The solid wall simulations capturing dam-break flow formed the basis for a parametric study numerically investigating the effect of changing the permeability of the porous material on the fluid-structure interaction between dam-break flood waves and a permeable flood barrier. The ability of the MPM code to produce reliable results for wave impact on porous media is validated by extensive comparison of simulation results with numerical and experimental results published in Liu et al.’s 1999 paper “Numerical Modeling of Wave Interaction with Porous Structures”[94], and in Ren et al. (2016) “Improved SPH simulation of wave motions and turbulent flows through porous media”[126].
Fig. 4.17 Initial geometry of model validation simulations for wave impact on porous media, after Liu (1999) [94]. All dimensions are in metres.
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The geometry of the numerical simulations used for this verification study, based on the geometry employed by Liu (1999) [94], is shown in Figure4.17. The experiments conducted by Liu et al. (1999) used a water tank of 0.892m × 0.44m × 0.58m. A porous structure of 0.29m × 0.44m × 0.37m was built with crushed rocks (d50= 1.59 cm, n = 0.49) was placed at
the centre of the tank, from x = 0.3–0.59m. A gate was constructed 2cm (including the gate thickness and gap) away from the porous structure and pulled up manually within 0.1s at the beginning of the test. The initial water level d was 0.25m. The numerical set-up is the same as the experimental set-up except that there is a gap of 2cm between the water column and the near side of the porous dam, and no gate.
The permeability of the porous block based on the grain size of the material is described by the Kozeny-Carman equation:
κ = D2p
A n3
(1 − n)2 (4.5)
where κ is the permeability [m2], Dpis the effective grain size diameter [m], n is the soil
porosity [no units], and A is a constant, equal to 150 (Ergun, 1952) [40]. In these simulations, the soil porosity is fixed so that the permeability is varied only by altering the grain size.
In the double-point MPM code, the permeability of the soil body is directly related to the interaction force vector between the liquid and the soil particles, resulting from the drag force exerted on the solid particles by the fluid, originating from the relationship equation developed by Ergun ( [40]), and characterised by the Kozeny-Carman equation, as above. The initial geometry and permeability parameters (grain size and porosity) are the same as those of the experimental results published by Liu (1999):
κ = 0.0159 2 150 0.493 (1 − 0.49)2 = 7.623 × 10 −7m2 (4.6)
where the mean grain size diameter is 0.0159m and the initial porosity, n, is 0.49, resulting in a permeability of 7.623 × 10−7m2.
Figure4.18shows a comparison of the results obtained using MPM for the time history of the free-surface displacement at x = 0.445m (i.e. in the centre of the porous block, as indicated in Figure 4.17) to the experimental data published in Liu et al., 1999 [94], the VOF method results also published in Liu et al., 1999 [94] as well as the ISPH method results published in Akbari and Namin, 2013 [2] and the SPH model results published in Ren et al., 2016 [126].
Liu et al. (1999) considered that the flow in the porous structure was free of turbulence if the permeability of the porous medium was very small. The turbulence effect was only found to be significant if the pore size was comparatively large (Hsu et al., 2002 [55]). Shao (2010) [131] and Akbari and Namin (2013) [2] both identified that the turbulence might be
Fig. 4.18 Comparison of the MPM results for time history of free-surface displacement at x= 0.445m with various published data [126]
significant in the wave breaking zone, to simplify matching conditions with the porous flow region and the flow external to the porous structure, the turbulence effect was not incorporated in their ISPH models even in the flow outside the porous medium. Ren et al. (2014) [126] included a sub-particle-scale turbulence closure model in their WCSPH model in the external flow, but not in the flow inside the porous structures. Consequently, a large jump in the level of turbulence at the interface was observed. In their 2016 study, an improved WCSPH model was developed to investigate the wave motions and turbulent flow both in and around the porous structure. These results are also plotted in Figure4.18. The results obtained using MPM are in very close agreement with published results, particularly the VOF (volume of fluid) results obtained by Liu in 1999 and the SPH results without the turbulence model that were published in Ren et al., 2016. The SPH results with the inclusion of the SPS turbulence model show the closest match for the experimental results, except for a slight under-prediction at the early stages. The MPM results closely resemble the other numerical results with no turbulence model, i.e. Liu (1999)’s use of the VOF method and Akbari and Namin (2013)’s ISPH results and Ren et al.’s “SPH without turbulence model” simulations. These all slightly
4.3 Model Validation 109
Fig. 4.19 Time history of free-surface profiles compared with experimental and numerical results published in Liu (1999) [94]
overestimate the free-surface displacement between 0.4s < t < 0.8s. This discrepancy arises since significant turbulence is generated in the porous flow zone as the particles initially rush rapidly into the dam, and this turbulence is not accounted for in the model. This overestimation of the free-surface displacement becomes much less significant later in the simulations; the much broader gap between the water column edge and the leading edge of the dam gives the flow more time to develop. The MPM simulations produce a better match to the experimental simulations after this initial peak, whereas in the period t > 1.5s, the SPH models with and without turbulent effects overestimate the displacement. Since this investigation is focused on the maximum run-up height, the initial peak is not crucial and we can consider the method robust without additional turbulence models.
Figure4.19shows time-history comparisons of free-surface profiles for the flow passing through the porous dam at different points in time for a direct comparison with the numerical and experimental results published in Liu (1999). After the gate is opened, or the simulation is started, the water column immediately begins to collapse and fluid particles rush towards the porous dam. After t = 0.2s, fluid particles have filled the gap between the initial water column and the porous dam, and a dam-break flow begins to develop. Since the porous dam offers some resistance to the flow, the particles stack up and rise to form a small upward jet develops inside the leading edge of the dam at t = 0.4s. This jet can be observed in both the numerical and experimental results. Some of the impacting water is reflected by the porous boundary to form a wave travelling back towards the left side of the tank, reaching the tank wall at t = 0.8s before being totally re-reflected by the solid boundary back towards the dam. The free fluid domain on the left side of the tank shows a free-surface oscillation driven by the gap between the initial water column and the porous dam. The frequency of this oscillation is comparable to the natural frequency of the water body contained on the left side of the tank; similar to harbour oscillations [94]. This fluctuation gradually decreases as particles pass through the porous dam. Concurrently, fluid particles escape the far boundary of the porous dam and propagate towards the right tank boundary, reaching the wall at t = 1.0s and reflecting to form a similar oscillation on the right side of the dam.
The largest discrepancy between numerical and experimental results occurs at t = 0.2s, where the experimental data indicate a faster advancement of water particles near the bottom inside the porous dam. This could be partially explained by the gate-opening phenomena whereby the manual operation of the gate is non-instantaneous, taking around 0.1s and allowing water near the bottom of the tank to be released earlier. Additionally, the higher flow rate near the bottom of the tank could be caused by the larger porosity that results from the presence of a flat glass surface [94].
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As the simulation progresses, the initial discrepancy between the experimental and numeri- cal result diminishes. The pressure difference drives fluid particles through the porous dam, and at t = 1.2s the front reaches the far wall and is reflected, breaking on the porous dam at around t = 1.6s. Close agreement between the numerical and experimental results show that this phenomenon is captured authentically, although particle scattering is observed in the MPM simulations.
Overall, a very consistent agreement between the MPM simulations and the experimental and numerical results published in Liu (1999) is obtained. The lower free-surface displacement at the early stage (t < 0.4s) in the numerical results is likely to be caused by gate opening effects as discussed above. Later in time, numerical results from both sets of simulations compare well with image data, suggesting that the numerical model produces an accurate representation of flow data.