# Local Scour for Vertical Piles in Steady Currents: Review of Mechanisms, Influencing Factors and Empirical Equations

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^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Influencing Factors

#### 2.1. Intensity of Flow

#### 2.2. Flow Depth

_{s}depended on the relative size of the pile h/D and the relative medium sediments diameter $D/{d}_{50}$. When $h/D$ is less than 2, the volume of down-flow diverted into the scour hole diminished. When $h/D$ is larger than 3, the interactions between anti-clockwise damming near the water surface and the horseshoe vortex in clockwise will fade to zero. As a result, the horseshoe vortex can act on sediment scour completely. When $h/D$ is below 1, a nearly to the water surface height bar will be formed behind the pile, which will weaken the strength of wake vortex. For the influence of $D/{d}_{50}$, Raudkivi and Ettema [57] came to the conclusion that the scour depth was independent with $h/D$ when $D/{d}_{50}$ was in high values. However, the scour depth did not seem to stop increasing with $h/D=6$ for a given value of $D/{d}_{50}$.

#### 2.3. Sediments

_{g.}Dey [59] classified the sediment material into uniform sediment (${\sigma}_{\mathrm{g}}<1.4$) and non-uniform sediment (${\sigma}_{\mathrm{g}}>1.4$). Hoffmans [60] defined ${\sigma}_{\mathrm{g}}<1.3$ as uniform sediment, and stated that sediments non-uniformity ${\sigma}_{\mathrm{g}}$ was about 1.8 in natural rivers. Researchers found that the scouring rate and scour depth decreased significantly with ${\sigma}_{\mathrm{g}}$ increased [61,62,63]. Chiew and Melville [62], for an example, observed that the finer particles would be armored by coarse particles both on the upstream bed and at the scour hole. The scour depth decreased with ${\sigma}_{\mathrm{g}}$ when ${\sigma}_{\mathrm{g}}$ was less than 5. In accord with Figure 2a,b, Zanke et al. [29] obtained that the scour depth rarely depended on the sediments non-uniformity in high flow intensity cases. As is shown in Figure 2a, relative scour depths ${d}_{\mathrm{s}}/D$ were almost the same when flow intensity $U/{U}_{c}$ surpassed 4, though the sediment non-uniformity σ

_{g}were different. In non-uniform sediments, flow intensity was determined as the ratio of approach flow velocity U to the armoring velocity ${U}_{a}$. The armoring effects were found to increase with ${\sigma}_{\mathrm{g}}$ by Melville [64]. For this reason, the first threshold peaks (Figure 2a) tended to decrease with the increasing ${\sigma}_{\mathrm{g}}$.

_{50}on scour depths have been developed to a deeper understanding. In early years, many researchers believed that the relative scour depth ${d}_{s}/D$ increased with the increase of relative medium sediments diameter $D/{d}_{50}$ when $D/{d}_{50}$ was smaller than 25–50. When it surpassed 50, ${d}_{s}/D$ would be independent with $D/{d}_{50}$. Nevertheless, with relatively larger flume experiments and long-time scouring, Sheppard et al. [53] put forward a different view for $D/{d}_{50}>25-50$ cases. As is shown in Figure 5, d

_{s}/D decreased with D/d

_{50}increased. Later, a similar trend was found by Lee and Sturm [65]. However, what differs from Sheppard et al. [53] is that when $D/{d}_{50}>300-400$, ${d}_{s}/D$ is independent on it.

#### 2.4. Pile

_{c}affected the relative scour depth ${d}_{s}/D$ significantly. This section focused on the submerged pile characterized in height to diameter ratio ${h}_{c}/D$ and submergence ratio $h/{h}_{c}$. Although, these were not studied as early as those reviewed above, some progressive results were obtained by a few scholars. Table 2 is submerged experimental data collected from the literature. Except for data from Zhao et al. [15], they are all in clear-water scour conditions. To depict the experimental data concisely, tests numbers in Table 2 were numbered with their own tests numbers.

_{c}/D when ${h}_{c}/D$ is smaller than 3–4. Except the diamond data from Zhao et al. [15], which was in live-bed scour condition and fitted with a blue line, there is not a curve reflecting the independence of ${h}_{c}/D$ for ${d}_{s}/D$. Zhao et al. [15] found the maximum scour depth was independent with ${h}_{c}/D$ when ${h}_{c}/D$ was larger than about 2. The vortices shedding behind the pile was weakened when the height of the pile was small. No scour was seen behind the edge of the pile when ${h}_{c}/D$ was smaller than 0.25. With numerical simulations methods, Zhao et al. [15] obtained shear stress around the pile, and found a significant decrease in smaller value of ${h}_{c}/D$.

_{c}and d

_{s}/D for submerged cases are also investigated by a few researchers. According to Hunt et al. [1] and Tsutsui [70], there was a reverse flow and a separation bubble on the top of the pile instead of the anti-clockwise damming in front of the pile when the pile was submerged. The additional water down wash flowed on the side and behind the pile. Consequently, local scour for submerged piles or structures needs to be studied too. However, as far as the authors are aware, local scour researches in submerged vertical piles were only seen in [4,15,67,68,69,71]. Among them, influences of flow depth studies were even fewer. Dey et al. [67] observed that an increase in h/h

_{c}would reduce the block of flow, and thus the horseshoe vortex was weakened in strength and dimension. They also pointed out that the downward pressure gradient for the submerged pile was less than that of the unsubmerged pile. In addition, the interaction between the free surface and the wake region behind the pile was analyzed by Sarkar and Ratha [69]. They found that free surface was not seen to interact with the wake region when 2.3 < h/h

_{c}< 3. However, for h/h

_{c}< 2.3, upper layer mixes with wake and intensity increases with Froude number, resulting in a weakened vortices shedding behind the pile. For experimental data in Figure 8, the relative scour depth d

_{s}/D decreased rapidly with the increasing submergence of the pile h/h

_{c}, especially for clear-water conditions [67,68,69]. It does not seem to stop at some value of h/h

_{c}as data of [15].

#### 2.5. Time

#### 2.5.1. Finite Time

#### 2.5.2. According to Asymptotically Functions

#### 2.5.3. According to Critical Shear Stress

_{c}, above which the sediment could not be mobilized. Scouring would be in an equilibrium state when the sediment in scour hole was larger than the calculated d

_{c}. For non-uniform sediment, the equivalent sediment size of mixing layer d

_{m}was chosen to represent d

_{c}. Nevertheless, the critical shear stress of sediment particles in scour hole is usually measured indirectly in laboratory. An example of bed shear stress associated with turbulence horseshoe vortex was given by Li et al. [21]. With a 2D PIV system, Li et al. [21] measured the time-averaged flow field in approach flow and the instantaneous velocity in front of the pile. Instantaneous bed shear stress was obtained by $\tau =\rho (\upsilon +{\upsilon}_{T})\frac{{u}_{1}}{{y}_{1}}$, with $\upsilon $ is water dynamic viscosity, ${\upsilon}_{T}$ is the eddy viscosity in the approach flow and ${u}_{1}$ and ${y}_{1}$ are the streamwise velocity and vertical position of the first measurement point above the bed. The direct methods need to be helped with numerical simulations. Manes and Brocchini [16] assumed the length scale of large eddies approximates the maximum scour hole depth. Through dimensional analyses and drag force acting on the cylinder, Manes and Brocchini [16] proposed an approach that combined theoretical analyses with empirical considerations in equilibrium scour depth estimation. In clear-water scour condition, the equilibrium scour depth was expressed as ${d}_{se}~(\frac{{U}^{2}}{g})(\frac{\rho}{{\rho}_{s}-\rho}){({C}_{d})}^{2/3}{(\frac{D}{{d}_{50}})}^{2/3}$ (where ${C}_{d}$ is assumed a constant drag coefficient), which contained all the non-dimensional groups identified by [41,71,85]. The dimensionless equation obtained from Figure 4 in [16] can be written as $\mathrm{ln}(\frac{{d}_{se}g}{{U}^{2}})=\frac{2}{3}\mathrm{ln}(\frac{D}{{d}_{50}})+4.42$.

## 3. Empirical Equations

#### 3.1. Exponential Formulas

#### 3.2. Logarithmic Formulas

_{c}are restricted for the range of 0.4–1.0. In addition, the pile shape should be vertical circular piles. Nevertheless, as was stated by Melville [64], D and U must be expressed in a consistent system of units, e.g., meters and meters per second, respectively. The formula for calculating the equilibrium time of scour depth t

_{e}is dimensional inconsistency.

_{d}. Equation (6) is very meaningful, for it could be used for circular and rectangular piles in clear-water scour. Moreover, the sediments non-uniformity was included. As scour depth is an incremental function of time T in Equation (6), scour depth will be infinite when the time is infinite. Fortunately, they obtained Equation (9) for the equilibrium time of scour depth T

_{e}. However, some limitations were given by them too. Firstly, the dimensionless sediments gain sizes D

^{*}should be less than 15. Where ${D}^{*}={({g}^{\prime}/{\upsilon}^{2})}^{1/3}{d}_{50}$, ν is kinematic fluid viscosity. Secondly, approach flow depth h cannot be smaller than 5 cm in a rectangular channel. Thirdly, the ratio of pile width to channel width should be less than 0.05. What is more, the densimetric particle Froude number ${F}_{d}$ is required to be far more larger than the densimetric particle Froude number of scour entrainment ${F}_{d\beta}$. Lastly, Equation (6) could only be used in clear-water scour conditions with a threshold Froude number F

_{t}< 1.2.

_{50}is the sediments median grain size, t is the scour time, $T=t/{t}_{R}$ is the dimensionless time and ${t}_{R}={z}_{R}/[{\sigma}^{1/3}{({g}^{\prime}{d}_{50})}^{1/2}]$ is the relative time.

#### 3.3. Hyperbolic Functions

#### 3.4. Numerical Functions

_{bc}is the critical bed shear stress on flat bed.

_{w}to account for the wave effects.

## 4. Conclusions

- (1)
- A local scour around a vertical pile involves interactions between sediments and flow fields, which is a process with complicated three-dimensional turbulence. Down-flow in front of a pile and the existed incoming boundary layer are essential in forming the horseshoe vortex. Shear stress at the pile edges, which are produced by the concentrated streamlines, will be amplified to a range of 5–10 times compared to that of the approach flow. Due to this amplification, scour was found to start at the pile upstream corners. Interactions between the three-dimensional horseshoe vortex and vortices shedding are responsible for scour behind the pile.
- (2)
- The flow intensity $U/{U}_{c}$ in uniform sediments or $U/{U}_{ca}$ non-uniform sediments connects the approach flow velocity and sediment particles. In clear-water scour conditions, flow intensity is smaller than 1. Maximum scour depths were found at the transition from clear-water scour to live-bed scour conditions. The existing equilibriums of local scour in clear-water scour conditions indicated that shear stress in scour hole was smaller than the critical shear stress of sediment particle. In live-bed scour conditions, due to the supplements of sediments particles from upstream, equilibrium scour depth oscillates near the mean value. Both uniform and non-uniform sediments attained to their second scour depth peaks when the flow intensity was about 4.0.
- (3)
- Flow depth is an important factor in local scour. The unsubmerged vertical piles have been studied far more than that of submerged cases. In unsubmerged conditions, the maximum scour depth increased with the increasing flow depth until to a critical value. A curve (Figure 3a), which enveloped large amounts of experimental data, could give a reference in scour depth design when considering the flow depth effects. Due to the different flow fields, more studies are needed in submerged cases such as caissons, manifolds in offshore engineering.
- (4)
- Sediments parameters of medium size and non-uniformity in local scour were connected with flow intensity. Armoring effects, which decreased scour depth in non-uniform sediments, were found to increase with the increasing sediments gradation. However, reviews and conclusions in this paper were only for non-cohesive sediments.
- (5)
- Scour depth increases with pile width and pile height. In submerged cases, relative scour depth ${d}_{s}/D$ decreases with the increasing submergence ratio $h/{h}_{c}$. For live-bed scour conditions, the relative scour depth tends to be a constant value when the submergence ratio surpasses 2.
- (6)
- Certifications for equilibrium of scour depth used by researchers vary broadly. Categories of finite time, according to asymptotically functions and according to the shear stress in scour hole were classified. Different equations based on these certifications with experimental data have been adopted in literature. Empirical equations were categorized into four types: exponential formula, logarithmic formula, hyperbolic functions and numerical functions.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Brief sketch of the local scour mechanisms (adapted from Roulund et al. [6], with permission from Journal of Fluid Mechanics, 2005). δ is boundary layer thickness, ${x}_{s}$ is the size of horseshoe vortex.

**Figure 2.**Relative local scour depth variations with flow intensity. (

**a**) Reproduced from Melville [44], with permission from Fourth International Conference on Scour and Erosion, 2008. (

**b**) Experimental data from literature.

**Figure 3.**Relationships of water depth with maximum scour depth. (

**a**) Envelope line of maximum scour depth. (

**b**) Trends of the water depth with maximum scour depth.

**Figure 4.**Influence of ripples on maximum scour depth (Adapted from Melville [64], with permission from Journal of Hydraulic Engineering, 1997).

**Figure 5.**Dependence of d

_{s}/D on D/d

_{50}(Adapted from Sheppard et al. [53], with permission from Journal of Hydraulic Engineering, 2004).

**Figure 6.**Influence of pile width and height on scour depth. (

**a**) Influence of pile width on scour depth. (

**b**) Influence of pile height on scour depth (submerged).

**Figure 10.**Evolution of maximum scour depth (Reproduced from Lança et al. [81], with permission from River flow, 2010).

Source | ${\mathit{d}}_{50}\left(\mathbf{mm}\right)$ | ${\mathit{\sigma}}_{\mathit{g}}$ | $\mathit{U}/{\mathit{U}}_{\mathit{c}\mathit{a}}$ | $\mathit{h}/\mathit{D}$ | $\mathit{D}/{\mathit{d}}_{50}$ | t(hour) | ${\mathit{d}}_{\mathit{s}}/\mathit{D}$ |
---|---|---|---|---|---|---|---|

Askoy et al. [43] | 3.47 | 1.4 | 0.48–0.56 | 0.9–4.7 | 12–58 | 6.7 | 0.44–0.90 |

Ettema [45] | 0.84–7.8 | uniform | 0.50–0.95 | 0.2–21 | 13–188 | 9.7–250 | 0.32–2.09 |

Ettema et al. [46] | 1.05 | uniform | 0.80 | 2.5–15.6 | 61–387 | 24–48 | 1.07–1.73 |

Hancu [47] | 2.00 | uniform | 0.74–1.96 | 0.8 | 65 | / | 0.58–2.07 |

Jain and Fischer [48] | 0.25–2.5 | uniform | 0.90–4.22 | 1–4.9 | 20–406 | / | 1.61–2.54 |

Lança et al. [49] | 0.86 | uniform | 0.93–1.04 | 0.5–5 | 58–465 | 168–330 | 0.94–2.32 |

Melville [50] | 0.24–1.4 | 1.22–1.3 | 1.0–5.25 | 1.0–2.0 | 36–423 | / | 0.91–2.10 |

Melville and Chiew [51] | 0.8–0.96 | uniform | 0.4–0.96 | 0.6–12.5 | 18–222 | 3.3–119 | 0.10–2.56 |

Mia and Nago [34] | 1.28 | 1.29 | 0.71–0.82 | 2.7–5 | 47 | 2.3–5 | 1.18–1.77 |

Pandey et al. [52] | 0.4–1.8 | 1.17–1.2 | 0.69–0.87 | 1.4–2.5 | 37–288 | 24 | 1.21–1.60 |

Sheppard et al. [53] | 0.22–2.9 | 1.21–1.51 | 0.75–1.21 | 0.2–11.6 | 314–4136 | 41–616 | 0.76–1.73 |

Sheppard and Miller [35] | 0.27–0.84 | 1.32–1.33 | 0.60–6.10 | 1.3–3.2 | 181–563 | 0.3–332 | 0.72–2.24 |

Shen et al. [54] | 0.24 | uniform | 1.08–4.87 | 0.8–1.0 | 633 | / | 1.23–1.82 |

Yanmaz and Altinbilek [55] | 0.84–1.07 | 1.13–1.28 | 0.44–0.76 | 0.7–3.5 | 44–80 | 3–6 | 0.56–2.66 |

Source | Tests | d_{50} (mm) | D (cm) | h (cm) | h_{c}/D | U/U_{c} | h/h_{c} | t (hour) | d_{s}/D |
---|---|---|---|---|---|---|---|---|---|

Dey et al. [67] | 16–24 | 1.86 | 8 | 25 | 0.38–8.33 | 0.90 | 1.0–8.3 | 48 | 0.79–1.96 |

Euler and Herget. [68] | 12–15 | 0.76 | 3 | 10.1 | 0.33–1.33 | 0.65 | 2.53–10.1 | 22 | 0.07–0.73 |

Euler and Herget. [68] | 20–23 | 0.75 | 3 | 9.7 | 0.33–1.33 | 0.6 | 2.4–9.7 | 24 | 0.07–0.97 |

Sarkar and Ratha [69] | 4, 8, 12, 16, 20 | 0.26 | 11.5 | 20 | 0.43–1.30 | 0.89 | 1.33–4 | 10 | 0.52–0.80 |

Sarkar and Ratha [69] | 41, 45, 49, 53, 57 | 0.52 | 5.4 | 20 | 0.93–2.78 | 0.89 | 1.33–4 | 10 | 0.96–1.35 |

Zhao et al. [15] | 8–14 | 0.40 | 6 | 50 | 0.33–8.33 | 1.02 | 1–25 | 4.5 | 0.48–1.06 |

Zhao et al. [15] | 22–28 | 0.40 | 6 | 50 | 0.33–8.33 | 1.25 | 1–25 | 2.35 | 0.62–1.04 |

**Table 3.**Parameters of Test 2 in Lança et al. [81].

Parameters | d_{50} (mm) | σ_{g} | D (cm) | U (cm/s) | U/U_{c} | h (cm) | t (hour) | d_{s} (cm) |
---|---|---|---|---|---|---|---|---|

Test 2 | 0.86 | 1.40 | 8 | 27.0 | 0.86 | 16 | 1094.5 | 19.55 |

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**MDPI and ACS Style**

Liang, B.; Du, S.; Pan, X.; Zhang, L.
Local Scour for Vertical Piles in Steady Currents: Review of Mechanisms, Influencing Factors and Empirical Equations. *J. Mar. Sci. Eng.* **2020**, *8*, 4.
https://doi.org/10.3390/jmse8010004

**AMA Style**

Liang B, Du S, Pan X, Zhang L.
Local Scour for Vertical Piles in Steady Currents: Review of Mechanisms, Influencing Factors and Empirical Equations. *Journal of Marine Science and Engineering*. 2020; 8(1):4.
https://doi.org/10.3390/jmse8010004

**Chicago/Turabian Style**

Liang, Bingchen, Shengtao Du, Xinying Pan, and Libang Zhang.
2020. "Local Scour for Vertical Piles in Steady Currents: Review of Mechanisms, Influencing Factors and Empirical Equations" *Journal of Marine Science and Engineering* 8, no. 1: 4.
https://doi.org/10.3390/jmse8010004