density filter, oscillations are smoothed out by filtering density and then re-assigning a new density to each particle [

The two dimensional (2D) SPHERIC Benchmark test case of dam break evolution over a wet bed has been used to investigate the impact of several parameters when performing free surface flow simulations. The choice of EOS, the artificial viscosity constants, a dynamic versus a static smoothing length, SPH particle resolution and the FEM mesh scaling of the boundaries were investigated. The numerical results showed generally a tendency to be ahead of the experimental results, i.e. to have a greater wave front velocity. This was argued to be an effect of using a frictionless contact algorithm between the SPH particles and the FEM boundaries. The choice of EOS, FEM mesh scaling as well as using a dynamic or a static smoothing length showed little or no significant improvement to the result. However, all EOSs showed larger than physical pressure levels in the order of 10^{6} Pa and large pressure oscillations. This is believed to be an effect of not reducing the speed of sound in the calculations, which is usually done. The SPH particle resolution and the choice of artificial viscosity constants had a major impact. The method used for comparing numerical and experimental results showed increased mean error for both highly resolved cases and artificial viscosity constants set equal to 0.1 and 0. However, by visual observation, it was noted that increasing the number of particles representing the system increased the number of flow features resolved and that a highly viscous solution was obtained by setting the artificial viscosity constant to unity. Thus, the relation between the artificial viscosity constants and the particle resolution and its impact on the behavior of the fluid needed to be further investigated.

The research presented was carried out as a part of “Swedish Hydropower Centre―SVC”. SVC has been established by the Swedish Energy Agency, Elforsk and SvenskaKraftnät together with Luleå University of Technology, The Royal Institute of Technology, Chalmers University of Technology and Uppsala University. www.svc.nu