The writers appreciate this discussion. However, in the original paper, the writers recognize the local scour processes around the bridge element (i.e., primarily downflow and horseshoe vortices) as a leading cause of dune height and the wash-out parameter, ΔFd, of dune wave-length. Local scour is strongly dependent on the densimetric Froude number, Fd, according to Oliveto and Hager (2002). ΔFd can be viewed as a measure of the sediment transport capacity by the flow behind the bridge element. In clear water conditions, the closer ΔFd is to zero (at the approach bed upstream of the bridge element), the greater is the sediment transport capacity by the flow behind the bridge element (i.e., the exit-dune tends to get flatter and lengthen faster). Conversely, the influence of the wake vortices appears negligible. The writers would like to reinforce these findings. The analysis is restricted to the case of cylindrical piers and does not involve the effect of time, for the sake of simplicity. The same symbols as in the original paper and the discussion will be adopted. Fig. 1 shows a typical exit-dune for an experiment conducted at the University of Basilicata, Italy, by the first author of the original paper. The bed sediment was sand with d50 1⁄4 1.75 mm and σ 1⁄4 1.5. The discharge was 80 L=s, the approach flow depth ho 1⁄4 0.25 m, and the pier diameter D 1⁄4 0.12 m. This experiment was intentionally planned to allow for a gradual and controlled dune evolution over time. To this purpose, the parameter ΔFd 1⁄4 Fdi − Fd was controlled. ΔFd (at the approach bed) was 1.7, definitely greater than 0, to avoid a too fast development or even extinction of the dune. On the other hand, ΔFd was not too high to allow for a sufficiently deep scour hole and a suitable sediment deposition. The eroded material around the pier was deposited immediately downstream of the scour hole and formed a dune, which progressively flattened, gradually enlarged, and expanded downstream by the flow [Figs. 1(a–c)]. Dune presence often generates secondary erosion processes immediately downstream of the dune itself, as in this case [Fig. 1(d)]. Assuming the dune crest as the reference bed level, the value of ΔFd behind the pier is lower than that at the approach bed and is estimated to be 0.8 at 1.5 h [Fig. 1(a)] and 0.9 at 3 and 5 h [Figs. 1(b and c), respectively]. The incipient motion over exit-dunes appears to occur at lower stages than those for plane beds, probably because shear stresses increase at crest locations and/or the action of wake vortices. However, the latter action was not so evident from visual observations during this and other experiments. The writers also compare the maximum dune-crest height, hD;max, and the scour depth, z, observed at the same time. The parameter hD;max was found proportional to z, although the data are quite spread (r 1⁄4 0.75, SSE 1⁄4 0.04 m). This implies that 75% of the variation in hD;max are explained by the downflow and the horseshoe vortices, whereas the remaining 25% are attributed to unknown variables, including ΔFd and wake vortices. The following functional relationship Eq. (1), according to the original paper, was considered to mainly cause the effects of Fd and
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Jun 1, 2024
C Kogias + 7
Open Access