**5. Discussion**

Predicting scour depth at complex piers has always been a problem due to the difficulty of analyzing the complexity of flow field and pier geometry, as reviewed extensively by Ettema et al. [38]. As a result of the limitation of contemporary computing capability, numerical modelling has not been practical and efficient enough to provide reliable scour simulation. Thus, engineers still rely heavily on empirical or semi-empirical equations based on dimensional analysis to calculate the equilibrium scour depth at piers, although errors may be significant and modifications are usually needed for unusual conditions. To date, a few prediction methods have been well developed and

widely adopted around the world for complex piers, including the Coleman method (Coleman [16]), the FDOT (Florida Department of Transportation) method (Sheppard and Renna [39]), HEC-18 method (Arneson et al. [40]), and the Sheppard-Melville (S-M) method (Sheppard et al. [41]). The S-M method is a meld of the equations of Sheppard and Miller [42] and Melville and Chiew [43]. Ettema et al. [38] has noted that the S-M method is more readily adapted than the other methods as further information becomes available. However, the S-M method was originally developed for single piers and does not deal with geometric complexity. Yang et al. [28] proposed a modification for clear-water scour at complex piers by replacing the pier diameter in the original S-M method with the equilibrium pier width calculated by other equations. Yang et al. [44] extended the modification mentioned above to the live-bed flow range. The newly extended method can be expressed as below.

When *U*/*Uc* ≤ 1:

$$\frac{d\_{\rm sc}}{D\_{\rm c}} = 3.17 f\_1 f\_2 f\_3 \tag{1}$$

$$f\_1 = \tanh\left[\left(\frac{y\_0}{D\_\varepsilon}\right)^{0.65}\right] \tag{2}$$

$$f\_2 = 1 - 1.2 \left[ \ln \left( \frac{lI}{lI\_c} \right) \right]^2 \tag{3}$$

$$f\mathfrak{z} = \frac{D\_{\mathfrak{e}}/d\_{\mathfrak{5}0}}{69.25(\,D\_{\mathfrak{e}}/d\_{\mathfrak{5}0})^{-0.34} + 0.14(\,D\_{\mathfrak{e}}/d\_{\mathfrak{5}0})^{1.41}}\tag{4}$$

When 1 < *U*/*Uc* ≤ *Ulp*/*Uc*:

$$\frac{d\_{sc}}{D\_c} = f\_1 \left\{ 2.3 \left[ 1 - \left( \frac{\frac{lI}{lL\_c} - \frac{lI\_{lp}}{lL\_c}}{\frac{lI\_{lp}}{lL\_c} - 1} \right) \right]^{1.1} + 3.17 f\_3 \left( \frac{\frac{lI}{lL\_c} - \frac{lI\_{lp}}{lL\_c}}{\frac{lI\_{lp}}{lT\_c} - 1} \right)^{1.1} \right\} \tag{5}$$

When *U*/*Uc* > *Ulp*/*Uc*:

$$\frac{d\_{\rm sc}}{D\_{\rm c}} = 2.3f\_1 \tag{6}$$

$$\mathcal{U}\_{lp} = \max \{ \mathbf{5} \mathcal{U}\_{c\prime} \, 0.6 \, \sqrt{\mathbf{g} \, \mathbf{y}\_0} \} \tag{7}$$

Specifically, *Ulp* represents the flow velocity where the bed-forms are washed out, i.e., the transitional flat-bed stage. The equivalent pier width *De* is calculated by the method of Coleman [16] for aligned complex piers and that of Sheppard and Renna [38] for skewed complex piers.

The performance of Equations (1)–(7) is compared with the HEC-18 and FDOT methods in Figure 10. Supplementary experiments were also performed to investigate the scour at stand-alone piers and the results are included in the figure for comparison. In Figure 10a, for the side-by-side arrangemen<sup>t</sup> with aligned flow the live-bed scour is aggravated by the proximity of another pier, while no significant difference exists under the clear-water flow regime. In contrast, in Figure 10b with a 30◦ skew angle, the scour depths at both the piers are reduced by the presence of another pier, although the horizontal extent of the merged scour hole is typically somewhat larger than the hole produced by any individual complex pier. It is counter-intuitive that a larger flow blockage usually leads to greater scour depth. A possible explanation is that the presence of a downstream pier causes a greater flow separation that starts from a longer distance upstream of the pier site, then the attacking angle of the flow is actually reduced due to enhanced flow diversion. More analysis, both physical and numerical, is needed to better describe the mechanism of scour attenuation in similar situations.

**Figure 10.** Scour comparison with single complex pier without proximity and performance of predictors. (**a**) side-by-side: aligned; (**b**) side-by-side: skewed; (**c**) staggered; and (**d**) tandem.

For staggered and tandem pier arrangements, in Figure 10c,d, the upstream pier is not significantly affected by the downstream pier. The modified and extended S-M method then provides an adequate but not excessive safety margin even if the scour depth is aggravated due to an adjacent pier, and the natural "descend-ascend" trend with increasing flow velocity in the live-bed regime can also be captured with good accuracy. In contrast, although the FDOT method and the HEC-18 method show fair accuracy for clear-water scour, they both tend to over-simplify the scour trend under the live-bed flow regime and may lead to significant errors. The HEC-18 method uses Froude number (rather than *U*/*Uc* as used in other popular methods) to account for the influence of flow velocity and this may be the reason for the excessive overestimation that can be observed in Figure 10. Therefore, it is suggested, for engineers and researchers, to use the modified and extended S-M method expressed by Equations (1)–(7) to predict the scour depth at the upstream pier when aligned to the approaching flow. Attention should also be paid to special conditions that may lead to scour aggravation due to the skew angle as shown in Figure 10a. In the meantime, the information provided in Figure 8 can be used to evaluate the regional bed topography, the potential location for severe scour damage, or bed level stability during bed-form migration.

It should be noted that the scour pattern at bridge sites can also be significantly influenced by other artificial structures or natural topography, e.g., weirs, sluice gates or shoals, which may even be far enough away to be neglected in error sometimes. Wang et al. [45] studies the scour aggravation caused by an adjacent submerged weir set downstream of the pier under live-bed conditions. The scour pattern for a staggered or tandem pier arrangemen<sup>t</sup> will be more prone to be affected by the downstream weir. In addition, the existence of the weir leads to upstream bed aggradation and consequently changes the relative pile-cap elevation to the original bed level. The relative pile-cap elevation has been considered as a key factor affecting the scour development and equilibrium scour depth at complex piers (Coleman [16]; Ataie-Ashtiani et al. [18]; Moreno et al. [22]; Yang et al. [28]). Similar effects can also be caused by general bed degradation that usually occurs downstream of sluice gates. Furthermore, natural topography, e.g., shoals, bends, and sand bars, may cause oblique flow that much complicate the scenario for scour depth analysis.

Generally, the analysis of scour at complex bridge piers is complicated. A reliable prediction can be obtained only when the influence of pier proximity is accurately integrated with the influences of many other factors that are not addressed by this paper. The current study is still limited and only investigates the general scour pattern with typical pier arrangements and a fixed pile-cap elevation. More research needs in the future include testing extensively the influence of pier complexity, sediment type, gradation, large flow blockage due to consecutive piers, and unconventional pier arrangements, which may help creating an updated scour prediction framework that is not discussed by the current study.
