**3. Results**

#### *3.1. Time Series for Test E3*

In Figure 4, a time series for test E3 is given. The tests were conducted under the conditions for test E3 (see Table 1 for details) for 1, 2, 4, 8, 16, and 24 h, after which bed profile measurements along the symmetry plane at *Z*/*D* = 0 were acquired. It can be seen that the depth of the scour hole upstream of the cylinder increased incrementally with time throughout the first 24 h of the experiment. However, between 16 and 24 h, the profiles in the upstream region were very similar, showing minimal di fferences. In terms of the relative scour depth, *dse*/*D*, and its relation to the foundation head, this indicates that 24 h can be seen as an acceptable time to equilibrium, *te*, under the described experimental conditions.

**Figure 4.** Bed profile measurements in the *XY* (symmetry) plane at *Z*/*D* = 0 for *t* = {1, 2, 4, 8, 16, 24} h.

Downstream of the cylinder, it can be seen that the dune-like primary deposit increased in height and length with time until 16 h elapsed. At 24 h, the profile further increased in length, but the height of the dune decreased slightly when compared with the dune formed after 16 h. It can be seen that changes in the scour formation in the downstream region were more significant than changes in the upstream region with time; however, since the primary quantity for the foundation head design is *dse*/*D* located near the upstream face of the cylinder, the best indication of an equilibrium condition with respect to design can be taken from this region.

This is further indicated by comparison of the scour profiles in the *XY* plane between tests E1 and E3 in Figures 5 and 6. For this pair of tests, the relative coarseness *D*/*d*50, flow shallowness *h*/*D*, and blockage ratio *D*/*b* were very similar (refer to Table 1); only the duration of the tests (48 h for test E1 compared with 24 h for test E3) differed among the two tests.

**Figure 5.** Bed profile measurements in the *XY* (symmetry) plane at *Z*/*D* = 0 for tests E1 and E3.

**Figure 6.** Bed profile measurements in the *XZ* plane at *Y*/*D* = 0 for tests E1 and E3.

As was indicated by the time series shown in Figures 4–6, when all other scour-governing parameters are held constant, the scour formation upstream of the cylinder does not change significantly as time progresses beyond 24 h. From the bed profiles in the *XY* plane, the scour hole profiles in the upstream region were very similar between tests E1 and E3. Downstream of the cylinder, the length of the dune increased, and the height of the dune decreased with time. This was also observed in the time series profiles in Figure 4.

The contour profiles of the scour formation in Figure 6 similarly show that the geometry of the scour hole did not change significantly with time upstream of the cylinder. Downstream of the cylinder, the width of the dune changed slightly. However, as previously mentioned, since changes in the

downstream region do not affect design of the foundation head, the equilibrium state is evaluated primarily based on the region in which the depth of scour is at a maximum (i.e., upstream).

#### *3.2. E*ff*ect of Blockage Ratio D*/*b on Equilibrium Scour Formation*

Figures 7 and 8 show profiles of the equilibrium scour formations for tests E1 and E2. For this pair of tests, all scour-governing parameters (*U*/*Uc*, *D*/*d*50, and *h*/*D*) were held constant. Movable sidewalls were installed in the flume in order to alter flume width *b* and, therefore, blockage ratio *D*/*b* as well. Test E1 was conducted for 48 h, and test E2 was conducted for 24 h. However, the results of the previous section indicated that the scour formation in the vicinity of the cylinder was virtually unchanged beyond 24 h, and any significant changes were only observed in the downstream region. Therefore, the following discussion will pertain mainly to the scour formation upstream and close to the cylinder.

In Figure 7, the depth of the scour hole is shown to be higher for the test with the lower blockage ratio. This is indicative of the strong effect of sidewall proximity on the mechanism of local scour. In test E1, the channel was significantly wide (*b* = 1.22 m) such that the spanwise pressure gradient imposed by the sidewalls did not significantly affect the flow field mechanisms surrounding the cylinder (i.e., the HSV and the wake vortices). Therefore, scour was allowed to progress unimpeded.

In contrast, the width of the channel for test E2 was approximately one-third of that of test E1 (*b* = 0.4 m). It is then very likely that the secondary flows (boundary-induced currents at right angles to the main flow) were capable of interacting with the flow field mechanisms surrounding the cylinder, particularly the wake vortices. Furthermore, the aforementioned spanwise pressure gradient that is a noted feature of horizontally confined flows was more likely to influence the cylinder as the confinement increased. All of these effects of sidewall proximity served to disrupt and weaken the removal and deposition of sediment, reducing the size of the scour hole and the primary deposit.

**Figure 7.** Bed profile measurements in the *XY* (symmetry) plane at *Z*/*D* = 0 for tests E1 and E2.

**Figure 8.** Bed profile measurements in the *XZ* plane at *Y*/*D* = 0 for tests E1 and E2.

Although the above discussion is restricted to the upstream region due to the previously mentioned time differences between tests E1 and E2, the alteration in the dune formation for test E2 can also be attributed to the effect of channel blockage. Beyond the crest of the dune for test E2, the form of the primary deposit differs from that of test E1. Since the wake region is in closer proximity to the sidewalls, it is reasonable that the effect of horizontal confinement would affect the deposition of sediment downstream of the cylinder in addition to the removal of sediment in the vicinity of the cylinder as previously discussed.

This is further illustrated in Figure 8 which shows the effect of *D*/*b* on the extents of the scour formation in the horizontal plane at *Y*/*D* = 0. It can be seen that the size of the scour hole was much smaller in plain view for test E2 with *D*/*b* = 0.15 than for test E1 with *D*/*b* = 0.05. Again, the increase in sidewall proximity appears to have suppressed the lateral progression of scour. This is also observed downstream of the scour hole, where the primary deposit became narrow as horizontal confinement increased.

#### *3.3. E*ff*ect of Blockage Ratio D*/*b on the Progression of Local Scour*

In Figure 9, a description of the progression of relative scour depth *ds*/*D* with dimensionless time *t*/*te* (where *t* is time and *te* is the time to equilibrium) is given. Test E3 of the present investigation was included as well as two tests from the investigation of Yanmaz and Altinbilek [22]. All tests had similar values of *D*/*d*50 and *h*/*D*; only *D*/*b* differed within the presented data set. For all tests, it can be seen that *ds*/*D* increased rapidly, attaining most of the maximum relative scour depth within half of the time to equilibrium. The values of the measured equilibrium scour depth (at *t*/*te* ≈ 1) are shown to be very similar among all tests as well. The figures also include curves calculated using predictive methods for the progression of *ds*/*D* with time from Melville and Chiew [21] (Equation (2)) and Aksoy et al. [23] (Equation(3)).

$$d\_s/d\_{sc} = \exp\{-0.03|(\mathcal{U}\_\mathcal{L}/\mathcal{U})\ln(t/t\_\varepsilon)|^{1.6}\}\tag{2}$$

$$d\_s/D = 0.8 \times (l\text{\textdegree\textdegree\textdegree C}/l\text{\textdegree\textdegree T}\_c)^{3/2} \times (h\text{\textdegree D})^{0.15} \times (\log T\_s)^{0.6} \tag{3}$$

$$T\_s = td\_{50} \times (\Delta g d\_{50})^{0.5} / \text{D}^2 \tag{4}$$

**Figure 9.** Comparison of *dse*/*D* with *t*/*te* for test E3 and data from [22] with *D*/*d*50 ≈ 76 as well as estimation curves [21,23] and points [19] from literature.

In Equation (4), Δ = (ρ*s* − ρ)/ρ, where ρ*s* is the density of the bed material, and ρ is the density of water. In Figure 9, the experimental parameters for test E3 were used to calculate the predictive curves. In Figure 10, the parameters for the test by D'Alessandro [17] were used for the prediction. The value of *dse*/*D* calculated using the predictive method presented in Williams et al. [19] (Equation (5)) was also included for each test as shown by the starred data points.

$$d\_{\rm s\%} \text{D} = 0.76 k\_{\rm c}^{1.69} \times \text{(h\%)}^{0.32} \tag{5}$$

In Equation (5), *kc* is the ratio between the velocity along the separating streamline *Us* and *Uc*. In Figure 9, it can be seen that there were small differences in *dse*/*D* observed in the middle of the scour process (i.e., approximately between 0.1 < *t*/*te* < 0.7). However, there was no specific trend which can be noted between the development of scour and blockage ration *D*/*b*. In the described middle section of the scour process, the scour depth was slightly lower for *D*/*b* = 0.1 when compared with *D*/*b* = 0.05; however, *dse*/*D* was slightly higher for *D*/*b* = 0.09 than either of the other tests. (Although unlikely, this could possibly be attributed to the small changes in *D*/*d*50 or *h*/*D* among the tests.). In general, for *D*/*d*50 < 100, it can be concluded that changes in *D*/*b* have a very small influence on both the progression of local scour as well as the equilibrium depth of scour.

The predictive method described by Equation (2) is represented by the solid curve in Figures 9 and 10, and this method slightly over-predicted the depth of scour (*ds*) throughout the time to equilibrium. The method described by Equation (3) over-predicted the depth of scour for *t*/*te* < 0.5, beyond which the curve approached the results of Yanmaz and Altinbilek [22]. Equation (2) showed good prediction of *dse*/*D* for all tests.

**Figure 10.** Comparison of *dse*/*D* with *t*/*te* and data from [17,27] with *D*/*d*50 ≈ 137 as well as estimation curves [21,23] and points [19] from literature.

However, the results presented in Figure 10 indicate that the effect of *D*/*b* was amplified as *D*/*d*50 increased. The results presented in Figure 10 had similar values of *U*/*Uc* and *D*/*d*50. The value of *h*/*D* was also altered among the tests; however, the literature indicated that the effect of flow depth on *dse*/*D* is minimal when *h*/*D* > 1.4 [3]. Therefore, in terms of influencing parameters, only *D*/*b* differed among the tests. In contrast to the data set presented in Figure 9, there were significant changes noted with changing *D*/*b*. For *D*/*b* = 0.09, the depth of scour was significantly higher throughout the progression to equilibrium when compared with *D*/*b* = 0.06. This was also in contrast to the results of the previous section which indicated that the relative scour depth increased as *D*/*b* decreased. This indicates that the effect of blockage ratio on local scour is also dependent on the value of *D*/*d*50, and the effect is amplified when *D*/*d*50 > 100.

This is in agreemen<sup>t</sup> with the results of Tejada [18] who conducted a series of experiments with varying sizes of bed material (*d*50) and similarly reported that the effect of *D*/*b* was minimal when *D*/*d*50 < 100 when compared with *D*/*d*50 > 100. This is reasonable, since larger values of relative coarseness are usually indicative of smaller sediment particles which would be more susceptible to changes in the flow field surrounding a cylinder.

Interestingly, Equation (2) provides a good estimation of the progression of scour for the results of Chabert and Engeldinger [27], while Equation (3) provides a better estimation for the test of D'Alessandro [17]. Furthermore, Equation (5) shows a much closer prediction of the test of Chabert and Engeldinger [27] than D'Alessandro [17]. This also indicates that further investigation on higher values of *D*/*d*50 is required in order to establish the role of *D*/*b* on local scour.
