# Effect of Pressure Fluctuations and Flow Confinement on Shear Stress in Jet-Driven Scour Processes

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{q}, e

_{H}, e

_{g}, e

_{d}and e

_{ρ}are free exponents, K is a coefficient, ρ and ρ

_{s}are water and sediment densities, respectively, q is the unit discharge for 2D case or the total discharge for the 3D counterpart, g is the gravitational acceleration, d is the characteristic sediment diameter (i.e., mean diameter d

_{50}for uniform materials) and H is the drop height. The authors of [5,6] also showed that the variable Δ + D is a monotonic increasing function of q, H and ρ/(ρ

_{s}−ρ), whereas it decreases with d for both 2D and 3D cases. In addition, the variable Δ + D is slightly affected by the non-uniformity coefficient σ = (d

_{84}/d

_{16})

^{1/2}, where d

_{n}is the material size for which n% is finer. Figure 1 shows two pictures taken during an experimental test.

_{s}for the analyzed configuration. Finally, this study also furnishes another unprecedented result, i.e., it corroborates the approach based on the phenomenological theory of turbulence recently developed by [23], showing that it can be successfully applied to estimate maximum shear stress.

## 2. Experimental and Theoretical Background

_{50}= 2.25 mm and density ρ

_{s}= 2214 kg/m

^{3}). Two jet inclinations with respect to the horizontal were tested, i.e., α = 45° and 60°. The ranges of tested parameters are reported in Table 1, including those relative to the discharge Q, water depth D above the original sediment level and average shear stress τ

_{av}. Figure 2 illustrates the main hydraulic and geometric parameters characterizing the scour hole and jet configuration.

_{1}and F

_{2}can be expressed as F

_{1}≈ F

_{2}≈ 0.5ρgD

^{2}B, resulting in F

_{1}− F

_{2}≈ 0 N. As for force resultant F

_{i,i+1}acting on the scour hole profile segment i−i+1, it is reasonable to assume a linear variation of the pressure distribution between two successive sections of measurement (Figure 3b). Consequently, F

_{i,i+1}can be expressed as:

_{i}is the value of the pressure measured at the axial point i. Note that, in this specific case, i is an integer varying between 1 and 8 (number of measurement points in Figure 2). In Equation (2), l

_{i,i+1}indicates the length of the segment i−i+1, whose inclination α

_{i,i+1}with respect to the horizontal was estimated from the plots of dynamic equilibrium axial profiles (Figure 3a). The application of the linear momentum conservation law to the selected CV leads to the following force component balances (Figure 3b), where T

_{v}and T

_{h}represent the vertical and horizontal components of the vector T, whose magnitude is ${\left({\mathrm{T}}_{\mathrm{v}}^{2}+{\mathrm{T}}_{\mathrm{h}}^{2}\right)}^{1/2}$:

_{w}+ ρ

_{s}g V

_{s}is the weight of the CV, with V

_{w}and V

_{s}indicating the volume of water and rotating material in the CV, ρ and ρ

_{s}are the water and sediment densities, g is the gravity acceleration, M

_{1}= ρQV

_{j}and M

_{2}= ρQ

^{2}/(BD) are the momentum fluxes of the jet and of the flow exiting the CV, respectively, and V

_{j}is the jet velocity. The authors of [30] showed that the magnitude of the resultant force T distributed on the scour surface for the segment i−i+1 (T

_{i,i+1}) scales with h

_{Gi,i+1}and x

_{Gi,i+1}as follows:

_{Gi,i+1}and x

_{Gi,i+1}are the vertical and longitudinal coordinates of the center of mass G

_{i,i+1}of each segment i−i+1 (Figure 3a), and the origin O′ of the reference system is located at the intersection of the vertical plane containing the maximum scour depth with the water surface. It is worth mentioning that Equation (5) also reflects the findings of [4,12,16,23,25,27,28,29], who showed that shear stress reduces with the water depth and longitudinal distance from the jet impact zone. Furthermore, [30] showed that T

_{i,i+1}and, consequently, the corresponding shear stress τ

_{i,i+1}acting on the scour profile segment i−i+1 can be computed as follows:

_{i,i+1}is the angle between the direction of the free vector T and the segment i−i+1. Thus, the average shear stress τ

_{av}can be expressed as:

## 3. Estimation of Pressure Fluctuation Effect on Average Shear Stress

_{τav}. For each test, we assume that the sum of W and M

_{1}sin(α) is strictly constant (variance equal to 0) at equilibrium. Notably, W+M

_{1}sin(α) does not vary with time, depending only on the equilibrium morphology, the amount of rotating material and characteristics of the jet, that can be reasonably considered to be constant at equilibrium. Therefore, from Equation (3), we can derive the following Equation (9) valid for dynamic equilibrium condition:

_{v}and F

_{5}, respectively. Experimental evidences show that T

_{h}<< T

_{v}and, consequently, T

_{v}≈ T (see Table 2), resulting in:

_{v}≈ T is also valid for α = 45°.

_{5}, the following equation applies:

_{i,i+1}= 90°−|α

_{i,i+1}| is the inclination of the force F

_{i,i+1}with respect to the horizontal and 0° ≤ β

_{i,i+1}≤ 90°. For measurements taken with piezometers, pressure oscillations cannot be correctly assessed because of the water column inertia. Therefore, in order to give a conservative estimation of the standard deviation of τ

_{av}, we assume that:

^{2}

_{pmax}represent the variances of the pressure measured at the section i and at the section of maximum scour depth by the pressure transducer. According to [4,28,29,33,34,35,36,37,38], the maximum oscillation, with respect to the mean pressure head value, occurs in the jet impingement zone, i.e., in correspondence with the section of maximum scour depth. Re-arranging Equation (11) as follows:

_{1}, A

_{i+1}(for i varying between 1 and 6) and A

_{8}as:

_{i}can be reasonably assumed as independent variables [33], by combining Equations (9), (10), (12) and (17), we obtain:

_{i,i+1}only depends on geometric characteristics of the scour hole profile, that can be reasonably assumed to be constant at equilibrium. Furthermore, Equation (7) may be re-written as:

_{i,i+1}and τ

_{i,i+1}. Again, as D

_{i,i+1}only depends on geometric characteristics of the scour hole profile at equilibrium, we can reasonably assume that it is constant. Thus, considering Equation (8), we finally obtain:

_{τav}and, consequently, the relative error e

_{%}= 100?σ

_{τav}?/τ

_{av}. The methodology illustrated in this section does not depend on sediment bed gradation and scale. In fact, the average shear stress is a result derived from the application of the angular momentum conservation law, which takes into account the torque contributions given by the shear stress distribution and the momentum flux of the impinging jet. The approach proposed by [30] is fully consistent with that based on the phenomenological theory of turbulence developed by [23], which is valid regardless of the gradation of the granular material and scale. However, further investigations are needed to validate its applicability in the generality of the cases.

## 4. Estimation of Maximum Shear Stress

_{j}. Its length, termed J

_{p}by [15], is measured from the jet impingement onto the water surface along the jet inclination, and can be estimated as J

_{p}= C

_{d}

^{2}y

_{0}. C

_{d}is the diffusion constant (assumed equal to 2.6 by [15]) and y

_{0}is the jet thickness assumed to be equal to the nozzle diameter D

_{p}. Beyond this distance, the maximum velocity decays and becomes equal to V along the jet centerline, whereas the entire flow velocity field is characterized by a lower velocity due to the jet diffusion. Therefore, as the maximum shear stress τ

_{m}can be related to the maximum diffused jet velocity V in the plunge pool (see also [16,17]), [15] proposed the following two expressions for τ

_{m}:

_{p}(resulting in a constant τ

_{m}within the potential core), and

_{p}. Equation (26) evidences that τ

_{m}decreases with J, i.e., with the distance along the centerline from water surface impingement to the granular bed. C

_{f}is the friction coefficient and, based on Blasius flow assumption, it can be expressed as follows:

_{m}, calculated using Equation (26), and average shear stress τ

_{av}measured by [30]. τ

_{m}was evaluated by Equation (26), as experimental tests are characterized by J

_{e}> J

_{p}, where J

_{e}is J at equilibrium. The maximum scour depth generally occurs in correspondence with the impingement point of the jet on the granular bed. Thus, for the following comparison, it can be reasonably assumed to be coincident with the intersection of the jet centerline with the scour hole surface, i.e., J

_{e}≈ (Δ + D)/sin(α) (see Figure 2). The mentioned comparison shows that the values of τ

_{m}predicted by [15] are lower than those of τ

_{av}measured by [30]. This confirms the prominent effect of rotating material on shear stress distribution and the effect of flow confinement due to the scour hole configuration.

^{*}= D, where H

^{*}is the impingement height. In particular, they showed that τ

_{m}is much bigger than that predicted by [28] and can be expressed by Equation (28):

_{e}is equal to H

^{*}(i.e., D in Figure 2) for vertical jet impinging on a flat plate. However, in the presence of a scour hole, H

_{e}is equal to J

_{e}, i.e., H

_{e}= Δ + D for the vertical jet (as sin(α) = 1). The authors of [25] found that the shear stress coefficient C

_{s}can be assumed equal to 0.38. In this regard, it is worth noticing that [28] proposed C

_{s}= 0.16 for non-confined jets. The significant difference between C

_{s}values is mainly due to flow recirculation. In fact, flow recirculation within a scour hole is much more prominent than that evidenced by [25] (note the considerable amount of rotating sediment in Figure 1a). Namely, [25] tested confined jets whose confinement ratio (box area to nozzle area) was 13,950. Whereas, tests conducted by [30] are characterized by LB/(πD

_{p}

^{2}/4) ranging between 200 and 450, where L is the scour hole length. This aspect is crucial to understand and explain the mentioned difference. In addition, [7] showed that the maximum scour depth (or, equivalently, Δ + D, D being constant) decreases with the amount of rotating material in the scour hole, and [23] demonstrated that the shear stress scales with 1/(Δ + D). Therefore, apart from flow confinement effect, the presence of rotating material within the scour hole provides a further explanation of the deviation between τ

_{m}values calculated using Equations (26) and (28) and τ

_{av}values measured by [30]

_{s}and τ

_{m}for tests conducted by [30], assuming the longitudinal shear stress distribution proposed by [25]. This last assumption appears to be reasonable considering that hydrodynamics of the phenomenon analyzed by [25] is similar to that occurring in the scour hole. Note that [25] “replicated conditions expected in a jet erosion process”. In so doing, they showed that the shear stress distribution in the radial direction proposed by [47] does not represent the real stresses in a confined environment, and proposed the following empirical TKE-based expression for shear stress distribution:

^{*}≤ 0.38, where r is the radial distance from jet centerline (Figure 2). Whereas, for λ > 0.38, they showed that the shear stress distribution follows that by [27]. Furthermore, they evidenced the similitude in terms of non-dimensional shear stress distribution among the different approaches, including those of [27,29,47]. Such approaches result in non-dimensional curves characterized by different shear stress peaks, but similar non-dimensional ratio τ

_{m}/τ

_{av}. Therefore, it is reasonable to assume a negligible effect of flow confinement on the ratio τ

_{m}/τ

_{av}. Consequently, the shear stress longitudinal distribution τ

_{r}relative to experimental tests performed by [30] was computed by using Equation (29) and assuming H

_{e}= J

_{e}. Due to the (quasi-) longitudinal symmetry of the scour hole, r ranges between 0 and 0.5 L. Figure 5a shows an example of shear stress distribution relative to test 11 of [30], in which τ

_{m}, computed with Equation (29), is equal to 29 N/m

^{2}, and τ

_{av}, estimated by using the mentioned distribution, is 11.8 N/m

^{2}, with r ranging between 0 m and 0.27 m.

_{m}/τ

_{av}≈ 2.5. Based on this deduction, τ

_{m}values of present tests (jets confined by a scour hole with rotating material) were contrasted with those predicted with Equation (28), resulting in C

_{s}= 3.5 (R

^{2}= 0.83), whereas C

_{s}= 0.16 for non-confined jets and C

_{s}= 0.38 for confined jets. Such difference highlights one of the novelties brought by our analysis in the assessment of maximum shear stress for jet-driven scour processes.

_{m}are either empirical or semi-theoretical and do not take into account the granulometric characteristics of bed material. Therefore, their validity could be limited to the tested range of parameters. Vice versa, according to [17], “a relation based on fundamental principles of physics, calibrated by using measured scour data, could overcome this problem.” In this regard, [23] developed the following scaling expression for τ:

_{m}(=2.5τ

_{av}) are contrasted against τ

_{B}values calculated using Equation (30), allowing us to derive the following predicting relation:

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Data availability

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**Figure 2.**Diagram sketch of the experimental apparatus with the indication of the main hydraulic and geometric parameters, including location of piezometers and pressure transducer.

**Figure 3.**Diagram sketch indicating (

**a**) the selected CV, the main geometric characteristics of the scour hole and the scheme of the forces acting on the CV; (

**b**) the polygon of forces acting on the CV along with the components T

_{h}and T

_{v}of the vector T and the force F

_{i,i+1}acting on the scour hole profile segment i−i+1.

**Figure 4.**Comparison between τ

_{m}calculated using Equations (26) and (28) and τ

_{av}derived by [30].

**Figure 6.**Comparison between values of the variable τ

_{B}calculated using Equation (30) and estimated values of maximum shear stress τ

_{m.}

**Table 1.**Range of variability of the main parameters for experimental tests conducted by [30].

Test | τ_{av} (N/m^{2}) | Δ (m) | D (m) | Q (m^{3}/s) | α (°) |
---|---|---|---|---|---|

1–6 | 99–237 | 0.090–0.155 | 0.020–0.150 | 0.00115–0.00165 | 60 |

7–12 | 64–180 | 0.010–0.180 | 0.020–0.150 | 0.00115–0.00165 | 45 |

Test | T_{v} (N) | T (N) |
---|---|---|

1 | 56.4 | 56.7 |

2 | 21.5 | 21.5 |

3 | 16.8 | 16.8 |

4 | 26.1 | 26.1 |

5 | 16.6 | 16.8 |

6 | 17.2 | 17.5 |

7 | 28.5 | 28.5 |

8 | 17.1 | 17.3 |

9 | 12.3 | 12.8 |

10 | 65.7 | 66.1 |

11 | 32.9 | 33.1 |

12 | 12.2 | 15.7 |

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**MDPI and ACS Style**

Pagliara, S.; Palermo, M.
Effect of Pressure Fluctuations and Flow Confinement on Shear Stress in Jet-Driven Scour Processes. *Water* **2020**, *12*, 718.
https://doi.org/10.3390/w12030718

**AMA Style**

Pagliara S, Palermo M.
Effect of Pressure Fluctuations and Flow Confinement on Shear Stress in Jet-Driven Scour Processes. *Water*. 2020; 12(3):718.
https://doi.org/10.3390/w12030718

**Chicago/Turabian Style**

Pagliara, Simone, and Michele Palermo.
2020. "Effect of Pressure Fluctuations and Flow Confinement on Shear Stress in Jet-Driven Scour Processes" *Water* 12, no. 3: 718.
https://doi.org/10.3390/w12030718