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47 = (a) Experiment (Hjorth, 1975) =3 5 3 7 9 11 1 Flow -1 -0.5 0 0.5 X/B (b) Numerical Figure 6.1. Comparison of bed shear stress (N/m2) distribution around a circular pier as calculated from an experiment by Hjorth (1975) and numerical computations (B = 0.075 m, V = 0.15 m/s, H = 0.1 m). shown in Figure 6.5. In the experiment, the thickness of the tions on the influence of the pier diameter on the bed shear boundary layer was about 0.06 m for all cases. This observa- stress; these observations were based on experiments. tion was also found in Gudavalli's experiments (1997). The Figure 6.9 shows the pressure field induced by the pier. If velocity vector around the pier is shown in Figure 6.6 for a the pressure field is sufficiently strong, it causes a three- shallow water case and a deep water case. The difference in dimensional separation of the boundary layer, which in turn velocity field can be observed on the figure, especially near rolls up ahead of the pier to form the horseshoe vortex sys- the base of the pier where the horseshoe vortex is much tem. A blunt-nosed pier, for example, is a pier for which the stronger in the shallow water case. induced pressure field is sufficiently strong to form the horse- Figures 6.7 and 6.8 show the distribution of shear stress shoe vortex system. around the pier for different relative water depths of H/ B = 0.2 and H/ B = 2. As can be seen in the figures, the shear stress is 6.6 SHALLOW WATER EFFECT ON MAXIMUM higher in the case of the shallow water depth. This is SHEAR STRESS explained as follows. When the water is deep, the velocity profile has the conventional shape shown in Figure 6.5. When The maximum shear stress max is the maximum shear stress the water depth becomes shallow, and if the mean depth that exists on the riverbed just before the scour hole starts to velocity is kept constant, the velocity profile must curve faster develop. One way to present the data is to plot max/max(deep) as towards the bottom of the profile because of the lack in verti- a function of H/ B (Figure 6.10). The parameter max(deep) is cal distance forced by the shallow water condition. This leads the value of max for the deep water case and is given by Equa- to a higher gradient of velocity near the bottom and, therefore, tion 6.1. The shallow water correction factor, kw, is the ratio to a higher shear stress since is proportional to the veloc- max/max(deep). The data points on Figure 6.10 correspond to the ity gradient. Johnson and Jones (1992) made similar observa- results of the four numerical simulations.