Vertical Velocity Profiles

7.2.2.1 Fully Developed Velocity Profile

The vertical velocity profiles from the experimental work and the numerical model with the RSM model were also compared 0.15 m upstream of the pier and 0.15 m beside the pier. Figure 7-1 shows that the vertical velocity profiles for the tests with the round nosed pier exemplify the well-recognised logarithmic relationship, as discussed in Section 2.5.1 and illustrated in Figure 2-6. They also illustrate the difference between the physical and numerical modelling more plainly and confirm the previous observation that the velocities calculated by the numerical model are underestimated, particularly as the flow intensity increases. As observed in Section 7.2.1, the velocities beside the pier are slightly larger than those upstream of the pier on account of the separated flow, and an equal increment in velocity distribution is observed for each flow.

A prominent dissimilarity is the shape of the logarithmic velocity profile. The numerical model yields a velocity profile that is distinctly underpredicted near the water surface with the maximum velocity simulated at a nondimensional flow depth of 0.55. This behaviour could be attributed to a numerical overchute of the fully developed profile due to the boundary condition of the proposed numerical model. The free surface has been treated as a shear free rigid lid by a symmetry boundary while the law-of-the-wall was employed to establish the shear stress for the boundary layer formation at the surface of the packed bed. Conversely, the entrance length condition for fully developed flow (in the order of 5 to 9 m) may not have been satisfied in the physical model, particularly for larger flow velocities, whereby the sediment bed was only placed in the flume for a distance of 5 m upstream of the pier with a 1:5 slope to facilitate a gradual transition of the velocity profile. Nevertheless, the hydrodynamic model performs reasonably well in reproducing the velocity profile from the laboratory closer to the bed.

Figure 7-1: Vertical velocity profiles for the round nosed pier (a) upstream of the pier and (b) beside the pier for a

0.00 0.10 0.20 0.30 0.40 0.50 No nd ime ns io na l f lo w d ep th v (m/s) Physical model - 56 L/s Physical model - 62 L/s Physical model - 68 L/s Physical model - 74 L/s ∀Water level

∀Fixed bed level

1.00 0.75 0.50 0.25 0.00 (a) 0.00 0.10 0.20 0.30 0.40 0.50 0.60 v (m/s) Numerical model - 56 L/s Numerical model - 62 L/s Numerical model - 68 L/s Numerical model - 74 L/s ∀Water level

∀Fixed bed level

The approach for modelling the free surface as a rigid lid is widely adopted to limit computational costs and is considered acceptable for flows with a small Froude number because modelling the free surface would not affect the morphodynamic results for small Froude numbers typically less than 0.2 or even 0.4 (Roulund et al., 2005; Lui & Garcia, 2008; Baykal et al, 2015). The maximum Froude number that was investigated in this study was 0.26, and even though the bow wave at the pier, shown in Figure 7-2

,

cannot be modelled, it only counteracts the horseshoe vortex in shallow waves (Melville, 1975). Once the morphodynamic model is resolved, future studies could investigate the feasibility of modelling air as an additional phase to capture the free surface of floods with high Froude numbers, for example with the VOF multiphase model.

Figure 7-2: Photograph of the bow wave forming in front of the cylindrical pier in the laboratory

For hydraulically rough and fully developed flow, the logarithmic velocity distribution in Figure 2-6 can be fitted by the wall function in Equation 2-29 where 𝐶𝐶+_{= 0. Curve-fitting the vertical} velocity profiles gives the nondimensional wall distance 𝑌𝑌+ _{= (𝑦𝑦 − 𝑦𝑦1)/𝑦𝑦0 and the roughness} length 𝑦𝑦0= 0.05𝑑𝑑 where 𝑦𝑦1 is the flow depth of 0.2 m and 𝑑𝑑 is the particle diameter. In other words, 𝑦𝑦0 is 0.01 m for sand, 0.04 m for the crushed pips and 1×10-5 _{m for the fixed bed in}

the laboratory. This is similar to the roughness length described by Mohammed et al. (2016) as 𝑦𝑦0 = 𝑘𝑘𝑠𝑠/30 where the roughness 𝑘𝑘𝑠𝑠 can be approximated as 3𝑑𝑑. Curve-fitting of the wall function also produces shear velocity values of 0.03 to 0.22 m/s for the fine sand, and 0.01 to 0.12 m/s for the crushed pips (in the same order as experimental values from Bagnold, 1941). In comparison, the numerical model simulates shear velocities as large as 0.31 and 0.21 m/s for the fine sand and the crushed peach pips, respectively. From curve-fitting to the numerical model, it is also evident that the roughness length is typically double to that from the physical modelling. The increased roughness and shear velocities simulated at the surface of the bed could justify the underestimated velocity profile by the numerical model. Salaheldin et al. (2004) also observed that the 𝑘𝑘-ε _{turbulence model generally overestimates} the bed shear stress and potentially the area of scour initiation. Numerical models with the 𝑘𝑘- ε _{turbulence model have been criticised for being constrained by the 𝑘𝑘-}ε model because the velocities, particularly the 𝜕𝜕-velocities, are underestimated. However, the proposed RSM model also underestimated the velocity profiles, albeit less, and produced slightly higher shear velocities and bed shear stresses.

Wake

Eddies from detached shear layers

Bow wave

7.2.2.2 Coupled Behaviour of Sediment Transport in a Flow Field

The vertical velocity profiles between a fixed bed and a sediment bed for different approach flows are compared in Figure 7-3 0.15 m upstream of the pier and 0.15 m beside the pier (see Figure 4-10). The reduced and formalized cross-sections of the flat rigid bed generally have increased and more uniform velocities than the movable sediment bed which trap and stabilize the vortices (Williams, 2014). Note that no velocity measurements were possible in the vicinity of the boundary layer by the ADV in the laboratory. Although commercial software exists to model sediment transport, they are not fully coupled. The interaction between fluid and sediment is coupled because sediment transport modifies the local flow patterns but also the bed in terms of elevation, slope and roughness, which in turn alters the local velocities and flow field in the vicinity of the pier, over different time scales. The presence of saltating particles in particular modifies the velocity profile inside the moving layer (Schneiderbauer & Pirker, 2013).

Figure 7-3: Vertical velocity profiles measured for the round nosed pier with fine sand (a) upstream and (b) beside the pier from the physical model

7.2.2.3 Repeatability

As discussed in Section 6.2.1, the 4 tests involving the cylindrical pier and crushed peach pips were duplicated 3 times to evaluate the repeatability and reliability of the results. Figure 7-4 shows the vertical velocity profiles measured upstream and beside the pier during the 3 tests that were repeated for 4 flows. Test 3 for the 40 l/s flow yielded underpredicted and unreliable results for both points, indicating that an incorrect flow less than the 40 l/s was tested, emphasizing the importance of repeat tests and monitoring the flow during the laboratory tests. 0.00 0.10 0.20 0.30 0.40 0.50 No nd ime ns io na l f lo w d ep th v (m/s) Movable bed - 56 L/s Movable bed - 62 L/s Movable bed - 68 L/s Movable bed - 74 L/s ∀Water level

∀Initial bed level

(a) 0.00 0.10 0.20 0.30 0.40 0.50 0.60 v (m/s) Fixed bed - 56 L/s Fixed bed - 62 L/s Fixed bed - 68 L/s Fixed bed - 74 L/s ∀Water level

∀ Initial bed level

(b) 1.00 0.75 0.50 0.25 0.00

Erroneous values that deviate from the repeat tests and logarithmic velocity profile are easily identified and discarded. For example, upstream of the pier, the 34 l/s Test 2 obtained a velocity 0.035 m/s lower (17% error) than the other tests at 0.15 m below the water level. However, the rest of the results are near identical, only deviating within a range of 0.009 m/s (a maximum error of 6%). This error could be attributed to small fluctuations in the flow rates that could have been caused occasionally by vacillating pumps or the ADV probe oscillated indistinctly with the flow and could not resolve the erratic vectors, particularly near the packed bed or water surface. Alternatively, the increased velocities measured in the laboratory could be caused by electrical noise evident in the ADV even under no flow conditions.

Figure 7-4: Vertical velocity profiles measured for the cylindrical pier with crushed peach pips (a) upstream of the pier and (b) beside the pier for three repeat tests

In document Coupled fully three-dimensional hydro-morphodynamic modelling of bridge pier scour in an alluvial bed (Page 138-141)