Riprap Drag Coefficient

While there is a lack of universally accepted drag coefficient determination equation, the literature has numerous empirical equations, but the equations all have certain drawbacks (Wang et al., 2018). The particle shapes that are significantly investigated for drag coefficients comprise sand particles and spheres at low and high Reynolds number. Moreover, regular shapes such as cubes, cylinders and prisms are generally studied as well (Wang et al., 2018).

The author of this thesis could not find specific literature that attempted to determine the drag coefficient or the settling velocity of angular riprap rocks. Langmaak and Basson (2015) obtained a drag coefficient of 1.66 for large riprap rocks. However, it was not clear whether the randomly chosen riprap rocks used in the settling velocity tests were a mixture of both subangular and angular rocks or only the angular rocks were tested in the settling velocity tests. Since the shape plays a critical role in the determination of the drag coefficient, it was incumbent that a drag coefficient value was specified for pertinent riprap rock shapes.

3-68 According to (Cheng, 1997), under high particle Reynolds number, the CD is 0.4 for spherical particles. However, for natural sediments, the CD value is between 1 and 1.2 (Cheng, 1997). Cheng (1997) made the previously mentioned recommendations on the CD value based on the findings of six previous studies: Sha (1956), Concharov (1962), Zhang (1989), Van Rijn (1989), Raudziki (1990), Zhu and Cheng (1993).

It is possible that no specific study has addressed the issue of the drag coefficient of angular or crushed rocks, as it could not be found from open source academic literature. As a result, Armitage (2002) recommended the determination of the drag coefficient of irregular shapes be physically tested at the hydraulics laboratory. This is important for the MN analysis approach since the results are sensitive to the drag coefficient or settling velocity.

Therefore, it was imperative that the drag coefficient of the angular stone was determined experimentally for this study. The previously mentioned drag coefficients were important as they were used as a reference to evaluate the drag coefficient and settling velocities found at the laboratory for this study.

The first step that was taken to determine the drag coefficient of the angular rocks was to determine a sample of rocks to be tested for the settling velocities. The following rock sizes were tested for the settling velocity tests:

• 0.026 m - 0.038 m • 0.038 m - 0.053 m • 0.053 m - 0.075 m

For each sample range, a total of 40 stones was taken from the piles in the hydraulics laboratory. The stones in the samples were measured for the three dimensions “a”, “b” and “c” as defined in the literature for use in the determination of the Corey Shape Factor. Figure 24 displays a typical dimension measurement of the stones using a 30 cm steel ruler.

3-69 Figure 24: Measurement of the rock shape factor dimensions

After determining all the rock dimensions, each rock was dropped individually into the settling tank shown in Figure 25. The water from the bottom of the tank up to the top was measured and recorded for each test. A nylon fish line with a heavy material at the tip was dropped in the tank, then a mark at the top was made to mark the top surface of the water in the tank. The fish line length was measured with a measuring tape on the floor to determine the water depth of the tank.

3-70 Figure 25: Settling tank used at the hydraulics laboratory (sourced from Langmaak (2013))

The time taken for a single stone to settle was recorded using a stopwatch. Langmaak (2013) used a reliable method in the determination of the settling velocities of the stones. Langmaak (2013) took advantage of the inspection windows, at the top and at the bottom of the tank. The distance travelled by the rock divided by the difference in the settling times between the two inspection windows were used to determine the settling velocities. However, in this study, the water in the tank was not sufficiently clean to take advantage of the inspection windows. As a result, only the time recorded with a stopwatch and the distance each rock travelled from the surface of the water to the bottom of the settling tank was used to determine the settling velocity, as shown below:

𝑣𝑠𝑠 =

𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑟𝑎𝑣𝑒𝑙𝑙𝑒𝑑 𝑏𝑦 𝑟𝑜𝑐𝑘

𝑇𝑖𝑚𝑒 𝑟𝑒𝑐𝑜𝑟𝑑𝑒𝑑 𝑡𝑜 𝑡𝑟𝑎𝑣𝑒𝑙 𝑓𝑟𝑜𝑚 𝑡𝑜𝑝 𝑡𝑜 𝑡ℎ𝑒 𝑏𝑜𝑡𝑡𝑜𝑚 𝑜𝑓 𝑡𝑎𝑛𝑘

The 0.053-0.075 m stone size rock sample was tested on a separate day from the other rock sample sizes. The 0.026-0.038 m and 0.038-0.053 m stone sample settling velocity tests were conducted on the same day. When the 0.026-0.038 m and 0.038-0.053 m stone samples were tested, the water in the tank was refilled. Thus, the settling water depths are different in the tables in Appendix B. For the 0.053-0.075 m the settling water depth was 4.905 m, while for the other two samples the settling water depth was 4.960 m.

3-71 One may argue that the disadvantage with the settling velocity determination method executed for this study was the influence of human reaction time. However, the settling tank height was significantly long. Therefore, the height of the tank (thus distance travelled) significantly contributed to the reduction of the error associated with the human reaction time. Alternatively, if the rock’s settling distance were shorter, then the error associated with the reaction time when pressing the stopwatch would be significant.

A summary of the rock Corey shape factors as well as the settling velocity and respective drag coefficients were tabulated in Appendix B. A total of 45 successful settling velocity tests were accomplished, while 120 rocks were dropped into the settling tank. The settling velocity test was deemed unsuccessful if the following occurred:

• if the rock touched a steel beam that was welded somewhere in the middle inside the settling tank; or

• if the rock touched the internal sides of the settling tank before reaching the bottom of the tank.

Using the same approach as Langmaak (2013), the Corey shape factor and drag coefficient for each of the 45 rocks was plotted in the graph as seen in Figure 26. A trend similar to that of Langmaak (2013) was obtained. In Figure 26, as the drag coefficient decreases, the Corey shape factor increases. Most of the data plotted between a drag coefficient of 1 and 3.5, with few outliers greater than CD of 3.5. One significant observation was that the drag coefficients were all greater than one.

The drag coefficient was calculated from the measured settling velocities for each rock in the

samples. Equation 3, 𝑣_𝑠𝑠 = √4₃(𝜌𝑟−𝜌𝑤)

𝜌𝑤

𝑔𝐷50

𝐶𝐷 , was used to calculate the drag coefficient for

each rock in Appendix B. the rock density was assumed to be 2700 kg/m3 and the density of the water was assumed to be 1000 kg/m3. The “b” dimension of each rock was assumed to be D50 in Equation 3, since it was a logically more representative value for D50. With the previously mentioned parameters known, it was possible to calculate the respective drag coefficient with Equation 3. The drag coefficient and the respective settling velocity were plotted onto Figure 26.

From Figure 26, a general trend was observed, whereby the decreasing drag coefficient of angular rocks relates to an increase in the settling velocities of the angular rocks. This is in correlation with the fact that drag force is a resistant force, thus more resistance from the rock

3-72 should result in smaller settling velocities, whereas smaller resistant forces should result in higher settling velocities.

Figure 26: Graph showing the relationship between the Corey shape factor versus the drag coefficient & the settling velocities of angular riprap rock.

According to Rooseboom and Mulke (1982), the drag coefficient at high particle Reynolds number must be constant for large particles. For the 0.026-0.038 m, 0.038-0.053 m and 0.053- 0.075 m stone size samples in Appendix B, the average drag coefficients were determined to be 2.001, 2.204 and 2.270 for each stone range. There was a slight difference due to the irregular nature of the rocks, but the results agree with the argument by Rooseboom and Mulke (1982).

The average Corey shape factor for the 45 data points was found to be 0.529, and the corresponding drag coefficient using the trendline equation was 2.17. Therefore, the assumed rock drag coefficient for angular riprap rock was determined to be 2.17. This value was acceptable since it was relatively close to the drag coefficient of 1.66 found in Langmaak’s (2013) study which used similar rock shapes.

However, the discrepancy in the drag coefficients of the two studies may be attributed to the irregular nature of riprap rocks. Other reasons could be attributed to the slight difference in the methods followed by Langmaak (2013) and in this study in determining the settling time. Nonetheless, it was expected that the drag coefficient determined in this study must be greater than the 1 to 1.2 drag coefficient recommended by Cheng (1997) for natural sediment.

CD = -2.6057(CSF) + 3.5503 CD = -3.0639(Vss) + 4.4885 0.00 1.00 2.00 3.00 4.00 5.00 6.00 0.00 0.20 0.40 0.60 0.80 1.00 1.20 D ra g Co ef fi ci en t (C D )

Drag Coefficient vs Corey Shape Factor & Settling