# The Phenomenological Theory of Turbulence and the Scour Evolution Downstream of Grade-Control Structures under Steady Discharges

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{c}). Likewise, to model jet driven scour processes, Hoffmans [11,12] applied the linear momentum equation to a selected control volume. Furthermore, in this case, the analytical closure was obtained by introducing a non-dimensional parameter calibrated with experimental data. Therefore, both approaches can be classified as semi-theoretical.

- (1)
- the basic assumptions of the PTT-theory are still valid, notwithstanding the fact that such structures are partially submerged in general.
- (2)
- the scour evolution is consistent with that of plunging jets, unsubmerged vertical weirs, and Piano Key weirs, that are non-linear weirs often used for stream rehabilitation projects [18].
- (3)
- two different phases can be distinguished, i.e., the developing and the developed phases. (It is worth remarking that this result was not obvious a priori. In fact, according to Tregnaghi et al. [19], generally scour processes are characterized by an “affine transformation of a scour hole profile into another” during scour evolution, but this does not imply that a homothetic expansion may occur (geometric similitude). Namely, the affine transformation is a necessary but not sufficient condition for a homothetic expansion of the scour hole.)
- (4)
- (5)
- the scour evolution equation developed by Bombardelli et al. [16] furnishes reasonably good estimations of the scour evolution, without tuning the coefficients and exponents derived for scour processes caused by jets.

## 2. PTT Approach: Literature Background

_{eq}) for both 2D and 3D configurations, respectively:

_{1}and K

_{2}are constants (equal to 0.3 and 0.5, respectively, see [16]), h is the total head energy of the jet, d is the characteristic diameter of the movable bed (assumed to be equal to d

_{50}, with d

_{xx}indicating the material size for which xx% is finer), ρ is the density of water, ρ

_{s}is bed sediment density, g is the gravitational acceleration, and Q is the unit discharge in Equation (1) and the total discharge in Equation (2). In Figure 1, we show a schematic of the scour hole along with the main geometric and hydraulic parameters, with α indicating the impinging angle of the jet on the water surface with respect to a horizontal plane.

_{test}, were varied in the range 45° ≤ α ≤ 90°. The pipe was located either at the center of the channel (full-model arrangement), or close to the channel glassed wall (half-model arrangement). It is worth mentioning that Bombardelli et al. [16] corroborated the findings of Pagliara et al. [20,21], showing that results obtained with the half-model setup are consistent with those of the full-model when an equivalent jet diameter D* = 2

^{0.5}D

_{test}is adopted. (Under such assumption, the mean jet velocity V of the half-model is equal to that pertaining to the full-model, thus resulting in the same densimetric Froude number F

_{dxx}= V/(g’d

_{xx})

^{0.5}, with g’ = g(ρ

_{s}−ρ)/ρ indicating the reduced acceleration due to gravity.) In addition, the non-dimensional time was assumed to be equal to T = t/T

_{r}, where T

_{r}= D

_{test}/(g’d

_{90})

^{0.5}for full-model arrangement, and T

_{r}= D*/(g’d

_{90})

^{0.5}for half-model arrangement [16,19]. According to Oliveto and Hager [22], the reference time T

_{r}accounts for the geometry of the structure, the approaching flow condition, and granulometric characteristics of the bed material. It can be expressed as T

_{r}= L

_{R}/V

_{R}with L

_{R}and V

_{R}= (g’d

_{90})

^{0.5}, indicating the reference length and velocity, respectively.

_{T}) between the two phases occurs for T = T

_{T}= t

_{T}/T

_{r}, estimated as follows [16]:

_{test}(or D/D*) ≤ 12.9; 30° ≤ α ≤ 90°; 13.0 ≤ F

_{d}

_{90}≤ 46.4.

_{1}= m

_{2}= 1.5 are coefficients (Foster et al. [23]), and K

_{3}= 8 and K

_{4}= 80 are two multiplicative constants. It is worth recalling that Bombardelli et al. [16] developed Equations (4) and (5) under the following hypotheses: (1) D is constant during the scour process; (2) Δ/D << 1; (3) the suspended/rotating sediment in the scour hole does not affect the energy transfer of the scour evolution; (4) the scour hole enlarges homothetically during the developed phase; (5) the scour volume is simplified as a cylinder for the 2D case and as a sphere for the 3D counterpart; (6) scalings for the shear stress τ proposed by Bombardelli and Gioia [13,14] and Gioia and Bombardelli [15] for the 2D and 3D cases are valid during the entire evolution process. Note that all the mentioned assumptions fully reflect the physics of the scour process and their validity will be also shown in the following sections. In particular, the hypotheses (3) and (6) were also validated by Palermo et al. [17,24]. Namely, Palermo et al. [24] analyzed the effect of the suspended/rotating material in the scour hole. They conducted specific experimental tests and proposed a novel, independent theoretical approach based on the conservation of angular momentum. Very interestingly, Palermo et al. [24] derived a theoretical expression for the shear stress that is analytically identical to that proposed by Bombardelli and Gioia [13,14], thus providing a further validation of the hypotheses (3) and (6). More recently, Palermo et al. [17] showed that the scalings for the shear stress τ hold true at all times at prototype scales. In addition, they corroborated the findings of Bombardelli et al. [16] at equilibrium, showing that the proposed approach is valid regardless of the bed sediment gradation and is not affected by scale effects. Finally, it is worth remarking that all the hypotheses adopted in our model are valid independently of the sediment bed gradation.

## 3. On the Applicability of the PTT to Grade-Control Structures

_{r}as the non-dimensional time, where D

_{test}(or D* for the half-model arrangement) was used to estimate T

_{r}. Recently, Palermo et al. [17] have analyzed the scour evolution downstream of a PK weir. For such structure typology, Palermo et al. [17] showed that it is appropriate to replace the diameter D

_{test}(or D*) in T

_{r}with the length of the weir crest. In so doing, they certified the validity of Equation (3), evidencing that the transition between the two phases is consistent with that observed in jet-driven scour processes. Based on these findings, we will assume T

_{rB}= B/(g’d

_{90})

^{0.5}, with B denoting the width of the channel (equal to the width of the analyzed grade-control structures). Therefore, we first address the occurrence of the two phases (developing and developed) for scour processes downstream of grade-control structures. Then, we show that an (almost) homothetic enlargement of the scour hole takes place in the developed phase.

#### 3.1. Developing and Developed Phases

_{rB}for selected tests by Ben Meftah and Mossa [3,26]. (Note that data of Ben Meftah and Mossa, [3], refer to scour downstream of the first of three consecutive bed sills.) In order to highlight the transition between the developing and developed phase, in Figure 2a we only included data for t/T

_{rB}< 10,000. Conversely, in Figure 2b, we show the entire scour evolution pertaining to same tests in a semi-logarithmic plot. Ben Meftah and Mossa [26] conducted experimental tests with three different grade-control structures, having the same total drop height. More specifically, the upstream surface slope of all the three structures was kept constant and equal to 1H:1V. Also, three different inclinations of the downstream face were simulated, i.e., vertical, 1H:1V, and a 3H:1V. Tests were undertaken using one uniform bed material and under different hydraulic conditions. A diagram sketch of the tested structures is shown in Figure 2 of [26]. It should be noted that for runs R25, R20 and R3 of Ben Meftah and Mossa [26] shown in Figure 2, the inclination of the downstream surface of the grade-control structure was equal to 18° (i.e., 3H:1V slope), 45° (i.e., 1H:1V slope), and 90° (i.e., vertical face), respectively. In addition, data pertaining to runs R20 and R25 were extracted from Figure 6 of Ben Meftah and Mossa [26]. Thus, consistent with other tests done by the same authors and developed under similar conditions, we assumed that the equilibrium time is equal to 3 h and 14 h for runs R20 and R25.

_{s}and inclinations, respectively. Ben Meftah and Mossa [26] pointed out that data pertaining to phases I and III exhibit two distinct and consistent trends, regardless of the analyzed structure configuration. Conversely, scour evolution pertaining to phase II depends on structure geometry. More specifically, the inclination of the downstream surface of the structure (assumed to be equal to the impingement angle α of the jet on the water surface) affects jet diffusion and flow characteristics in the downstream basin. Notably, experimental evidence allowed Ben Meftah and Mossa [26] to conclude that scour evolution is similar for α ranging between 45° and 90°, whereas scour evolves more slowly for smaller values of the angle α, thus reducing the difference of slopes of data trend pertaining to phase I and III. In particular, they observed that scour depth increases faster with the drop height for vertical structures. (Note that the impact angle of the jet increases with the drop height being water discharge and tailwater depth constant.) These results suggest that the kinetics of scour evolution downstream of grade-control structures mainly depends on the jet impact angle on the water surface. Such conclusion is fully consistent with the findings of Bombardelli et al. [16], who indicated that T

_{T}decreases with α (Equation (3)).

_{T}is also well predicted by Equation (3), as shown by the vertical lines reported in Figure 2a. In this regard, we would like to remark that we calculated T

_{T}= t

_{T}/T

_{rB}assuming that α is equal to the inclination of the downstream structure surface for runs R25 and R20 (i.e., α = 18° and 45°, respectively), whereas we estimated that α = 80° for run R3 relative to a vertical structure. In turn, α < 30° is outside the range of applicability of Equation (3). Nonetheless, experimental evidence seems to confirm that Equation (3) provides reasonable estimations of the parameter T

_{T}, regardless of the structure configuration and the impingement angle α. Likewise, data pertaining to tests with bed sills [3] were also plotted in Figure 2. In this case, the non-dimensional transition time was calculated with Equation (3) assuming α = 0°, as the flow over the bed sill is almost horizontal and the drop height was relatively small. Also in this case, the transition time appears to be in good agreement with the kinetics of the scour evolution.

_{rB}for some selected runs, along with the transition time calculated with Equation (3). In the absence of specific indications by the authors of [1], for these runs we estimated that α ranges between 0° and 10°. Once again, the existence of two distinct phases is confirmed in Figure 3. The predicted transition time (using α equal to 0° and 10° in Equation (3)) seems to be consistent with the kinetics of the scour process, i.e., the transition occurs in correspondence with the change of the slope. More specifically, the evolution of the non-dimensional variable Δ/B in the first instants of scour process (i.e., for t/T

_{rB}< 8000) is shown in Figure 3a. Overall, it is worth noticing that the transition time between the two phases (corresponding to the change of the slope of experimental curves of scour evolution) is well approximated by Equation (3).

_{f}) and impermeable (configuration B

_{f-imp}) upstream filtering layer (see Figure 1 and Table 1 in [7] for diagram sketch and details of tested configurations, respectively). Likewise, Pagliara and Palermo [7] simulated a stepped gabion weir structure, made by different superimposed layers of prismatic gabions. The length and height of the steps of the structure were the same, resulting in a downstream slope of the pseudo-bottom equal to 45°. Four different upstream configurations of the structure were tested, i.e., without and with an upstream filtering layer (configurations GW

_{0}and GW

_{imp}, and GW

_{f}and GW

_{f-imp}, respectively). In Figure 2 of [7], the diagram sketch of the simulated structure is shown and in Table 1 of [7] the different upstream configurations are summarized. Note that Pagliara and Palermo [7] published scour depth data at equilibrium. Therefore, in the following, we present scour evolution data of tests conducted by Pagliara and Palermo [7] with rock, grade-control structures not included in [7]. In Figure 4a, we show the scour evolution of selected tests with rock, grade-control structures (B2 configuration) conducted with similar downstream water levels and different discharges. In Figure 4b, data for stepped gabion weirs are presented (GW

_{f-imp}configuration). For these datasets, by using a camera located in front of the glassed wall of the channel, the authors were able to distinguish the two phases and identify the beginning of the homothetic expansion of the scour hole (empty symbols in Figure 4). Consequently, we tested the predicting capability of Equation (3) by assuming α equal to the inclination of the pseudo-bottom (i.e., α = 45°). As shown in Figure 4, for both structures, Equation (3) provides a reasonably good estimation of the transition time, thus confirming the consistency of the physics of scour evolution.

^{3}/s and straight rock sill is shown in Figure 5b, for which we found that α ≈ 55°. Similarly, for curved rock sills tests, we estimated that α was almost equal to 70°. It is worth remarking that α only depends on the structure configuration, whereas it does not vary significantly with Q in the tested range of parameters. Consequently, we calculated the transition time with Equation (3) assuming α = 55° and α = 70° for straight and curved sills, respectively. Overall, in Figure 5a, we show that Equation (3) provides a reasonably good estimation of the transition between the two phases.

#### 3.2. Homothetic Expansion of the Scour Hole

_{rB}, considering that a homothetic expansion occurs when Δ/L tends to a constant value, with L indicating the scour length. For all the analyzed structures (and consistently with the definitions given in all the studies employed in this paper), the scour length L(t) was extrapolated from available longitudinal profiles at different instants. In other words, L was assumed to be equal to the distance between the points where the longitudinal scour profile (passing through the point of maximum scour depth) intersects the bed level (i.e., distance between origin and end point of the longitudinal profile). For scour downstream of rock, grade-control structures and stepped gabion weirs under steady flow conditions, Pagliara and Palermo [7] showed that the length of the scour increases linearly with its depth at equilibrium, thus corroborating the findings of Pagliara et al. [20,21], who concluded that Δ/L is almost constant during the developed phase. Furthermore, considering the similitude of non-dimensional scour profiles pointed out by Pagliara and Palermo [7], it is reasonable to assume that Δ/L is almost constant during the entire developed phase for rock, grade-control structures and stepped gabion weirs as well.

_{rB}for some selected tests conducted in the mentioned studies. In particular, data pertaining to bed sills are plotted in Figure 6b and labeled according to the studies from which they were taken. Subscripts s1, s2 and s3 indicate the first, second and third sill of the series, respectively. In Figure 6a, a detailed view of the same data for t/T

_{rB}< 12,000 is shown.

_{rB}< 6000 is shown.

_{T}, i.e., a homothetic expansion of the scour hole occurs during the developed phase, regardless of the structure configuration. This represents an unprecedented result and confirms the validity of one of the hypotheses of the PTT-based scour evolution model presented by Bombardelli et al. [16].

## 4. Validation of the PTT-Evolution Model

#### 4.1. Generalities on the PTT-Evolution Model and Its Applicability to Grade-Control Structures

_{in}for t = t

_{in}, where t

_{in}indicates the initial time, and the asymptotic value of the scour depth Δ

_{eq}at equilibrium. For the following analysis, we assumed Δ

_{in}equal to the first experimental value of the scour depth and for t

_{in}the corresponding time. In so doing, we validated Equation (5) for the entire scour evolution, including the developing phase. In addition, we assumed Δ

_{eq}= Δ

_{fin}, with Δ

_{fin}indicating the experimental value of the scour depth at equilibrium.

_{in}as above and Δ

_{eq}= Δ

_{eq}

_{,calc}, with Δ

_{eq}

_{,calc}indicating the scour depth at equilibrium estimated with empirical formulas proposed by the authors of each analyzed study. Numerical solutions were obtained initially by using the coefficients K

_{4}and m

_{2}proposed by Bombardelli et al. [16], and by assuming d = d

_{50}in Equation (5). In the following section, we show the result of such comparisons for selected tests derived from different authors and pertaining to various structures.

_{4}and m

_{2}. More specifically, in the following section, we will also show that by only tuning the coefficient K

_{4}, the deviation between predicted and measured values of the variable Δ decreases. (The value of the coefficient m

_{2}will be kept equal to 1.5, as suggested by Yalin [31] and Julien [32]). However, we chose not to do so to preserve the generality of the model. This behavior can be explained by considering that dΔ/dt increases with K

_{4}(see Equation (5)), i.e., scour evolution becomes slower for smaller values of K

_{4}, especially during the developing phase. As scour in grade-control structures are characterized by a longer duration of the developing phase in comparison with pipe jet flows, because of lower jet flow inclination and velocity, we expect that, by selecting smaller values of K

_{4}, the predicting capability of Equation (5) improves for the analyzed structures.

#### 4.2. Validation with Data of Gaudio and Marion [1]

_{fin}was assumed to be equal to the average value of scour depths after 20 h (as suggested by Gaudio and Marion, [1]). Under such assumptions, we found that the deviation between predicted and measured scour depths is less than 20% (Figure 7b,d). Higher deviations can be observed during the initial phase of the process (i.e., for t < 60,000 s, as shown in Figure 7a,c. However, the reduction of the model performance in the initial phase of the scour evolution should not surprise, as it was rigorously derived for the developed phase.)

_{eq}= Δ

_{eq}

_{,calc}, obtained with Equation (8) of Marion et al. [34] for uniform bed sediment and non-interfering bed sills. In Figure 7a,c, we show that the deviation between experimental data and predictions is generally higher in the initial phase of the process, whereas it reduces during the developed phase (i.e., deviation less than 25%, as shown in Figure 7b,d).

_{4}to 0.3 and assuming Δ

_{eq}= Δ

_{fin}(the dotted line in Figure 7). As mentioned above, the purpose of this comparison is to show that model predictions could be even better by tuning the coefficient K

_{4}.

#### 4.3. Validation with Scour Evolution Data Not Included in Pagliara and Palermo [7] and Data of Palermo and Pagliara [27]

_{eq}= Δ

_{fin}and Δ

_{eq}= Δ

_{eq}

_{,calc}calculated with Equations (3) and (6) of Pagliara and Palermo [7], valid for rock, grade-control structures and stepped gabion weirs, respectively. For straight and curved rock sills, we only contrast numerical solutions of Equation (5) obtained assuming Δ

_{eq}= Δ

_{fin}, as the authors did not provide any specific equation for such structure typology. Finally, for the sake of comparison, we also included the numerical solutions of Equation (5) obtained assuming Δ

_{eq}= Δ

_{fin}and K

_{4}= 0.3 for all the analyzed structures.

_{eq}= Δ

_{eq}

_{,calc}.

#### 4.4. Validation with Data of Lu et al. [25]

_{r}provided by Lu et al. [25] for each run. In addition, the value of Δ

_{eq}

_{,calc}was computed using Equation (2) of Lu et al. [25], in which the maximum scour depth is expressed as a function of the structure height, d

_{50}, the tailwater depth, and a non-dimensional number accounting for the effect of the water depth at the toe of the structure. Such a non-dimensional number is similar to the densimetric Froude number introduced by the authors. In Figure 9, model predictions obtained assuming Δ

_{eq}= Δ

_{eq}

_{,calc}and Δ

_{eq}= Δ

_{fin}are contrasted against data for steady tests SM2R7-6 and SM1R4-5 of Lu et al. [25]. In all cases, the theoretical model predicts data reasonably well, especially for t > 2000 s (deviation less than 20%).

_{4}to 0.3 and assuming Δ

_{eq}= Δ

_{fin}(dotted line in Figure 9).

## 5. Summary and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Gaudio, R.; Marion, A. Time evolution of scouring downstream of bed sills. J. Hydraul. Res.
**2003**, 41, 271–284. [Google Scholar] [CrossRef] - Lenzi, M.A.; Marion, A.; Comiti, F.; Gaudio, R. Local scouring in low and high gradient streams at bed sills. J. Hydraul. Res.
**2002**, 40, 731–739. [Google Scholar] [CrossRef] - Ben Meftah, M.; Mossa, M. Scour holes downstream of bed sills in low-gradient channels. J. Hydraul. Res.
**2006**, 44, 497–509. [Google Scholar] [CrossRef] - Guan, D.; Melville, B.W.; Friedrich, H. Live-bed scour at submerged weirs. J. Hydraul. Eng.
**2015**, 141, 04014071. [Google Scholar] [CrossRef] - Wang, L.; Melville, B.W.; Guan, D.; Whittaker, C.N. Local scour at downstream sloped submerged weirs. J. Hydraul. Eng.
**2018**, 144, 04018044. [Google Scholar] [CrossRef] - Wang, L.; Melville, B.W.; Whittaker, C.N.; Guan, D. Temporal evolution of clear-water scour depth at submerged weirs. J. Hydraul. Eng.
**2020**, 146, 06020001. [Google Scholar] [CrossRef] - Pagliara, S.; Palermo, M. Rock grade control structures and stepped gabion weirs: Scour analysis and flow features. Acta Geophys.
**2013**, 61, 126–150. [Google Scholar] [CrossRef] - Pagliara, S.; Palermo, M.; Kurdistani, S.M.; Hassanabadi, L.S. Erosive and hydrodynamic processes downstream of low-head control structures. J. Appl. Water Eng. Res.
**2015**, 3, 122–131. [Google Scholar] [CrossRef] - Bormann, E.; Julien, P.Y. Scour downstream of grade control structures. J. Hydraul. Eng.
**1991**, 117, 579–594. [Google Scholar] [CrossRef][Green Version] - Beltaos, S.; Rajaratnam, N. Plane turbulent impinging jets. J. Hydraul. Res.
**1973**, 11, 29–59. [Google Scholar] [CrossRef] - Hoffmans, G.J.C.M. Jet scour in equilibrium phase. J. Hydraul. Eng.
**1998**, 124, 430–437. [Google Scholar] [CrossRef] - Hoffmans, G.J.C.M. Closure problem to jet scour. J. Hydraul. Res.
**2009**, 47, 100–109. [Google Scholar] [CrossRef] - Bombardelli, F.A.; Gioia, G. Towards a theoretical model for scour phenomena. In River, Coastal, and Estuarine Morphodynamics, Proceedings of the RCEM 2005, 4th IAHR Symposium on River, Coastal and Estuarine Morphodynamics, Urbana, IL, USA, 4–7 October 2005; Parker, G., Garcìa, M., Eds.; CRC Press: Boca Raton, FL, USA, 2005; Volume 2, pp. 931–936. [Google Scholar]
- Bombardelli, F.A.; Gioia, G. Scouring of granular beds by jet-driven axisymmetric turbulent cauldrons. Phys. Fluids
**2006**, 18, 088101. [Google Scholar] [CrossRef][Green Version] - Gioia, G.; Bombardelli, F.A. Localized turbulent flows on scouring granular beds. Phys. Rev. Lett.
**2005**, 95, 014501. [Google Scholar] [CrossRef][Green Version] - Bombardelli, F.A.; Palermo, M.; Pagliara, S. Temporal evolution of jet induced scour depth in cohesionless granular beds and the phenomenological theory of turbulence. Phys. Fluids
**2018**, 30, 085109. [Google Scholar] [CrossRef] - Palermo, M.; Bombardelli, F.A.; Pagliara, S.; Kuroiwa, J. Time-dependent scour processes on granular beds at large scale. Environ. Fluid Mech.
**2021**, 21, 791–816. [Google Scholar] [CrossRef] - Palermo, M.; Crookston, B.; Pagliara, S. Analysis of Equilibrium Morphologies Downstream of a PK Weir Structure. Proccedings of the World Environmental and Water Resources Congress 2020, Henderson, NV, USA, 17–21 2020; pp. 43–51. [Google Scholar] [CrossRef]
- Tregnaghi, M.; Marion, A.; Gaudio, R. Affinity and similarity of local scour holes at bed sills. Water Resour. Res.
**2007**, 43, W11417. [Google Scholar] [CrossRef] - Pagliara, S.; Hager, W.H.; Unger, J. Temporal evolution of plunge pool scour. J. Hydraul. Eng.
**2008**, 134, 1630–1638. [Google Scholar] [CrossRef][Green Version] - Pagliara, S.; Amidei, M.; Hager, W.H. Hydraulics of 3D plunge pool scour. J. Hydraul. Eng.
**2008**, 134, 1275–1284. [Google Scholar] [CrossRef] - Oliveto, G.; Hager, W.H. Temporal Evolution of Clear-Water Pier and Abutments Scour. J. Hydraul. Eng.
**2002**, 128, 811–820. [Google Scholar] [CrossRef] - Foster, G.; Meyer, L.; Onstad, C. An erosion equation derived from basic erosion principles. Trans. ASAE
**1977**, 20, 678–682. [Google Scholar] [CrossRef] - Palermo, M.; Pagliara, S.; Bombardelli, F. Theoretical Approach for Shear-Stress Estimation at 2D Equilibrium Scour Holes in Granular Material due to Subvertical Plunging Jets. J. Hydraul. Eng.
**2020**, 146. [Google Scholar] [CrossRef] - Lu, J.-Y.; Hong, J.-H.; Chang, K.-P.; Lu, T.-F. Evolution of scouring process downstream of grade-control structures under steady and unsteady flows. Hydrol. Process.
**2013**, 27, 2699–2709. [Google Scholar] [CrossRef] - Ben Meftah, M.; Mossa, M. New Approach to Predicting Local Scour Downstream of Grade-Control Structure. J. Hydraul. Eng.
**2020**, 146, 04019058. [Google Scholar] [CrossRef] - Palermo, M.; Pagliara, S. Effect of unsteady flow conditions on scour features at low-head hydraulic structures. J. Hydro-Environ. Res.
**2018**, 19, 168–178. [Google Scholar] [CrossRef] - Pagliara, S. Influence of sediment gradation on scour downstream of block ramps. J. Hydraul. Eng.
**2007**, 133, 1241–1248. [Google Scholar] [CrossRef] - Guan, D.; Melville, B.W.; Friedrich, H. Local scour at submerged weirs in sand bed channels. J. Hydraul. Res.
**2016**, 54, 172–184. [Google Scholar] [CrossRef] - Ben Meftah, M.; De Serio, F.; De Padova, D.; Mossa, M. Hydrodynamic structure with scour hole downstream of bed sills. Water
**2020**, 12, 186. [Google Scholar] [CrossRef][Green Version] - Yalin, M.S. The Mechanics of Sediment Transport; Pergamon Press: Oxford, UK, 1977. [Google Scholar]
- Julien, P.Y. Erosion and Sedimentation, 2nd ed.; Cambridge University Press: Cambridge, UK, 2010. [Google Scholar]
- Gaudio, R.; Marion, A.; Bovolin, V. Morphological effects of bed sills in degrading rivers. J. Hydraul. Res.
**2000**, 38, 89–96. [Google Scholar] [CrossRef] - Marion, A.; Tregnaghi, M.; Tait, S.J. Sediment supply and local scouring at bed sills in high-gradient streams. Water Resour. Res.
**2006**, 42, W06416. [Google Scholar] [CrossRef]

**Figure 1.**Diagram sketch of scour hole evolution, along with the main hydraulic and geometric parameters, with α indicating the impinging angle of the jet on the water surface with respect to a horizontal plane.

**Figure 2.**Δ/B versus t/T

_{rB}for (

**a**) t/T

_{rB}< 10,000 and (

**b**) total test duration for tests T07, T08, T10 and T13 of Ben Meftah and Mossa [3] and tests R25, R20 and R3 of Ben Meftha and Mossa [26], along with the transition times calculated using Equation (3). Tests of Ben Meftah and Mossa [3] pertain to bed sills (for which we adopted α = 0°). Tests of Ben Meftah and Mossa [26], pertain to grade-control structures with inclined downstream surface for which we adopted α = 18° (Run R25), 45° (R20), and 80° (R3).

**Figure 3.**Δ/B versus t/T

_{rB}for Test 1 (Q = 0.0122 m

^{3}/s), Test 7 (Q = 0.0179 m

^{3}/s), and Test 9 (Q = 0.0144 m

^{3}/s) of Gaudio and Marion [1] pertaining to bed sills (for which we estimated 0° < α < 10°), along with the vertical lines indicating the transition times calculated using Equation (3) with α = 0° and α = 10° for (

**a**) t/T

_{rB}< 8000 (detailed view) and (

**b**) total test duration.

**Figure 4.**Δ/B versus t/T

_{rB}for (

**a**) rock, grade-control structures (data not included in Pagliara and Palermo [7]) and (

**b**) stepped gabion weirs tests (Palermo and Pagliara [27]), along with the vertical line indicating the transition time calculated using Equation (3) with α = 45°. Filled symbols pertain to the developing phase, whereas empty symbols belong to the developed phase.

**Figure 5.**(

**a**) Δ/B versus t/T

_{rB}for straight and curved rock sills (Palermo and Pagliara, [27]), along with vertical lines indicating the transition time calculated using Equation (3) with α = 55° and 70°. (

**b**) Picture of a test conducted with a straight rock sill (Q = 0.010 m

^{3}/s).

**Figure 6.**Δ/L versus t/T

_{rB}for selected tests pertaining to bed sills (Test 1 of Gaudio and Marion [1] and tests T10-s1, T10-s2 and T10-s3 of Ben Meftah and Mossa [3]): for (

**a**) t/T

_{rB}< 12,000 (detailed view) and (

**b**) total test duration; Δ/L versus t/T

_{rB}for selected tests pertaining to grade-control structures (tests R3, R19 and R26 of Ben Meftah and Mossa, [26]): for (

**c**) t/T

_{rB}< 6000 (detailed view) and (

**d**) total test duration.

**Figure 7.**Comparison of numerical solutions of Equation (5) with experimental data pertaining to Test 1 (

**a**) for t < 60,000 s (detailed view) and (

**b**) total test duration, and Test 9 for (

**c**) t < 60,000 s (detailed view) and (

**d**) total test duration (data from Gaudio and Marion [1]). Numerical solutions were obtained assuming Δ

_{eq}= Δ

_{fin}(dashed line), Δ

_{eq}= Δ

_{eq,calc}(continuous line), and Δ

_{eq}= Δ

_{fin}and K

_{4}= 0.3 (dotted line).

**Figure 8.**Comparison of numerical solutions of Equation (5) with experimental data pertaining to: (

**a**) rock, grade-control structures (data not included in Pagliara and Palermo [7]); (

**b**) stepped gabion weirs (data from Palermo and Pagliara [27]); (

**c**) straight and curved rock sills (data from Palermo and Pagliara [27]). Numerical solutions were obtained assuming Δ

_{eq}= Δ

_{fin}(dashed line), Δ

_{eq}= Δ

_{eq,calc}(continuous line), and Δ

_{eq}= Δ

_{fin}and K

_{4}= 0.3 (dotted line).

**Figure 9.**Comparison of numerical solutions of Equation (5) with experimental data pertaining to: (

**a**) test SM2R7-6 and (

**b**) test SM1R4-5 of Lu et al. [25]. Numerical solutions were obtained assuming Δ

_{eq}= Δ

_{fin}(dashed line), Δ

_{eq}= Δ

_{eq,calc}(continuous line), Δ

_{eq}= Δ

_{fin}and K

_{4}= 0.3 (dotted line).

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Di Nardi, J.; Palermo, M.; Bombardelli, F.A.; Pagliara, S. The Phenomenological Theory of Turbulence and the Scour Evolution Downstream of Grade-Control Structures under Steady Discharges. *Water* **2021**, *13*, 2359.
https://doi.org/10.3390/w13172359

**AMA Style**

Di Nardi J, Palermo M, Bombardelli FA, Pagliara S. The Phenomenological Theory of Turbulence and the Scour Evolution Downstream of Grade-Control Structures under Steady Discharges. *Water*. 2021; 13(17):2359.
https://doi.org/10.3390/w13172359

**Chicago/Turabian Style**

Di Nardi, Jessica, Michele Palermo, Fabián A. Bombardelli, and Stefano Pagliara. 2021. "The Phenomenological Theory of Turbulence and the Scour Evolution Downstream of Grade-Control Structures under Steady Discharges" *Water* 13, no. 17: 2359.
https://doi.org/10.3390/w13172359