TIPO/CLASE PREDOMINANTE
3.1. M OTILIDAD ACELERADA
The proposed hydro-morphodynamic model was examined for parameter instabilities and sensitivities, and was calibrated to accommodate the sediment material as well as the mesh resolution, to obtain a numerically stable and accurate solution. It was established that a very fine computational mesh in the order of 1 to 2 mm is required to resolve the horseshoe vortex that is crucial for bridge pier scour. However, the sediment transport submodels of the proposed numerical model are very sensitive to the mesh resolution and result in an instability of the packed bed with an unrealistic and irregularly shaped scour hole as shown by Figure 7-13 (Vonkeman et al., 2017). The scour hole was simulated by a model setup with parameters identical to those of Sawadogo (2015) for hydrodynamic entrainment but with a minimum mesh size of 1.8 mm. In spite of the scour instability, the model demonstrated that it had the potential to predict bridge pier scour as a maximum scour depth of 0.125 m was simulated which is of the same order as the 0.116 m measured in the laboratory for a cylindrical pier with a 34 l/s flow.
In a sensitivity analysis for dam bottom outlet flushing, Sawadogo (2015) also recognized that the choice of the mesh size is crucial for the proposed hydro-morphodynamic model and that a finer mesh can cause an irregularly shaped scour hole. Schneidernauer & Pirker (2014) also used a fine mesh with a minimum size of 2.5 mm. Consequently, the parameters that were different in the model study by Schneidernauer & Pirker (2014) and Sawadogo (2015) were compared in Table 7-2. The parameters used by Shao & Li (1999) are also compared because most of the sediment transport algorithms were based on their model study. However, Shao & Li (1999) adopted a different approach whereby the motion of individual saltating particles were tracked as opposed to the proposed hydro- morphodynamic model that modelled the deformation of the particulate bed.
Figure 7-13: Longitudinal section and contour plot in plan of irregular bed deformation pattern at a bridge pier for
a fine mesh with a large diffusion coefficient for a flow of 34 l/s
Table 7-2: Comparison of physical and calibration parameters used by different model studies
Model study Shao & Li (1999) Schneiderbauer & Pirker
(2014)
Sawadogo
(2015) Current study Current study
Model description Lagrangian tracking of individual aeolian sand particles Aeolian snow particle transport Bottom outlet flushing of peach pips Bridge pier scour of peach pips Bridge pier scour of fine sand
Turbulence model LES closed by 𝑘𝑘-ε 𝑘𝑘-ε 𝑘𝑘-ε RSM RSM
Fluid density (kg/m3) 1.225 1.225 1000 1000 1000
Particle density (kg/m3) 2700 917 1350 1275 2629
Particle size (mm) 0.200 0.3 0.740 0.740 0.214
Min mesh size (mm) 1.0 2.5 17.0 1.8 1.8
Max mesh aspect ratio 4.5 2.0 3.0 9.0 9.0
Number of cells 1 800 000 1 996 800 90 210 1 052 647 1 052 647 Time step (s) 0.005 0.001 0.001 0.001 0.001 Fluid velocity (m/s) 1.40 6.40 3.30 0.23 0.37 Entrainment coefficient 𝐂𝐂η′ 1.73×10-3 0.52×10-4 0.35 1.50 0.73 Diffusion coefficient λ𝑏𝑏 (m2/s) - 0.01 10.00 0.01 0.01 Scale factor 𝑺𝑺𝒕𝒕 50 20 60 60 60
Angle of repose 𝝋𝝋 (degrees) - 35 45 44 45
Packing ratio - 0.63 0.5 0.5 0.5
Creeping parameter 𝒑𝒑𝑭𝑭 - 0.5 0.1 0.5 0.5
Impact velocity coefficient 𝒄𝒄𝑰𝑰 10 10 1 10 10
Ratio of initial ejection
velocity to shear velocity 𝒉𝒉𝒉𝒉𝒎𝒎 0.5 0.4 0.75 0.5 0.5
Extensive sensitivity testing of the proposed model indicated that the main nonphysical parameter directly related to the mesh resolution and responsible for the scour hole irregularity is the diffusion coefficient λ𝑏𝑏 in equations (3-33) and (3-34) of the IB method. A
diffusion coefficient of 10 was selected by Sawadogo (2015) and 0.01 by Schneiderbauer & Pirker (2014) who attempted to ascribe a physical meaning to the parameter by first relating
λ𝑏𝑏 to the scale factor and then approximating it as 0.01 X the fluid density (Sawadogo,
-0.20 -0.15 -0.10 -0.05 0.00 0.05 -0.25 -0.05 0.15 0.35 0.55 Bed lev el (m )
Longitudinal axis A-A (m)
∀Initial bed level
Physical model Pier Numerical model y ( m ) x (m) ∆z (m) Flow Direction A B B
2015). However, a diffusion coefficient of 10 yielded erratic scour results and it was found that a value ≤ 0.01 should be implemented instead. This is in accord with Schneiderbauer & Pirker (2014) who implemented the same diffusion coefficient for a 2.5 mm minimum mesh size similar to that of the proposed study. Because the dimension for the diffusion coefficient is area per unit time, it is recommended that future studies of the proposed hydro- morphodynamic model select a diffusion coefficient with an order of magnitude based on the guideline
which relates the volume fraction of the IB method to the mesh resolution or grid cell volume 𝑉𝑉. A reduction of the diffusion coefficient value dramatically reduces the erosion of the sediment bed and should be compensated by an increase in the dimensionless entrainment coefficient 𝐶𝐶η′ which is considered the chief calibration parameter. Shao & Li (1999) used 1.73×10-3 for the aerodynamic entrainment of sand particles, Schneiderbauer & Pirker (2014)
used 0.52×10-4 for the aerodynamic entrainment of snow particles, while Sawadogo (2015)
selected 0.35 for the hydrodynamic entrainment of peach pips. The entrainment coefficient for the hydrodynamic transport of particles is much larger than that for the aerodynamic transport of particles because it is related to the fluid and particulate densities. No experimental investigations have been conducted to explain the different values presented for the nonphysical proportionality factor but it is proposed that the following approximation, based on Shao & Li (1999), should be used to establish the order of magnitude for the entrainment coefficient in equation (3-15)
where 𝑠𝑠 is the particle density relative to the fluid. Based on the relative density of sand (𝑠𝑠 = 2.63) or peach pips (𝑠𝑠 = 1.28), the value of 𝐶𝐶η′ was selected as 0.73 and 1.5 for the fine sand and crushed peach pips, respectively. Furthermore, a saturated angle of repose of 45° and 44° was selected from Table 4-1 for the sand and peach pips. The angle of repose is an extremely sensitive parameter causing numerical instabilities with unrealistic scour patterns. Second to the entrainment coefficient, the creeping parameter 𝑝𝑝𝑟𝑟 of equations (3-16) and (3- 17) should be considered a principal calibration parameter. It defines the probability (between 0 and 1) that a particle with no initial velocity is entrained into the surface rolling or saltating transport mode. For simplicity, a median value of 0.5 for the creeping parameter was found to be the best at replicating the scour results from the laboratory.
The packing ratio in the shear slides equation (3-31) and the IB method equation (3-33) also dramatically destabilized the model. A value of 0.5 as well as an impact velocity coefficient of 10 and a velocity ratio ℎ𝑠𝑠𝑎𝑎 of 0.5 were selected from the original study by Shao and Li (1999) because they offered improved numerical stability while the observed effect of the latter two parameters on the scour hole were negligible. Finally, a scaling factor of 60 was selected because the time scale of scour to approach equilibrium is in the order of minutes or hours, and the time scale of turbulence fluctuations is within the order of seconds. According to Lui & Garcia (2008), the time scale disparity makes coupled hydro- morphodynamic simulations stiff, possibly leading to numerical instabilities.
𝑂𝑂(λ𝑏𝑏) = 𝑉𝑉𝐼𝐼𝑖𝑖𝑚𝑚
𝑑𝑑∆𝑡𝑡
� , (7-1)
The final scour depth results obtained from the improved and calibrated hydro- morphodynamic model are shown in Figure 7-14 relative to the experimental work for a cylindrical pier with a flow of 34 l/s and an initial bed level of 0 m.
Figure 7-14: 3D isometric contour plots of the bridge pier scour for a flow of 34 l/s (a) from experimental work, (b) from improved numerical model and (c) with numerical instability for a fine mesh