Table 1
Changes in scouring based on flow intensity at 30°, 60°, 90°, 120°, 150° angles of the curved channel.
The impact of abutment length on scour depth
As shown in Fig. 7 (a-j), the impact of the abutment length on scouring along the curved channel is different at each bend angle. In a straight channel, scour depth increases as the abutment length increases (Kayaturk, 2005). However, in the present study, the scour depth in the curved channel decreased as the abutment length on the outer bank increased. In general, less scouring was observed for an abutment length of 12 cm. This showed that the scour formed as a result of the velocity distribution at the outer bank of a curved channel decreased in line with a reduction in the effect of secondary flow as the abutment length increased. When the bridge abutment is placed in the curved channels, the scouring at the inner and outer banks differs. Local scour depth occurs more often at the outer bank due to the effect of secondary flow and maximum velocity path while deposition occurs at the inner bank. Therefore, at the outer bank of the curved channels, a local scour takes place due to the effect of both the secondary flow and downflow. As shown in Fig. 7(a-j), the scouring at the outer bank is greater than the scouring at the inner bank. Thus, it is important to know in advance the properties of the flow in the curved channel when selecting a hydraulic structure such as a water intake structure.
Fig. 7. Changes in t /tm andds /La based on differentLa values for α =30°, α =60°,α =90°, α =120° and α =150°
The impact of bend angle on scour depth
It was generally observed that, for scour depths at bend angles of theLa = 8 cm abutment located on the outside bank in a 180° curved channel, the maximum scour increased in line with the increasing bend angle. Thus, maximum scour usually occurs at a 150o bend angle. This is because the maximum velocity in a linear channel occurs around the channel axis close to the water surface. The local flow slows down when entering the curve and gains momentum after passing the curve turning point. In a 180-degree curved channel, the maximum velocity orbits diverge from the normal orbits, first going towards the inner bank. When α = 45°, they start to flow towards the outer bank and when α = 60°, they start to settle at the outer bank. The maximum velocity orbit located at the outer bank turns towards the channel axis after α = 120° and can be observed near the channel axis at the curve exit. The scour depths at the bend angles of a La = 8 cm abutment placed on the inner bank of the curved channel demonstrated that the maximum scour also increased in line with increasing bend angle. On the inner bank, the maximum scour occurred at a 150° bend angle (Fig. 8). Based on all bend angles, a variation of between 5% and 15% was observed for the maximum scour depths formed.(Fig. 8).
In the second half of the curved channel, the scour depths formed on the bridge abutments were greater than in the first half. At the entrance of the channel, where the material transferred from the upstream compensates the carried material, there was less scouring at 30°. Therefore, if the bridge abutment is placed at the beginning of the curve, there will be less scour depth.
Fig. 8. Variation on outer and inner bank scour depths forα =30°, α =60°, α =90°, α =120°, α =150° bend angles.
The active length of the tailwater vortexes is longer than that of the horseshoe vortexes. However, because the velocity of horseshoe vortexes is greater than the velocity of the tailwater vortexes, the maximum scour is observed in the abutment upstream. Another reason for this is that the accumulation of particles unraveling from the upstream face in a downstream direction due to the falling flow intensity. In the present study, the maximum scour occurred around the upstream wall of the abutment (see Fig. 9).
Fig. 9 Variations in scour depth on the inner bank at 8 cm abutment.
Comparison of outer bank and inner bank scour depths
Figure 10 (a-o). A comparison of inner bank and outer bank maximum scour depths in the curved channel at bend angles of 30°, 60°, 90°, 120°, and 150° demonstrated that the maximum scour depth in all bend angles of the curved channel was higher on the outer bank than on the inner bank. This was because the material moving as a result of the secondary flow causes scouring on the outer bank and accumulation of the sediment on the inner bank. Sanjou and Nezu (2009) stated that there is a strong relation between the horizontal vortices and secondary flows. Furthermore, Minor et al. (2007) reported that secondary flows are caused by turbulence anisotropy at the wall in narrow channel bends.
Variations in the scour depth between the inner and outer banks of the main channel at bend angles α = 30°, 60°, 90°, 120° and 150° of the curved channel (based on the dimensionless parameters denominated with ds/La on the y-axis andt/tm on the x-axis) are displayed in Figure 11 (a-o). Examination of the scour depths along the curved channel shows that the scour depth at each bend angle was greater on the outer bank than the inner bank. This was due to the formation of secondary and helicoidal flow in the curved channel. The secondary flow also occurs due to the porousness of the wall and pressure oscillations. In meander oscillations, the secondary flow effect is strengthened due to the centrifugal force. The outer bank is scoured while the sediment is accumulated on the inner bank. As described by Onen and Agaccioglu (2013), this is because the secondary movement starts before the curve entrance in a curved channel, accelerates in the curvature, and decreases toward the curvature exit. One of the most important characteristics of the curved flow is the helicoidal flow; another is the maximum velocity path motion. The helicoidal flow is caused by friction, and by centrifugal and inertia forces. The helicoidal flow is defined as the ratio of the mean kinetic energy of the secondary action in a given cross-section to the total kinetic energy of the flow. Onen and Agaccioglu (2013) stated that the velocity of fluid particles near the channel bed is significantly delayed by the boundary resistance. Fluid that moves with a lower velocity near the bed is forced to follow a sharper curved orbit to establish an equilibrium between centrifugal and pressure forces; conversely, the orbits of fluid particles closer to the surface move towards the channel bed due to the higher inertia caused by higher speed. To maintain the fluid mass, the water moves towards the bed at the outer bank and towards the surface from the bed at the inner bank. Thus, in addition to the longitudinal velocity component, a radial velocity component perpendicular to the channel axis is formed. This forms the secondary flow in the cross section plan.
Fig. 10 Variation on outer and inner bank scour depths
The variation in dimensionless maximum scour depth with dimensionless time was investigated for different median sediment grain sizes (d 50) and constant abutment length, and for constant flow intensity for values of bend curvature angles, as shown in Fig. 11. This clearly shows that as the grain diameter decreases, the depth of scour increases.
Fig. 11 ds /Laversus t /t m ford 50=1.16, d 50=1.34,d 50=3.72 mm: (a) α=60°, (b) α=120°
Empirical correlations predicting the equilibrium scour depth were developed for the oblong bridge abutment at the inner banks of the curved channel with bend curvature α = 30°, 60°, 90°, 120°, and 150°. The resulting correlation is given in Eq. (13) and the values of the constants and the correlation coefficients (R) are presented in Table 2. Eq. (14) gives the relation between the inner and outer bank based on the experimental data in the current study. For Eq. (14), the correlation coefficient is 0.98.
\(\left(\frac{d_{\text{se}}}{L_{a}}\right)_{\text{inner\ bank}}=a\left(\frac{L_{a}}{y}\right)^{b}\ \left(\frac{V}{V_{c}}\right)^{c}\left(F_{d}\right)^{d},\left(\frac{L_{a}}{B}\right)^{e}\left(\frac{L_{a}}{B_{a}}\right)^{f}\)(13)
where dse is the equilibrium depth (m),La is the length of abutment (m), y is the flow depth (m), V is the mean flow velocity (m/s),Vc is the critical velocity (m/s), Fd is the densimetric Froude number (-), Bis the width of channel (m), and Ba is the width of abutment (m).
\(\left(\frac{d_{\text{se}}}{L_{a}}\right)_{\text{outer\ bank}}=1.45\left(\frac{d_{\text{se}}}{L_{a}}\right)_{\text{inner\ bank}}\)(14)
The predicted equilibrium scour depth (dse )p is compared with the observed equilibrium scour depth (dse )o to yield the average percent error ε as
\(\varepsilon=\frac{100}{N}\sum_{i=1}^{N}\left\lfloor\frac{\left(d_{\text{se}}\right)_{o}-\left(d_{\text{se}}\right)_{p}}{\left(d_{\text{se}}\right)_{o}}\right\rfloor\)(15)
in which N is the number of data points. The average percent error ε is a function of the constant in the equilibrium scour depth equation. The average percent error values for Eq. (15) are given in Table 2. Good agreement is obtained between the observed equilibrium scour depth and the values computed from the predictive Eq. (13).