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A hydraulic jump is a rapid transition from supercritical flow to subcritical flow characterized by the development of large scale turbulence, surface waves, spray, energy dissipation and considerable air entrainment. Hydraulic jumps can be found in waterways such as spillways connected to hydropower plants and are an effective way to eliminate problems caused by high velocity flow, e.g. erosion. Due to the importance of the hydropower sector as a major contributor to the Swedish electricity production, the present study focuses on Smoothed Particle Hydrodynamic (SPH) modelling of 2D hydraulic jumps in horizontal open channels. Four cases with different spatial resolution of the SPH particles were investigated by comparing the conjugate depth in the subcritical section with theoretical results. These showed generally good agreement with theory. The coarsest case was run for a longer time and a quasi-stationary state was achieved, which facilitated an extended study of additional variables. The mean vertical velocity distribution in the horizontal direction compared favorably with experiments and the maximum velocity for the SPH-simulations indicated a too rapid decrease in the horizontal direction and poor agreement to experiments was obtained. Furthermore, the mean and the standard deviation of the free surface fluctuation showed generally good agreement with experimental results even though some discrepancies were found regarding the peak in the maximum standard deviation. The free surface fluctuation frequencies were over predicted and the model could not capture the decay of the fluctuations in the horizontal direction.

Fluid mechanics of large hydropower plants are characterized by very high flow rates and large physical dimensions both in production and spill waterways. The hydraulic head harvested in production needs to be handled when spillways are engaged. In open spillways, this is done by accelerating the flow and then using the dissipative features of a hydraulic jump. Hydraulic jumps are an effective way to eliminate problems caused by high velocity flow, e.g. erosion. The lower velocities past the jump may also create beneficial flow conditions for migrating species such as salmonoids [

Hager, Bremen and Kawagoshi [

in the range

Several authors have investigated the velocity field in the abovementioned regions using different experimental techniques, e.g. [

where x is the distance downstream of the inlet and

where

in the range

Modelling of highly disturbed aerated free surface flows such as hydraulic jumps, is complex when grid based method is used [

A few papers have been devoted to SPH modelling of hydraulic jumps. López, Marivela and Garrote [

Present study will focus on the general behavior of hydraulic jumps when using the meshless, Lagrangian particle method SPH. Special attention will be given on how the spatial resolution of the SPH particles impacts the overall behavior of the jump and the conjugate depth. Apart from the geometrical parameters such as depth, the internal velocity field and its impact on the free surface will be studied. Based on the averaged velocity field the jump length will be determined and compared to experiments. The instantaneous velocity field showed large coherent vortices which affected the free surface. The fluctuations of the free surface will be studied extensively and numerical results will be compared to experimental data, e.g. the standard deviation and fluctuations frequencies.

In the SPH-method, the fluid domain is represented by a set of non-connected particles which possess individual material properties, e.g. density, velocity and pressure [

where

where the kernel function is

In both Equation (8) and (9), h is the smoothing length;

where

The NULL material model implemented in software package LS-DYNA defines the deviatoric viscous stress as,

where

where

Polynomial EOS implemented in LS-DYNA [

where

where

A first order time integration scheme is used and the time step is determined according to,

where

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

A two dimensional horizontal spillway channel and hydraulic jump were investigated in present work with a single phase (water) model. The schematic geometrical setup is shown in

box, SPH particles maintained the state from previous active time step or an externally imposed state and hence no governing equations were solved. By placing a fixed number of SPH particles in an ordered configurationoutside the computational box an inlet was obtained, see the black box at the inlet section in

A perfect agreement of numerical and theoretical results implies that

In

Cases (simulation time) | Number of particles | Number of cores | Compute time [h] |
---|---|---|---|

d_{1}/4 (30 s) | 36,879 | 8 | 114 |

d_{1}/5 (5 s) | 10,749 | 8 | 9 |

d_{1}/6 (5 s) | 15,479 | 16 | 26 |

d_{1}/10 (5 s) | 42,999 | 12 | 114 |

direction until it reached the weir, see

should not be interpreted as an actual boundary layer. It was more likely an effect of the truncated kernel domain due to the lack of SPH nodes outside the boundary. Considerable research has been devoted to this topic and a possible solution is the use of a different boundary condition such as the ghost particle [

In

To exclude the initial transient phase, data was collected in the time interval 2.5 s to 5.0 s for all cases and in the interval 15 s to 30 s for the coarsest case. Generally, coarser cases showed better agreement than finer cases. This behavior could be explained by the increased number of flow features resolved which as reported in [

As mentioned in the introductory section, several authors have commented on the existence and the implication of large vortex structures and its effect on the free surface in the roller region. The vortex paring mechanism and the merging with the stationary vortex reported in [

Case (interval) | A | P |
---|---|---|

d_{1}/4 (2.5 s - 5 s) | 0.969 | 0.049 |

d_{1}/5 (2.5 s - 5 s) | 0.986 | 0.041 |

d_{1}/6 (2.5 s - 5 s) | 0.968 | 0.077 |

d_{1}/10 (2.5 s - 5 s) | 0.936 | 0.119 |

d_{1}/4 (15 s - 30 s) | 1.041 | 0.045 |

vortices translated with increasing size in the downstream direction in agreement with [

However, when investigating the dimensionless maximum standard deviation

A Fast Fourier Transform (FFT) analysis of the numerical depth at several positions downstream the jump toe was conducted.

dimensionless distance downstream the jump toe

Two dimensional hydraulic jumps have been investigated in present work using the Meshfree, Lagrangian particle based method Smoothed Particle Hydrodynamics. Four cases with different spatial resolution of the SPH

particles were set up and as a general result more flow features were observed for the highly resolved cases. All cases showed a tendency to propagate in the upstream direction, which was assumed to be a consequence of the frictionless boundary condition used. Furthermore, a “artificial” boundary layer was observed close to the bottom which affected the incoming jet and was likely caused by the truncated kernel due to the lack of SPH nodes outside the boundary. The conjugate depth

fairly good with experimental results even though the length of the jump was under predicted by roughly 25%. The maximum velocity in horizontal direction indicated a too dissipative zone past the roller which was likely caused by the viscosity model used. The investigation of vortex structures and its effect on the free surface showed generally good agreement for the mean and the standard deviation, even though the peak in the standard deviation occurred further downstream as compared to the experimental results. However, when comparing the maximum standard deviation as function of the Froude number a favorable result was obtained. The investigation of free surface fluctuation frequencies indicated a general over prediction of frequencies and that the longitudinal decay was not captured by the SPH model. Also, a minor under estimation of the Strouhal number was obtained even though the outcome was within the range of experiments. This work has shown that it is possible to investigate the dynamics of the internal velocity field and its impact on the free surface in a hydraulic jump using a relative simple and coarse SPH model. However, a future study focusing on highly refined cases and a more sophisticated viscosity model would be interesting.

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 Svenska Kraftnät together with Luleå University of Technology, KTH Royal Institute of Technology, Chalmers University of Technology and Uppsala University www.svc.nu.

Patrick Jonsson,Pär Jonsén,Patrik Andreasson,T. Staffan Lundström,J. Gunnar I. Hellström, (2016) Smoothed Particle Hydrodynamic Modelling of Hydraulic Jumps: Bulk Parameters and Free Surface Fluctuations. Engineering,08,386-402. doi: 10.4236/eng.2016.86036