**4. Live-Bed Scour Experiments**

#### *4.1. Mean Scour Depth at Complex Piers*

A live-bed flow regime occurs when the mean flow velocity exceeds the threshold of sediment incipient motion. The general sediment motion on the bed surface leads to the formation of bed-forms and there is general transport of sediment downstream. Under a live-bed flow regime, the scouring process fluctuates due to the alternate arrival of the crests and troughs of the bed-forms and thus the instantaneous bed level data are time-averaged to obtain a mean scour depth. A dynamic equilibrium

may be reached with mean scour depth constant over time. The equilibrium mean scour depth is dependent on the flow intensity ratio (*U*/*Uc*) and may show a "descend-ascend" trend with increasing *U*/*Uc*, as reported by Chee [34], Chiew [35], and Melville [36].

Figure 7 shows the mean scour depths as functions of flow intensity ratio *U*/*Uc* at selected measuring locations, including the most scoured pile of each complex pier and the upstream, downstream, and middle sections of the space between piers. It should also be noted that the bed-forms observed in the present study were dunes or washed-out dunes (for large flow intensity) migrating towards downstream. Other bed-form patterns, including ripples and anti-dunes, were not observed due to the characteristics of non-ripple-forming coarse sand (usually with *d*50 > 0.6-mm).

**Figure 7.** Relationship between equilibrium mean scour depth and flow intensity ratio at various locations. (**a**) side-by-side: aligned; (**b**) side-by-side: skewed; (**c**) staggered; and (**d**) tandem.

Figure 7a shows the scour variation of a side-by-side pier arrangemen<sup>t</sup> with aligned flow. The maximum scour depth at the two adjacent complex piers stays reasonably constant with increasing flow intensity, while the peak value at the transitional flat-bed stage (*U*/*Uc* = 4 ∼ 5) is smaller than the peak at the clear-water threshold, which is in accord with the conclusions of Chee [34], Chiew [35], and Melville [36] for non-ripple-forming coarse sediment. The typical "descend-ascend" trend with flow intensity is absent for scour between the two piers, and the scour peak occurs for *U*/*Uc* = 2 ∼ 3, when the approaching bed-form size is usually maximized. The location of maximum scour between the piers moves to the downstream section with flow intensity increasing above 3.5.

When there is a 30◦ skew angle between the piers and the flow direction, the mean live-bed scour depth at the downstream pier exceeds that at the upstream pier (Figure 7b). It is conjectured that the incoming sediment rapidly erodes away without further accumulation between the piers due to the strong wake vortices and the contracted flow. Thus, the scour at the inner side of the downstream pier (upstream pier flank) will be subjected to less influence of the approaching bed-forms, while, in contrast, a stronger filling effect exists at the outer side of the upstream pier. Furthermore, the

"descend-ascend" trend is present for scour between the two piers, which indicates that, compared with Figure 7a, this scour more resembles that in a unified scour hole, i.e., the two scour holes have merged to form a single extensive scour hole.

The scour trend of the staggered pier arrangemen<sup>t</sup> is shown in Figure 7c. Under a clear-water flow regime, the scour at the downstream pier is significantly reduced by the lee e ffect and by the sediment supply from the upstream pier. However, the scour depth at the upstream pier drops sharply with increasing flow after entering the live-bed flow regime due to the approaching dunes, while, in contrast, the downstream pier is much less subjected to the influence of bed-forms with a live-bed peak larger than that at the clear-water threshold. The live-bed scour depths at the two piers are generally close but the downstream pier is slightly more scoured, which is in contrast to the trend with clear-water scour. A possible explanation is that with high flow intensity the lee protection of the upstream pier disappears while the diverted flow acting directly on the downstream pier with a certain skew angle actually strengthens the downstream scouring capacity. This is also in accordance with the observed diverted flow path behind the upstream pier. In addition, the attenuation e ffect of downstream vortex shedding becomes negligible due to the stronger flow.

In Figure 7d, the scour trend at the upstream pier in the tandem arrangemen<sup>t</sup> is similar to the staggered arrangement, which indicates that, for both arrangements, the upstream piers are not significantly a ffected by the downstream pier and still maintain the same scour features as a stand-alone pier. In contrast, the scour at the downstream pier is greatly attenuated. The downstream pier, as well as the measuring locations between the piers, sugges<sup>t</sup> similar and uniform trends of slightly more scour with increased flow intensity and show a live-bed scour peak greater than that at the clear-water threshold. This feature is similar to the downstream pier in staggered arrangemen<sup>t</sup> and presumably tends to disappear with increasing distance between the two piers. Generally, for staggered or tandem arrangements with significant upstream/downstream distinction, the most hazardous scour overall may occur at the leading piles of the upstream pier under a clear-water flow regime.

#### *4.2. Bed-Form Migration between Adjacent Piers*

Besides the live-bed scour at each complex pier, it is also important and useful to understand the scour pattern and the varying trend within the space between two bridges, where the altered hydrodynamic and morphological features may a ffect human activity or natural processes, e.g., debris drifting and accumulating, boating, etc. Figure 8 shows both the clear-water and live-bed scour features between two piers. The shaded areas represent the varying range of bed level under a live-bed flow regime, and the data with *U Uc* = 2.2 are plotted specifically because the bed-form size is usually maximized at this flow intensity with a medium celerity, and thus the features are usually most typical. In addition, the mean magnitude of bed level fluctuation at each measuring location is also shown in the small embedded graphs within Figure 8.

In Figure 8a, for side-by-side arrangemen<sup>t</sup> with aligned approaching flow, the range of bed level between two piers under live-bed condition is always lower than that at the clear-water threshold and the most significant additional scour occurs at the downstream section.

An opposite relationship can be observed in Figure 8b when a 30◦ skew angle exists, where the scour depth at the clear-water threshold stays larger than live-bed scour for any flow intensity and the di fference is also maximized at the downstream section. A feature shared by the side-by-side arrangements with or without pier skewness is that the maximum live-bed in-between scour between the piers tends to occur at the central measuring location.

**Figure 8.** Live-bed scour range between adjacent piers. The embedded subplots show the trend of mean scour fluctuation at each measuring location. (**a**) side-by-side: aligned; (**b**) side-by-side: skewed; (**c**) staggered; and (**d**) tandem.

In Figure 8c, for the staggered arrangement, the increased flow intensity, and the corresponding approaching bed-forms attenuate the scour at the upstream end (adjacent to the rear of the upstream pier) but, in contrast, aggravate the scour at the downstream end (adjacent to the face of the downstream pier). In accordance with the conjecture in the last section, this phenomenon is also the outcome of the weakened lee-wake protection to the downstream pier with large flow intensity.

In Figure 8d, the scour depths under clear-water and live-bed flow regimes are similar. Only very minor scour occurs between the piers except close to the face of the downstream pier as part of the natural profile of the downstream scour hole.

For all the pier arrangements in the present study, the mean magnitude of the bed level fluctuation between the piers always decreases from upstream to downstream, regardless of the varying trend of the mean bed level, as shown by all the embedded graphs in Figure 8. This indicates that the bed-forms are damped when migrating downstream between the two adjacent complex piers. This damping effect can be influenced by other factors besides the bed level profile. This effect is discussed further below.

Cross-correlation analysis is used to determine the level of similarity of two data series. With time series (such as instantaneous bed-level data in the present study) a time lag between two series can be identified by a cross-correlation analysis performed after displacing one series relative to the other. Guan et al. [37] used cross-correlation analysis to calculate the migrating celerity of bed-forms between two locations, which can be determined by dividing the known distance between the locations by the calculated time lag of the correlation peak. In the present study, correlation analysis has been performed on data from pairs of adjacent measuring locations between the piers. Data with *U*/*Uc* = 2.2 (a typical value as mentioned above) are shown in Figure 9 and the trends with other flow intensity ratios are similar. In the figure, a higher correlation peak represents a greater similarity between the time series (i.e., less deformation of the bed-forms) and a smaller time lag represents a faster migrating celerity between the two adjacent measuring locations. The correlation coefficients are normalized by the largest peak value in each subplot.

**Figure 9.** Cross-correlation analysis for the four different pier arrangements in the present study (*U*/*Uc* = 2.2). (**a**) side-by-side: aligned; (**b**) side-by-side: skewed; (**c**) staggered; and (**d**) tandem.

In Figure 9a, for a side-by-side pier arrangemen<sup>t</sup> with aligned flow, bed-forms keep accelerating and are damped continuously when passing between the piers, although the bed level does not vary significantly as shown in Figure 8a. The acceleration is probably due to the flow contraction that is magnified with large flow intensity under a live-bed flow regime. In contrast, with a 30◦ skew angle (Figure 9b), bed-forms tend to pass between the piers with a constant celerity but are damped to a much greater extent than with the aligned arrangement. The enhanced damping effect can be attributed to the larger vertical (and also horizontal) extent of the scour hole, in which the approaching bed-forms may collapse and lose the original shape features. This is also in accord with the findings of our previous unpublished study on the relationship between mean scour fluctuation and scour depth.

In Figure 9c for the staggered pier arrangement, a process of repeated acceleration and deceleration can be observed. A possible explanation is that the migrating bed-forms firstly leave the region dominated by the upstream pier and then enter the region that is significantly affected by the downstream pier. The flow tends to be more contracted and accelerated in the region closer to either one of the piers and therefore the celerity of the migrating bed-forms also tends to behave in the same way.

For the tandem arrangement, as shown in Figure 9d, the bed-forms tend to decelerate behind the upstream pier due to the lee-wake protection and then accelerate after entering the downstream scour hole before collapsing to the bottom of the leading piles. Damping of the bedforms is marked.

Generally, the bed-forms tend to accelerate in contracted flow near the piers regardless of the pier arrangement, and deceleration may occur when the lee-wake protection effect prevails. Furthermore, the magnitude of the bed-forms will always be damped during migration downstream while being eroded by the turbulence in the scour hole, which is also in accord with the decreasing mean fluctuation magnitudes shown in Figure 8.
