# Apron and Cutoff Wall Scour Protection for Piano Key Weirs

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{b}= streamwise length of the base of the PK weir, B

_{i}= the length of the inlet key, B

_{o}= the length of the outlet key, S

_{o}= slope of the outlet key, S

_{i}= slope of the inlet key, W

_{u}= cycle width, N = number of keys or cycles, P = weir height, T

_{s}= sidewall thickness, W

_{i}= width of the inlet key, W

_{o}= width of the outlet key, P

_{d}= height of the PK weir base relative to the invert of the channel, x

_{M}

_{1}, x

_{M}

_{2}, and x

_{M}

_{3}are streamwise lengths from the apron to sphere measurement locations, L

_{a}is the length of the apron, and L

_{c}is the length or depth of the cutoff wall. Note that a false floor was not included upstream of the PK weir, as its presence would be negligible on the results of this study. We also note that the dimensions of the apron and cutoff wall are not indicative of actual thicknesses as may be structurally designed for specific applications.

_{90}= 20.00 mm, d

_{50}= 13.00 mm, uniformity coefficient σ = 1.54, density ρ

_{s}= 2604 kg/m

^{3}, and relative density G = ρ

_{s}/ρ = 2.61, where d

_{xx}is the diameter for which xx% is finer and ρ is the water density. Substrate 2 was a fine gravel with d

_{90}= 9.10 mm, d

_{50}= 6.50 mm, σ = 1.30, ρ

_{s}= 2604 kg/m

^{3}, and G = 2.61. A diagram sketch illustrating the main hydraulic and scour parameters is shown in Figure 2, with H indicating total hydraulic head (H = h

_{u}+ V

^{2}/2g) where h

_{u}is the depth of flow relative to the weir crest (with total weir crest length = L), g is the standard acceleration due to gravity, and V is an average upstream flow velocity. Figure 2 also shows the PK weir foundation height P

_{d}, the change in energy head, ΔH, h

_{d}the downstream flow depth, L

_{a}the length of apron in the streamwise direction, L

_{max}is the maximum scour length, X

_{max}is the streamwise distance from the end of the apron to the maximum scour depth Z

_{max}, and Z

_{max_ref}is the maximum scour depth without an apron, referred to herein as the reference maximum scour depth.

_{u}. Conversely, the downstream water level (and, consequently, the tailwater depth h

_{d}) was measured by using a Microsonic mic+130 ultra-sonic sensor (±1%) (Microsonic, Dortmund, Germany) [37,38] at 3.35 m downstream (x/P = 8). Initial investigations profiled the downstream water surface to select an appropriate location for computing h

_{d}.

^{TM}D435 depth camera (accuracy of ±1 mm) (Intel, Santa Clara, CA, USA) [30,39,40] processed with a Utah State University (USU) custom MATLAB (vR2020) script. The camera data was cross-checked with point measurements taken on a grid layout with a point gage assembly (accuracy of ±1 mm). This camera is capable of capturing the surface of solids or roughened or opaque water surfaces in three dimensions; its specifications are summarized in Table 2. Post-processing included 2D and 3D scour morphology plots and allowed quantifying maximum scour location (X

_{max}) and its depth (Z

_{max}), maximum length of the scour hole (L

_{max}), and other scour features.

_{a}) were tested, i.e., L

_{a}= 0P (reference configuration), 1P, 1.5P, and 2P, under three different combinations of Q and h

_{d}(Table 3). Note that all tests (see Table 4) were conducted with the lowest permissible tailwater depth in order to investigate the most critical hydraulic conditions. Each test began by slowly filling the headbox and substrate box until the weir was in a submerged condition. The target Q was set, the h

_{d}was lowered to the lowest permissible depth, and then the timer and video recording array were initiated. Note that for certain values of Q the horizontal bed prevented the target h

_{d}(~0.33P) from being achieved. Multiple experiments were conducted for more than 18 h each to confirm equilibrium conditions and necessary durations for the remainder of the test matrix. More specifically, to assess the time needed to reach the equilibrium configuration, this study conducted some selected longer tests. In particular, for Substrate 2 and Q = 600 L/s, the reference test 15 was conducted (lasting up 1170 min, i.e., 19.5 h, see Table 4). It was observed that the differences between the equilibrium morphologies at t = 6 h (i.e., the same time duration of Test 18, conducted under similar hydraulic conditions) and at 19.5 h were negligible. Thus, the duration of test 18 was long enough to reach the equilibrium configuration of the bed morphology. Similar considerations apply for all the tested discharges (e.g., for Q = 300 L/s test 14 was conducted with duration t = 1050 min).

_{c}was less than 1, where U and U

_{c}are the average flow velocity and the average threshold velocity, respectively, with U

_{c}calculated using the methodology developed by [42]. As for tests conducted with Q = 600 L/s and Substrate 1, tests were characterized by U/U

_{c}≈ 1; Conversely, for tests with Q = 600 L/s and Substrate 2 (i.e., tests 15, 18, 21, and 24), U/U

_{c}was slightly bigger than 1. Nevertheless, it was decided to also include these tests in these analyses, as their behavior in terms of the effect of apron geometry on scour characteristics was found to be essentially consistent with those of other tests. Therefore, they can provide useful information for practical applications.

## 3. Results

#### 3.1. PK Weir Hydraulics

_{c}− 1)

^{m}, with τ indicating the shear stress, τ

_{c}the critical shear stress, and m is a coefficient that is equal to 1.5 for non-cohesive materials according [43]. Therefore, the equilibrium configuration is achieved when τ = τ

_{c}and, consequently, Z = Z

_{max}, with h

_{d}being constant. The presence of a horizontal apron significantly modifies flow features in the downstream stilling basin. In particular, impinging jets are deflected horizontally and a more uniform distribution of the flow occurs downstream of the apron. In addition, a dissipative process may occur on the apron due to the formation of a submerged hydraulic jump. As a consequence, scour takes place downstream of the apron and is characterized by a significant reduction of the main lengths (Figure 3) in comparison with the reference case. However, depending on the tested conditions, the presence of scour may endanger the stability of the apron. Therefore, a cutoff wall is suggested to avoid undermining of the apron and a potential failure mode.

#### 3.2. Analysis of Scour Processes

_{90}). The overall observed trend was that scour depth and maximum scour dimensions significantly decrease with L

_{a}under identical hydraulic conditions (Figure 4 and Table 4) (note that for Run 10 no scour occurred in the stilling basin). Results evidenced that the effect of an apron is always prominent, regardless the bed material characteristics (i.e., for both Substrates 1 and 2). As mentioned in the previous section, this behavior can be explained when considering that the flow dynamics are radically different for protected basins. Both flow features and sediment transport mode exhibit similarities with that occurring downstream of an apron caused by submerged horizontal jets. Namely, scour starts in correspondence with the edge of the structure because of the excess of shear stress. During the developing phase, bed material is mobilized and kept in suspension. Successively, when the scour hole enlarges (developed phase), the jet exiting from the apron is deflected downward. Consequently, the sediment transport mode changes, involving both suspended and bed loads. Furthermore, this study corroborates the findings of [35], which showed that the maximum scour depth is a monotonic decreasing function of the non-dimensional apron length and average diameter of the bed material. To this end are presented Figure 4a,c,e (Substrate 1) and Figure 4b,d,f (Substrate 2) where longitudinal scour profiles pertaining to tests are shown. Please note that the apron length and cuttoff wall for each configuration are color matched to the corresponding maximum scour profiles. For example, in Figure 4, A L

_{a}/P = 1.5 is in red, but is overlapped by L

_{a}/P = 1.0 that is blue. The red scour profile (dots with dashed line) corresponds to L

_{a}/P = 1.5.

_{max}= 100(Z

_{max_ref}− Z

_{max})/Z

_{max_ref}, which is caused by the different apron configurations with Z

_{max_ref}indicating the maximum scour depth of the corresponding reference test (i.e., without an apron). Namely, tests were grouped according to the two substrate types and Z*

_{max}was plotted as function of the non-dimensional apron length L

_{a}/P for different densimetric Froude numbers F

_{rd}

_{90}= q/{h

_{u}[(G − 1)gd

_{90}]

^{1/2}} (Figure 5a,b). Note that q = Q/W and is the average unit discharge. A slight difference, depending on the inflow characteristics, can be pointed out between tests involving Substrate 1 and 2. Namely, for both the substrates Z*

_{max}is a monotonic increasing function of L

_{a}/P. However, for tests conducted with Substrate 1 and F

_{rd}

_{90}= 3.05 and 3.25, Z*

_{max}becomes almost constant for 1.5 ≤ L

_{a}/P ≤ 2, i.e., it only depends upon F

_{rd}

_{90}and the effect of L

_{a}/P becomes negligible (Figure 5a). In addition, the presence of the apron prevents the scour formation for L

_{a}/P = 2 and F

_{rd}

_{90}= 2.44 (i.e., Z*

_{max}= 100%, as shown in Figure 5a). A similar behavior can be pointed out for tests with Substrate 2. Overall, Z*

_{max}increases with L

_{a}/P, whereas it slightly decreases with F

_{rd}

_{90}(Figure 5b). These results corroborate the findings of [35,44,45], who found that scour processes occurring in protected stilling basins are essentially influenced by the geometry of the protection structures and the inflow conditions. The mentioned analysis allowed the derivation of the following predicting equation, valid for 0 ≤ L

_{a}/P ≤ 2:

_{max}and represents a simple tool that practitioners can use to estimate scour characteristics in protected basins.

_{a}was quantified regarding its effect on the maximum scour depth Z

_{max}and length X

_{max}, along with volume of sediment

**V**removed under different inflow conditions and for both the substrates. In particular, Figure 6 shows that, by adding a 1.0P = L

_{a}horizontal apron, the average reduction of the scour depth is approximately equal to 57%. This percentage increases with L

_{a}and ranges between 75% and 83% for L

_{a}= 1.5P and 2.0P, respectively (see also Table 5). It is worth remarking that the change in scour depth for L

_{a}= 1.5P and L

_{a}= 2.0P was on average 8%. It implies that, for practical applications and F

_{rd}

_{90}> 3, an additional protection (i.e., longer apron) may only minimize scour depth marginally. Likewise, the variable L

_{max}is also affected by L

_{a}(Figure 7). However, in this case a different behavior can be pointed out depending on the sediment bed gradation and, in particular, on d

_{50}. Namely, for Substrate 1, a general reduction of the scour hole length occurs by increasing L

_{a}, regardless of the inflow conditions. Conversely, for finer bed material, the maximum scour depth seems to be less affected by the apron length, especially for higher discharges. This behavior is similar to that observed by [45] downstream of block ramps, especially for low slopes of the ramp, for which the horizontal component of the total hydrodynamic force at the ramp toe is much larger than the vertical counterpart. Namely, in the presence of a horizontal, smooth apron, depending on the inflow conditions (i.e., especially for higher values of F

_{rd}

_{90}), a hydraulic jump may not occur. Thus, the scour downstream of the apron is essentially due to a horizontal, not submerged, jet. The shear stress due to jet generally exceeds the critical counterpart, resulting in a scour formation downstream of the apron. Nevertheless, for finer bed materials, the superficial layer can be more easily transported downstream, thus modifying the overall shape of the scour hole, which becomes more longitudinally extended (Figure 4b,d,f and Figure 7).

_{max}, Equation (1) can be re-written as follows:

_{rd}

_{90}≤ 5.3, 0 ≤ L

_{a}/P ≤ 2, 0.11 ≤ H/P ≤ 0.36, and 0.24 ≤ h

_{d}/P ≤ 0.59, but the results of this study trend to h

_{d}/P ≤1.03 (exceeds tested tailwater depth ratios presented herein). Note that the maximum scour depth in the corresponding reference test (Z

_{max_ref}) can be estimated using the following Equation (4) proposed by [27]:

_{c}= (q

^{2}/g)

^{1/3}is the critical depth, with q indicating the unit discharge, g the acceleration due to gravity, ΔH the change in energy head, and h

_{d}is the tailwater depth. Figure 8 show the comparison between measured and calculated (using Equation (3)) values of the variable Z

_{max}.

#### 3.3. Apron and Cutoff Wall Design Guidance

_{max}exceeds 75%, the effect of the apron length becomes negligible, i.e., a further longitudinal extension of the apron results in a very marginal additional reduction of the scour depth (Figure 5a,b). Therefore, for practical applications, it can be reasonably assumed that a design value of the apron length for which Z*

_{max}= 75. Consequently, Equation (2) can be rearranged as follows:

_{a}/P ≤ 2:

_{a}/P in order to maximize the scour reduction and minimize the construction cost. Note that Equation (6) is valid in the same ranges of parameters of Equation (3). It is worth remarking that Equation (6) does not depend on the tailwater level, which is one of the most important parameters in scour processes as the shear stress acting on the movable bed scales with the inverse of the sum of the water depth and scour depth. However, as mentioned in one of the previous sections, the range of variation of this parameter was relatively limited. Consequently, under tested conditions, its effect was found to be negligible. Overall, Equation (6) represents a useful tool as it can provide good estimations of the scour characteristics under most critical conditions (i.e., design condition). However, further investigations are needed to assess the effect of the tailwater depth. Figure 9 shows the plot of Equation (6), i.e., L

_{a}/P as function of the densimetric Froude number F

_{rd}

_{90}.

_{c}of a cutoff wall should be designed in such a way that it is larger than the maximum scour depth Z

_{max}calculated using Equation (3). To this end, the following is proposed:

_{max}, it is less than 30% for nearly all cases (see Figure 8) and thus a minimum FS would be 1.3. It is worth mentioning that the length of the cutoff wall L

_{c}accounts for both inflow conditions and geometry of the structure, as such variables are embedded in Z

_{max}. Further considerations for selection of L

_{c}might include depth to a nonerodable geologic feature to key into, any relevant field observations, and balancing risks and project economy (i.e., excavation, dewatering, control of water, etc.).

## 4. Conclusions

- Scour depths in the two tested noncohesive gravel materials without an apron and cutoff wall were significant.
- Jets ensuing from the PK weir structure diffused over the apron and cause horizontal jet scour downstream of the apron.
- Adding a 1.0P long apron can reduce scour by an average of 61%, adding a 1.5P long apron can reduce scour by an average of 75%, and adding a 2.0P apron can reduce scour by an average of 83%. Thus, apron length L
_{a}can be optimized considering PK weir hydraulics and geometry with geologic conditions at PK weir toe. - Equations (3) and (6) were created to estimate L
_{a}, Z_{max}downstream of an apron along with Equation (4), and Equation (7) was created to estimate L_{c}to protect the structure from scour undermining. Note that Equations (3) and (6) are considered valid for 2.4 ≤ F_{rd}_{90}≤ 5.3, 0 ≤ L_{a}/P ≤ 2, 0.11 ≤ H/P ≤ 0.36, and 0.24 ≤ h_{d}/P ≤ 1.03. All proposed equations should be validated using field data. - For this study, it was determined that there was an 8% difference in the reduction of scour from a 1.5P apron length to a 2.0P apron length, and a 1.5P apron may be an adequate and cost-effective length to minimize scour. Note that as the substrate diameter decreases the potential for scour depth and length downstream of the apron will increase.
- Although the facility tested at USU was quite large and might be considered a 1:5 to 1:10 scaled model, quantification of any scale effects requires additional investigation.

_{max}values as conservative design values. Another limitation was that only three Q and one h

_{d}conditions were tested in this study. To overcome this limitation, practitioners could use graphical means to interpolate potential Z

_{max}values based on the rating curves provided and might adjust their FS if tailwater depths are greater. Further, results may be scaled using the densimetric Froude number. Furthermore, only two relatively uniform substrate materials were studied. Practitioners could use methods similar to the Erodibility Index method of [47] to scale geologic material strengths in the field to a noncohesive laboratory-scale material, such as was selected and tested herein. Additionally, these results and equations only directly apply to horizontal aprons without a sill and very mild river slopes or energy gradients. Lastly, only one PK weir geometry was tested with q estimated as Q/W.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**Photographs from experiments under identical hydraulic conditions: (

**a**) Reference test (Run 14); (

**b**) Scour protection via horizontal apron and cutoff wall (Run 23).

**Figure 4.**Comparison of longitudinal bed profiles for runs: (

**a**) 1, 4, and 7; (

**b**) 13, 16, 19 and 22; (

**c**) 2, 5, 8 and 11; (

**d**) 14, 17, 20 and 23; (

**e**) 3, 6, 9, 9R and 12; and (

**f**) 15, 18, 21 and 24.

**Figure 5.**Z*

_{max}as function of F

_{rd}

_{90}for Substrate (

**a**) 1 and (

**b**) 2. Z*

_{max}as function of F

_{rd}

_{90}along with the plot of Equation (1) for F

_{rd}

_{90}= (

**c**) 2.44, (

**d**) 3.05, (

**e**) 3.99, and (

**f**) 4.98.

**Figure 6.**Effect of L

_{a}on Z

_{max}under different inflow conditions and Substrates (

**a**) 1 and (

**b**) 2. Note that 0P indicates corresponding reference tests.

**Figure 7.**Effect of L

_{a}on L

_{max}under different inflow conditions and Substrates (

**a**) 1 and (

**b**) 2. Note that 0P indicates corresponding reference tests.

**Figure 8.**Comparison between measured and calculated (using Equation (3)) values of the maximum scour depth Z

_{max}for protected basins, with 30% bounding lines referencing Z

_{max}Observed.

Parameter | Value |
---|---|

B | 1.04 m |

B_{b} | 0.52 m |

B_{i} = B_{o} | 0.26 m |

N | 4 |

P | 0.42 m |

P_{d} | 1.09 m |

S_{i} = S_{o} | 0.55 |

T_{s} | 0.025 m |

W_{i} | 0.248 m |

W_{o} | 0.194 m |

W_{i}/W_{o} | 1.280 m |

W_{u} | 0.49 m |

Feature | Detail |
---|---|

Global Shutter | 3 μm × 3 μm pixel size |

IR Stereo: FOV | 86° × 57° (±3°) |

IR Stereo: Resolution | 1280 × 720 |

RGB: FOV | 64°×41° × 77° (±3°) |

RGB: Resolution | 1920 × 1080 |

Substrate Type | Discharge Q | Headwater Ratio | Tailwater † h _{d} | Apron Length L _{a} |
---|---|---|---|---|

Substrate 1 d _{50} = 13 mmd _{90} = 20 mmσ = 1.54 Substrate 2 d _{50} = 6.5 mmd _{90} = 9.1 mmσ = 1.30 | 150 L/s | H/P = 0.11 | 16 cm = 0.38P | 2.0P = 0.84 m |

1.5P = 0.63 m | ||||

1.0P = 0.42 m | ||||

0.0P = 0.00 m * | ||||

300 L/s | H/P = 0.18 | 17 cm = 0.41P | 2.0P = 0.84 m | |

1.5P = 0.63 m | ||||

1.0P = 0.42 m | ||||

0.0P = 0.00 m * | ||||

600 L/s | H/P = 0.35 | 22 cm = 0.52P | 2.0P = 0.84 m | |

1.5P = 0.63 m | ||||

1.0P = 0.42 m | ||||

0.0P = 0.00 m * |

_{a}= 0.

Run (#) | Substrate (-) | t (min) | L_{a}/P(-) | Q (L/s) |
---|---|---|---|---|

1 * | 1 | 180 | 0.0 | 150 |

2 * | 1 | 240 | 0.0 | 300 |

3 * | 1 | 900 | 0.0 | 600 |

4 | 1 | 360 | 1.0 | 150 |

5 | 1 | 360 | 1.0 | 300 |

6 | 1 | 870 | 1.0 | 600 |

7 | 1 | 360 | 1.5 | 150 |

8 | 1 | 360 | 1.5 | 300 |

9 | 1 | 360 | 1.5 | 600 |

9R | 1 | 360 | 1.5 | 600 |

10 | 1 | no scour | 2.0 | 150 |

11 | 1 | 120 | 2.0 | 300 |

12 | 1 | 360 | 2.0 | 600 |

13 * | 2 | 450 | 0.0 | 150 |

14 * | 2 | 1050 | 0.0 | 300 |

15 * | 2 | 1170 | 0.0 | 600 |

16 | 2 | 840 | 1.0 | 150 |

17 | 2 | 780 | 1.0 | 300 |

18 | 2 | 720 | 1.0 | 600 |

19 | 2 | 240 | 1.5 | 150 |

20 | 2 | 840 | 1.5 | 300 |

21 | 2 | 990 | 1.5 | 600 |

22 | 2 | 480 | 2.0 | 150 |

23 | 2 | 600 | 2.0 | 300 |

24 | 2 | 720 | 2.0 | 600 |

_{a}= 0.

Run (#) | h_{d}(m) | Z_{max}(m) | X_{max}(m) | L_{max}(m) | V (m ^{3}) |
---|---|---|---|---|---|

1 * | 0.16 | 0.18 | 0.25 | 0.61 | 0.12 |

2 * | 0.17 | 0.33 | 0.42 | 1.07 | 0.36 |

3 * | 0.22 | 0.71 | 0.73 | 2.20 | 1.49 |

4 | 0.09 | 0.07 | 0.00 | 0.47 | 0.03 |

5 | 0.16 | 0.15 | 0.00 | 0.75 | 0.12 |

6 | 0.21 | 0.39 | 0.00 | 1.54 | 0.50 |

7 | 0.10 | 0.03 | 0.00 | 0.31 | 0.00 |

8 | 0.17 | 0.06 | 0.00 | 0.44 | 0.03 |

9 | 0.22 | 0.28 | 0.00 | 1.54 | 0.37 |

9R | 0.23 | 0.23 | 0.00 | 1.45 | 0.37 |

10 | 0.09 | No Scour | No Scour | No Scour | No Scour |

11 | 0.15 | 0.06 | 0.00 | 0.36 | 0.02 |

12 | 0.22 | 0.19 | 0.00 | 1.18 | 0.27 |

13 * | 0.10 | 0.28 | 0.40 | 1.07 | 0.39 |

14 * | 0.14 | 0.56 | 0.52 | 1.81 | 1.03 |

15 * | 0.22 | 1.01 | 1.16 | 3.19 | 3.36 |

16 | 0.11 | 0.11 | 0.00 | 0.61 | 0.07 |

17 | 0.16 | 0.20 | 0.00 | 1.15 | 0.26 |

18 | 0.24 | 0.45 | 0.00 | 1.83 | 0.74 |

19 | 0.11 | 0.06 | 0.00 | 0.47 | 0.03 |

20 | 0.15 | 0.12 | 0.00 | 1.28 | 0.15 |

21 | 0.25 | 0.34 | 0.00 | 2.37 | 0.56 |

22 | 0.10 | 0.04 | 0.00 | 0.45 | 0.02 |

23 | 0.15 | 0.11 | 0.00 | 1.02 | 0.12 |

24 | 0.24 | 0.25 | 0.00 | 3.05 | 0.45 |

_{a}= 0.

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## Share and Cite

**MDPI and ACS Style**

Lantz, W.; Crookston, B.M.; Palermo, M.
Apron and Cutoff Wall Scour Protection for Piano Key Weirs. *Water* **2021**, *13*, 2332.
https://doi.org/10.3390/w13172332

**AMA Style**

Lantz W, Crookston BM, Palermo M.
Apron and Cutoff Wall Scour Protection for Piano Key Weirs. *Water*. 2021; 13(17):2332.
https://doi.org/10.3390/w13172332

**Chicago/Turabian Style**

Lantz, Wyatt, Brian Mark Crookston, and Michele Palermo.
2021. "Apron and Cutoff Wall Scour Protection for Piano Key Weirs" *Water* 13, no. 17: 2332.
https://doi.org/10.3390/w13172332