Drag on pairs of square section obstacles in free-surface flows

The drag on pairs of square obstacles (of side D) in open channel flow is measured in experiments using a laboratory flume. The depth is uniform across the obstacles and the conditions in the flume are subcritical, with Froude number $\mathrm{Fr}<0.59$, Reynolds number Re = 4800 to 21900, and turbulence intensity $I_T\sim 8$ to 10%. The drag coefficient for an isolated square obstacle is found to be $C_D=2.11$, in agreement with previous studies, and independent of Re (for the range covered here). The root-mean-square variation in the drag coefficient for the single obstacle decreased monotonically with Re, defined in terms of hydraulic radius, and approached 0.241 at high Re in agreement with previous research. For two obstacles, standard tandem and side-by-side arrangements are studied first, followed by a full range of relative positions covering $-5\le s_x/D\le 20$ in the downstream direction and $0\le s_y/D\le 7.0$ in the cross-stream direction. The lowest drag coefficients are observed when the downstream obstacle is shielded directly behind the upstream obstacle (tandem arrangement) when negative drag coefficients are found. The largest drag coefficients are observed for nearly side-by-side arrangements, with the peak values found to be for the slightly upstream obstacle of a pair $(s_x/D=-1)$. The blockage ratio (D/B, the relative size of the obstacle compared to the channel width) is found to be an important factor. For D/B = 12.7% the largest drag coefficient is $C_D=3.82$, while for D/B = 6.3% the largest value is $C_D=2.85$. For tandem obstacles, the drag on one obstacle can largely be accounted for by the change in flow speed induced by the other obstacle, except at small separations $\left(\right|s_x/D|<3)$. The results will be useful in any applications where the force on multiple obstacles is required, such as the design of marine or riverine structures, or flood flows past buildings and vegetation.

Figure 1

Sketch of the flume used for the experiments.

Figure 2

Profiles of the normalized, temporally averaged, streamwise component of velocity in the flume with no obstacles present, at different downstream distances from the start of the main working section $x$ for $Q=0.0166{\mathrm{m}}^3{\mathrm{s}}^{-1}$, $H=158\mathrm{mm}$, and $S=0.0007$. (a) In the vertical direction at the center of the channel and (b) in the cross-stream direction at a height of H/3 below the free surface.

Figure 3

Sketch showing the arrangement of obstacles (here with $D/B=12.7\%$). The darker obstacle is mounted on a strain gauge in the center of the channel, while the lighter obstacle is moved to different positions. A negative value of $s_x$ indicates that the drag is measured on the upstream obstacle.

Figure 4

Normalized, temporally averaged, streamwise component of velocity on the channel centerline at a depth of H/3 below the free surface for flow past a single obstacle with $D/B=12.7\%$ and $\mathrm{Re}=11000$. The results from Lyn et al. [19] are also shown.

Figure 5

Measured drag force on a single obstacle plotted against the expected scale for the force. The diagonal line marks the linear regression $(C_D\approx 2.1)$. The vertical line in the top left corner represents the maximum measurement uncertainty in $F_D$.

Figure 6

Root-mean-square drag coefficient of an isolated obstacle at the channel center plotted against Reynolds number, based on the upstream hydraulic radius. The best fit (solid line) is given by Eq. (9).

Figure 7

Normalized, temporally averaged, streamwise component of velocity on the channel centerline for a tandem pair of obstacles with $D/B=12.7\%$, $\mathrm{Re}=16000$, and separations between obstacle centers of (a) 2.5D and (b) 10D.

Figure 8

Drag coefficient for the second obstacle in a tandem pair, positioned $s_x$ downstream of the first obstacle (so $s_x<0$ indicates the drag is measured on the upstream obstacle).

Figure 9

Comparison of predicted and measured drag coefficients at $D/B=12.7\%$. The drag was measured at $\mathrm{Re}=16000$ and the local velocity, used to compute $C_D{}^{\prime }$, was measured at $\mathrm{Re}=11000$. The continuous lines represent perfect agreement. (a) The measured drag coefficient plotted against the predicted drag coefficient. (b) The ratio of the predicted and measured drag coefficient plotted against the normalized streamwise distance. Dotted lines enclose an area of agreement between the predicted and measured values to within $\pm 10\%$.

Figure 10

Drag coefficient on one obstacle of a side-by-side pair. The drag is measured on an obstacle in the center of the channel, while the second obstacle would be against the channel sidewall at $s_y=3.5D$ (for $D/B=12.7\%$) or 7.4D (for $D/B=6.3\%$). The position of the sidewall in each case is indicated by the dashed lines. Reynolds number is 16000 for $D/B=12.7\%$ and 7500 for $D/B=6.3\%$.

Figure 11

Drag coefficient on a second obstacle as a function of relative position for (a) $D/B=12.7\%(\mathrm{Re}=16000)$ and (b) $D/B=6.3\%(\mathrm{Re}=7500)$.

Figure 12

Contours of drag coefficient on a second obstacle placed at different relative positions, with blockage ratios (a) $D/B=12.7\%(\mathrm{Re}=16000)$ and (b) $D/B=6.3\%(\mathrm{Re}=7500)$.