# Plain Stilling Basin Performance below 30° and 50° Inclined Smooth and Stepped Chutes

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Physical Model

#### 2.2. Instrumentation

_{R}≤ 1.5, where L

_{R}is the roller length derived from the flow depth measurements (Section 4.1.2). At each profile, the measurements were conducted from 0.006 m distance to the bottom up to the free surface. The sampling duration was 60 s per point at a frequency of 30 Hz.

## 3. Test Program and Inflow Conditions

_{1}= (1 − C

_{1})y

_{90}, (2) the inflow mean water velocity V

_{1}= q/h

_{1}, (3) the inflow Froude number F

_{1}= V

_{1}/(gh

_{1})

^{0.5}, (4) the inflow Reynolds number R

_{1}= q/ν, (5) the inflow Weber number W

_{1}= (ρV

_{1}

^{2}h

_{1})/σ, (6) critical flow depth h

_{c =}(q

^{2}/h

_{1})

^{1/3}, and (7) kinetic energy correction coefficient α, with C

_{1}as the mean (depth-averaged) air concentration at the inflow section as [15,16]:

_{90}is the characteristic flow depth defined up to y(C = 0.90), g as gravitational acceleration, ν as the kinematic viscosity of water, ρ as the water density and σ as the air-water surface tension.

^{2}/s. To assess the effect of approach flow aeration on the stilling basin performance, air concentrations C

_{1}at the inflow section were varied between 0.15 ≤ C

_{1}≤0.37. The lowest values of C

_{1}≈ 0.16 (Runs 6–9 and 22–24, Table 1) practically correspond to black-water at the stilling basin entrance. The latter was achieved for a smooth chute without bottom roughness or flow pre-aeration. Self-aeration of the flow upstream of the stilling basin entrance was provoked by roughening the bottom, resulting in C

_{1}≈ 0.26 (φ = 30°, Runs 4–6, Table 1) and C

_{1}≈ 0.28 (φ = 50°, Runs 19–21, Table 1). The highest approach flow aeration of C

_{1}≈ 0.32 (φ = 30°, Runs 1–3, Table 1) and C

_{1}≈ 0.36 (φ = 50°, Runs 16–18, Table 1) was achieved with a roughened smooth chute bottom combined with the jet-box pre-aeration. Detailed analysis of 30° smooth chute inflows is given in [5]. The air concentration profiles measured at the inflow section for 50° smooth chutes are shown in Figure 3a. They are compared with the advective diffusion model [17] and, similarly to 30° smooth chutes [5], showed a good agreement. The dimensionless velocity V/V

_{90}profiles, with V

_{90}as the velocity at y(C = 0.90), measured at the inflow section for 50° smooth chutes are shown in Figure 3b. They are approximated by:

^{2}= 0.92. The velocity profiles were almost similar to those measured for the 30° smooth chute inflows [5], as indicated by the similar exponent (N = 12.6 for the 30° smooth chutes).

^{2}/s. They corresponded to skimming flow with 2.7 ≤ h

_{c}/s ≤ 7.94. Quasi-uniform approach flow conditions were attained at the 30° stepped chute end with C

_{1}≈ 0.41 [5]. Detailed analysis of 30° stepped chute approach flows is given in [5]. The air concentration profiles measured at the inflow section for 50° stepped chutes are shown in Figure 4a. The measured range of C

_{1}and the shape of the air concentration profiles for different tests at the inflow section (Figure 4a) suggest that quasi-uniform flow was not fully attained at the stepped chute end for all 50° stepped chutes tests. This is shown in Figure 5, where measured values of C

_{1}are compared to the quasi-uniform mean air concentration values C

_{u}of [18] for the same chute slope and range of relative critical depths. For lower relative critical depths, i.e., h

_{c}/s = 2.71 and 3.36 (Runs 25 and 26, Table 1), the flow conditions at the chute end were practically quasi-uniform with C

_{1}/C

_{u}≈ 0.95 (Figure 5). On the other hand, for higher relative critical depths, i.e., for 3.97 ≤ h

_{c}/s ≤ 7.94 (Runs 27–30, Table 1), gradually varied flow conditions were attained. The advection diffusion model [17] well described the measured air concentration profiles, excepted close to the pseudo-bottom (Figure 4a). The dimensionless velocity V/V

_{90}profiles at the inflow section (Figure 4b) follow Equation (2) with N = 4.9 (R

^{2}= 0.87). Similarly to the results of other studies on steep chutes (e.g., [18,19]), a lower N was obtained as compared to 30° stepped chutes (N = 5.5 [5]).

## 4. Results and Discussion

#### 4.1. Free Surface Characteristics

#### 4.1.1. Mean Flow Depths

_{1}. The mean flow depths streamwise increased reaching a maximum in the “boiling” zone, herein defined as the roller length L

_{R,η}(Figure 6a). Downstream of the roller end, i.e., at x > L

_{R,η}, the flow depths decreased reaching a quasi-constant tailwater depth h

_{2}. As one may expect, higher approach flow Froude numbers F

_{1}at the 50° smooth and stepped chute ends (as compared to 30° chutes) resulted in higher mean flow depths η and longer jump rollers L

_{R,η}for quasi-similar unit discharge q.

_{2}/h

_{1}is plotted against F

_{1}, including the values for 30° and 50° chutes. They are compared to the basic solution of momentum conservation equation for classical hydraulic jumps:

_{1}, the sequent depth ratio is practically independent of the approach flow conditions (except F

_{1}) or chute slope φ. These results further strengthen the conclusions [5] that using equivalent clear-water parameters at the chute end lead to a fairly accurate prediction of the sequent depth ratio using the classical momentum principle. As mentioned in [5], these results indicate that Equation (3) is applicable even if the flow enters at a significant angle with the horizontal, which would not be expected a priori, because Equation (3) was derived for horizontal approach flows. Nevertheless, these results are in line with observations [20] for smooth chute inflows.

_{1})/(h

_{2}− h

_{1}) along the jump roller for 50° smooth and stepped chute approach flows are shown in Figure 7a,b, and compared to dimensionless flow depth data obtained with 30° smooth and stepped chute inflows. It can be seen that no major effect of the chute slope φ on the development of the mean flow depths Z occurred. Similarly to the 30° chutes [5], the dimensionless mean flow depths for 50° chutes are found to be higher below stepped chutes within the first half of the roller, i.e., x/L

_{R,η}≤ 0.5, as compared to smooth chutes, which is attributed to the relatively higher approach flow depth after stepped chutes. As such, the mean flow depths over the jump roller can be described with the self-similar function [5]:

#### 4.1.2. Roller Length

_{R}/h

_{2}for 50° (black symbols) smooth (Runs 16–24) and stepped (Runs 25–30) chute inflows, obtained from flow depth measurements L

_{R,η}and visual observations L

_{R,D}, are plotted against the F

_{1}in Figure 8a and compared to the dimensionless roller lengths obtained with 30° (gray symbols) smooth (Runs 1–9) and stepped (Runs 10–15) chute inflows. One can notice that the dimensionless roller lengths L

_{R,η}/h

_{2}(full symbols) are practically independent of the approach flow conditions or chute slope φ with a typical value of L

_{R,η}≈ 5.0h

_{2}. Similarly to the conclusions made in [5], the visually observed roller lengths (open symbols) are consistently shorter, with typical values of L

_{R,η}≈ 4.6h

_{2}.

_{R,η}/h

_{1}and compared with the roller length prediction developed by [5]:

_{R,η}and visually observed L

_{R,D}roller lengths, respectively. As can be noticed, the agreement is good. The resulting coefficient of determination for measured L

_{R,η}and visually observed L

_{R,D}roller lengths are R

^{2}= 0.92 and R

^{2}= 0.99, respectively.

#### 4.1.3. Flow Depth Fluctuations and Jump Length

_{1}. Irrespective of the chute slope, the surface fluctuations along the hydraulic jump showed a monotonic decrease in the streamwise direction. As expected, higher approach F

_{1}at the 50° chute end, as compared to 30° chutes, resulted in higher flow depth fluctuations.

_{J,η’}was deduced using the criteria introduced in [5], namely as the distance from the jump toe to the section where the surface fluctuations η’ are 1.1 times those measured in the tailwater zone. The resulting dimensionless jump lengths L

_{J,η’}/h

_{2}downstream of 50° smooth and stepped chutes are plotted in Figure 9b against F

_{1}, and compared with dimensionless jump length values obtained downstream of 30° smooth and stepped chutes and jump length prediction [20]. The jump lengths after smooth chutes were between 5.5 ≤ L

_{J,η’}/h

_{2}≤ 6.1, with an overall average value of L

_{J}= 5.8h

_{2}, in agreement with the recommendations of [20]. The hydraulic jumps occurring after stepped chutes consistently required an increased flow length x/h

_{2}, namely 6.5 ≤ L

_{J,η’}/h

_{2}≤ 7.0 with an overall average value of L

_{J,η’}/h

_{2}= 6.7h

_{2}. The chute slope had no major effect on the dimensionless jump lengths L

_{J,η’}/h

_{2}. These results further support the conclusion [5] that hydraulic jumps formed below stepped chutes required a longer normalized distance x/h

_{2}than those formed below smooth chutes.

_{H}′ = η′/H

_{K}with H

_{K}= αV

_{1}

^{2}/(2 g), plotted against the normalized streamwise coordinate x/L

_{J,η’}for all tests. The maximum fluctuation coefficients were observed in the vicinity of the jump toe, caused by the intense splashing [5]. Further downstream, the surface fluctuation coefficients tended to rapidly reduce, attaining quasi-constant values of C

_{H}’ ≈ 0.01 in the tailwater zone. No major effect of approach flow conditions or chute slope φ on the streamwise development of C

_{H}’ occurred. These results suggest that the surface fluctuations were mainly governed by the approach flow kinetic energy H

_{K}. As such the streamwise development of C

_{H}’ for 50° chutes can be described using the same equations developed for 30° chute inflows [5]:

^{2}= 0.57 and 0.97 for Equations (6) and (7), respectively.

#### 4.2. Bottom Pressure Characteristics

#### 4.2.1. Streamwise Pressure Distribution and Jump Length

_{m}, (2) fluctuating pressure characterized by standard deviation p’, (3) extreme maximum pressure p

_{max}and corresponding 99.9th percentile p

_{99.9}, (4) extreme minimum pressure p

_{min}and corresponding 0.1th percentile p

_{0.1}, (5) skewness as $S={{\displaystyle \sum \left({p}_{i}-{p}_{m}\right)}}^{3}/\left(n{\left(p\prime \right)}^{-3}\right)$ where p

_{i}is the pressure at a given instant and n is the number of samples, and (6) excess kurtosis defined as $K={{\displaystyle \sum \left({p}_{i}-{p}_{m}\right)}}^{4}/\left(n{\left(p\prime \right)}^{-4}\right)-3$ A typical streamwise bottom pressure development for 50° chutes is shown in Figure 11 (Run 22, smooth chute).

_{J}with respect to bottom pressures L

_{J,p}

_{′}and L

_{J,SK}were derived, namely as: (1) L

_{J,p}

_{′}distance from the jump toe to the section where pressure fluctuations p′ are 1.1 times those measured in the tailwater zone, and (2) L

_{J,SK}as a distance from the jump toe to the section where the pressure distribution followed a normal probability density function with skewness S and excess kurtosis K tending to zero (Figure 11b). The resulting jump lengths are plotted as L

_{J}/h

_{2}in Figure 12 against F

_{1}, along with jump lengths obtained with flow depth measurements L

_{J,η′}/h

_{2}(Figure 9). The pressure measurements indicated a similar range of dimensionless jump lengths as the flow depth measurements. The overall average values for smooth and stepped chutes were L

_{J}/h

_{2}= 5.8h

_{2}and L

_{J}/h

_{2}= 6.7h

_{2}, respectively. Moreover, if comparing the dimensionless jump lengths downstream of 30° and 50° chutes (Figure 12) it became noticeable that no major effect of the chute slope φ on the dimensionless jump length occurred. These results show that hydraulic jumps initiated below stepped chutes required an increased length x/h

_{2}, as compared to those below smooth chutes.

_{m}= (p

_{m}− h

_{1})/(h

_{2}− h

_{1}), (2) fluctuation coefficient C

_{P}’ = p’H

_{k}

^{−1}, (3) extreme maximum coefficient C

_{P}

^{max}= (p

_{max}− p

_{m})H

_{k}

^{−1}, (4), 99.9th percentile coefficient C

_{P}

^{99.9}= (p

_{99.9}− p

_{m})H

_{k}

^{−1}, (5) extreme minimum coefficient C

_{P}

^{min}= (p

_{m}− p

_{min})H

_{k}

^{−1}and (6) 0.1th percentile coefficient C

_{P}

^{0.1}= (p

_{m}− p

_{0.1})H

_{k}

^{−1}. In Figure 13a–h, the streamwise development of bottom pressure coefficients P

_{m}, C

_{P}’, C

_{P}

^{max}, C

_{P}

^{99.9}, C

_{P}

^{min}, C

_{P}

^{0.1}, S and K downstream of 50° smooth and stepped chutes are plotted against the normalized streamwise coordinate X

_{J}= x/L

_{J}. The latter bottom pressure coefficient developments for 30° chutes can be found in [5]. In following sub-chapters, the effect of chute slope on the streamwise development of bottom pressure coefficients is detailed.

#### 4.2.2. Streamwise Distribution of Mean Pressure

_{m}downstream of the 50° chutes, similarly to the 30° chutes [5], indicated the following flow zones (Figure 13a):

- deflection zone along 0 ≤ X
_{J}≤ 0.18, characterized by increased mean pressures due to the impact and flow curvature - transition zone along 0.18 < X
_{J}< 1, where mean pressures qualitatively follow the flow depths, and - tailwater zone along X
_{J}≥ 1, where mean pressures are quasi-hydrostatic.

_{m}for 30° and 50° smooth and stepped chutes are compared in Figure 14a,b, respectively.

_{J}≈ 0, up to 60% higher dimensionless mean pressures P

_{m}are observed downstream of 50° smooth chutes, as compared to those downstream of 30° smooth chutes.

_{J}≤ 0.18, the mean pressure with 50° smooth chutes decreases in a similar manner as on 30° smooth chutes, however, with slightly higher magnitudes caused by stronger flow curvature due to the more abrupt slope change. The local minimum below 50° smooth chutes is observed at X

_{J}≈ 0.18, where flow curvature greatly reduced, and beyond which the dimensionless mean pressure magnitudes coincide with those of 30° smooth chutes along the remaining part of the stilling basin, i.e., in the transition and tailwater zone. The influence reach of the flow curvature was thus somewhat longer below 50° smooth chutes, i.e., 0 ≤ X

_{J}≤ 0.18, as compared to those after 30° smooth chutes, i.e., 0 ≤ X

_{J}≤0.15 [5].

_{J}≈ 0.04, as compared to 30° stepped chute inflows at X

_{J}≈ 0. The maximum P

_{m}magnitudes downstream of 50° stepped chutes were up to 60% higher than maximum P

_{m}magnitudes observed downstream of 30° stepped chutes or two times higher for the similar streamwise position, i.e., X

_{J}≈ 0.04.

_{J}≈ 0. For the 50° chutes, only a small portion of the flow impacted near the jump toe (i.e., fictitious step edge) resulting in the more concentrated impact further downstream, i.e., at X

_{J}≈ 0.04. Downstream of the flow deflection point, within 0.04 < X

_{J}≤ 0.18, the mean pressures below 50° stepped chutes decreased in a similar manner as below 30° chutes, however with higher magnitudes caused by stronger flow curvature (Figure 14b). The mean pressure coefficients reached a local minimum at X

_{J}≈ 0.18, after which they coincide with 30° stepped chute inflows along the entire remaining stilling basin reach.

_{J}≈ 0 for smooth and X

_{J}≈ 0.04 for stepped chutes), one can notice up to 2.5 times higher magnitudes below smooth chute as compared to those below stepped chute (Figure 13a). As the mean pressure at this point was mainly governed by the approach flow kinetic energy [5], the mean pressure at the flow deflection point p

_{def}was normalized with the approach flow kinetic energy H

_{K}. The resulting mean pressure coefficients C

_{P}

^{def}= p

_{def}/H

_{K}are shown in Figure 15, along with C

_{P}

^{def}values for 30° chute. In case of the 50° chute, the mean pressures at the flow deflection point were, on average, 41% and 58% of the corresponding approach flow kinetic energy for smooth and stepped chutes. As detailed in [5], the higher magnitudes downstream of smooth chutes were caused by its lower approach flow depth, resulting in a more concentrated impact. Furthermore, the pressure coefficients C

_{P}

^{def}are approximately 20% higher below 50° smooth and stepped chutes, as compared to the corresponding ones for 30° chutes (Figure 15). This is clearly caused by the “sharper” angle of the flow impact.

_{m}magnitudes within 0.05 ≤ X

_{J}< 0.12, as compared to 50° smooth chute (Figure 13a). Further downstream, i.e., X

_{J}≥ 0.12, the mean pressure coefficients were practically independent of the approach flow conditions. The pronounced mean pressures in the flow deflection zone below 50° stepped (R

^{2}= 0.97) and smooth (R

^{2}= 0.95) chutes can be described as (Figure 13a):

_{1}on the streamwise mean pressure distribution occurred.

#### 4.2.3. Streamwise Distribution of the Pressure Fluctuation

_{P}’ development (e.g., [21,22,23]). The pressure fluctuations increased downstream of the jump toe reaching maximum values of C

_{P}’ ≈ 0.05 at X

_{J}≈ 0.12. Further downstream, they monotonically decreased towards quasi-constant tailwater magnitudes. Due to the flow deviation at the jump toe (X

_{J}= 0), the pressure fluctuations reached or exceed the magnitudes observed at X

_{J}≈ 0.12. No major effect of approach flow aeration C

_{1}on the streamwise development of C

_{P}’ occurred.

_{P}’ for 50° and 30° smooth chutes are compared in Figure 16a. No considerable effect of the chute slope φ on the pressure fluctuation development occurred.

_{P}’ = 0.09 (Figure 17). Downstream of the jump toe, the fluctuation magnitudes sharply increased reaching maximum values of C

_{P}’ = 0.15 at X

_{J}≈ 0.04. These values are up to 3 times higher as compared to the maximum values observed for smooth chute approach flows (Figure 13b and Figure 17). To further illustrate the severeness of pressure fluctuations downstream of stepped chute at this location, the absolute values of the pressure fluctuations p′ are compared for smooth (Run 24) and stepped chute (Run 30) inflows in Figure 18 with similar unit discharge of q ≈ 0.360 m

^{2}/s. Despite the significantly higher inflow Froude number F

_{1}for the smooth chute inflow (F

_{1}≈ 14), as compared to stepped chute inflow (F

_{1}≈ 8), the absolute values of pressure fluctuations p′ downstream of stepped chute were about 50% higher at the flow deflection point, as compared to the maximum values observed with smooth chute approach flow (i.e., in the zone of maximum pressure fluctuations at X

_{J}≈ 0.12). This shows that pronounced pressure fluctuations downstream of stepped chute are caused by the higher turbulence levels of the approaching flow, as compared to smooth chute approach flows. Downstream of the flow deflection point, i.e., X

_{J}> 0.04, the pressure fluctuation coefficients monotonically decreased up to X

_{J}≈ 0.13, after which they coincided with smooth chute approach flows magnitudes over the entire stilling basin reach (Figure 13b). No considerable effect of step size s occurred.

_{P}’ for 50° and 30° stepped chute approach flows are compared in Figure 16b. A major effect of the chute slope could be noticed. As a result of the relative downstream shift of the flow deflection point, the maximum pressure fluctuations for 50° stepped chutes occurred further downstream (X

_{J}≈ 0.04), as compared to the 30° chutes (X

_{J}≈ 0). At this point, up to 2 times higher C

_{P}’ magnitudes were observed, as compared to those of 30° stepped chute at the flow deflection point (X

_{J}≈ 0) (Figure 16b and Figure 17). Apart from the flow deflection point, the 50° stepped chute tended to increase the pressure fluctuations at the jump toe as well, with up to 30% higher C

_{P}’ magnitudes (Figure 16b and Figure 17). Downstream of the flow deflection point (X

_{J}≈ 0.04), as previously mentioned, the pressure fluctuations monotonically decreased and reached the 30° chute magnitudes at X

_{J}≈ 0.13, beyond which they coincided along the remaining downstream reach of the stilling basin.

_{P}’ and surface fluctuation C

_{H}′ are also compared in Figure 13b (only Equations (6) and (7) are included for clarity). As expected, the surface fluctuations were lower than the corresponding pressure fluctuations within the first half of the hydraulic jump, i.e., X

_{J}< 0.5. As for the 30° chute inflows [5], they coincided at X

_{J}≈ 0.5. At this point, the skewness of the pressure readings reached negative values, indicating a detachment of the bottom jet flow [5] (Figure 13g). The skewness values attended minimal values at X

_{J}≈ 0.75, beyond which they tended towards zero in the tailwater zone.

#### 4.2.4. Streamwise Distribution of Extreme Pressures

_{P}

^{max}and C

_{P}

^{min}downstream of 50° chute showed a similar streamwise distribution as the previously described pressure fluctuations (Figure 13c,d).

_{P}

^{max}= 0.42 and C

_{P}

^{min}= 0.29 at X

_{J}≈ 0.12 and X

_{J}≈ 0.18, respectively. Further downstream, they decreased attaining quasi-constant tailwater magnitudes. As a result of the flow deflection, the extreme pressure coefficients were of the same order of magnitudes at the jump toe, i.e., C

_{P}

^{max}≈ C

_{P}

^{min}≈0.32 (Figure 13c and Figure 20). In Figure 19a,c, the streamwise development of extreme pressure coefficients for 30° and 50° smooth chutes are compared, showing that the chute slope φ had no considerable effect.

_{P}

^{max}at the flow deflection point (i.e., X

_{J}≈ 0.04, Figure 13c), as compared to the 50° smooth chute, or up to two times higher compared to 30° stepped chute (Figure 19b). These values reached up to C

_{P}

^{max}≈ 1.26 (Figure 13c and Figure 20). Similarly, to the pressure fluctuation coefficients, pronounced values were also observed at the jump toe, where they reached up to C

_{P}

^{max}≈ 1.1 (Figure 13c and Figure 20). The peak in extreme negative pressures C

_{P}

^{min}was also observed at the flow deflection point, where they reached up C

_{P}

^{min}≈ 0.6 (Figure 13d and Figure 20). These values were up to 2 times higher as compared to 50° smooth or 30° stepped chutes (Figure 13d, Figure 19d and Figure 20). At the jump toe, the extreme negative pressures were similar to those of 50° smooth or 30°stepped chutes (Figure 13c, Figure 19d and Figure 20). Downstream of the flow deflection point, i.e., X

_{J}> 0.04, the extreme pressure coefficients for 50° stepped chutes decreased and were similar to 50° smooth chute approach flow magnitudes for X

_{J}> 0.13. Similar conclusions can be drawn for extreme pressure coefficients with 99.9% and 0.1% percentiles, but with typically two times lower magnitudes (Figure 13e,f).

_{J}> 0.13 or 30° stepped chutes for X

_{J}> 0.1. The pronounced pressure coefficients for 50° stepped chute inflows within X

_{J}≤ 0.13 are described as:

_{J}≤ 0.1 are described as [5]:

#### 4.2.5. Streamwise Distribution of Bottom Air Concentration and Cavitation Damage Protection

_{b}downstream of 30° and 50° smooth and stepped chutes are compared, including the bottom air concentration at the chute end, i.e., C

_{b},

_{ce}, illustrated at X

_{J}= −0.3.

_{b}in the downstream direction, reaching maxima C

_{b}≈ 0.11 at X

_{J}≈ 0.17, followed by a decrease towards zero at X

_{J}≈ 0.9. The bottom air concentration development was practically independent of the approach flow conditions or chute slope φ. The bottom air concentration development remained unaltered, despite the increased bottom air concentration at the 50° smooth chute end, i.e., C

_{b},

_{ce}values. This further strengthens the conclusion made in [12] that the bottom air concentrations downstream of smooth chutes were not influenced by the approach flow aeration C

_{1}due to the flow deviation at the jump toe, that generated high pressures (Figure 14a) and thus promoted de-aeration of the flow at the basin entrance.

_{J}≤ 0.4, as compared to 50° smooth chute approach flows, with values reaching up to C

_{b}= 0.18 at X

_{J}= 0.08 (Figure 21). The increased values after 50° stepped chute approach flows, similarly to those after 30° stepped chutes, can be attributed to the significantly higher C

_{b},

_{ce}and comparatively lower pressure magnitudes at the flow deviation point [12] (Figure 13a). Further downstream, the values of C

_{b}decreased up to X

_{J}≈ 0.4, beyond which they coincide with smooth chute approach flow values. Comparing the 30° and 50° stepped chute approach flows (Figure 21), it was noticeable that 50° stepped chute had a slightly higher air concentration (on average 2% more) within X

_{J}≤ 0.4, as compared to 30° stepped chute, despite the significantly higher bottom air concentration at the 50° stepped chute end (18%, on average) This was caused by the higher pressure magnitudes at the flow deviation point downstream of 50° stepped chute (Figure 13a), which lead to higher de-aeration rates at the basin entrance. Further downstream, i.e., X

_{J}> 0.4, the bottom air concentration development was independent of the chute slope φ

_{b}for 50° smooth chute approach flows was described by the equation developed for 30° [12] (Figure 21):

_{J}> 0.4. The pronounced bottom air concentrations for stepped chute approach flows can be approximated as (R

^{2}= 0.8, Figure 21):

_{J}≤ 0.5 and 0.15 ≤ X

_{J}≤ 0.4 considering 5% and 8% bottom air concentration limit [24,25,26], and (2) stepped chute inflows within 0 ≤ X

_{J}≤ 0.5 and 0 ≤ X

_{J}≤ 0.4 considering 5% and 8% bottom air concentration limit [24,25,26]. The pronounced extreme pressures downstream of 30° or 50° sloping stepped chutes within X

_{J}≤ 0.13 (Figure 19b,d) do probably not present a risk in terms of cavitation damage.

## 5. Conclusions

- The use of equivalent clear water parameters at the chute end leads to a fairly accurate prediction of the sequent depth ratio using classical momentum principle, irrespective of the approach flow conditions or chute slope φ, for a given approach Froude number F
_{1}. - The free surface characteristics along the plain stilling basin, such as dimensionless flow depths Z and dimensionless flow depth fluctuations C
_{H}’, are independent of the approach flow conditions or chute slope φ, either for smooth or stepped chutes. - The dimensionless roller lengths L
_{R}/h_{2}are independent of the approach flow conditions or chute slope φ. - The dimensionless hydraulic jump lengths L
_{R}/h_{2}are practically independent of the chute slope φ. Further, the results support the conclusions made by [5], stating that hydraulic jumps initiated below stepped chutes require an increased length x/h_{2}as compared to those below smooth chutes. - It is recommended that longer dimensionless plain stilling basin lengths are provided below stepped chutes irrespective of the chute slope, namely L
_{J}≈ 6.7h_{2}as compared to L_{J}≈ 5.8h_{2}below smooth chutes, plus a safety margin. - Increasing chute slope pronounces dimensionless mean bottom pressures P
_{m}, due to the stronger flow curvature, and slightly extends the influence reach of the flow curvature. The latter is more significant for stepped chute inflows. - Fluctuation and extreme pressure coefficients along the plain stilling basin invert are independent of the approach flow aeration C
_{1}or chute slope φ below smooth chutes. - Stepped chute inflows increase the fluctuating and extreme pressure coefficients at the plain stilling basin entrance, as compared to smooth chute inflows. Increasing stepped chute slope increases the extreme and fluctuating pressure coefficients within X
_{J}≤ 0.13. For 50° stepped chutes, these coefficients can reach up to 3 times higher magnitudes compared to smooth chute inflows or up to 2 times higher compared to the 30° stepped chute inflows. - The bottom air concentration development along the plain stilling basin below smooth chutes is independent of the chute slope φ, for X
_{J}≥ 0.08. Stepped chute inflows increase bottom air concentration within X_{J}≤ 0.4. Increasing stepped chute slope slightly increases the bottom air concentration within the latter region, namely by some 2% for 50° stepped chute compared to 30° chutes. - The cavitation protection length is independent of the chute slope, namely for: (1) smooth chute inflows within 0.1 ≤ X
_{J}≤ 0.5 and 0.15 ≤ X_{J}≤ 0.4 considering 5% and 8% bottom air concentration limit, and (2) stepped chute inflows within 0 ≤ X_{J}≤ 0.5 and 0 ≤ X_{J}≤ 0.4 considering 5% and 8% bottom air concentration limit. Evidently, stepped chute approach flows provide better cavitation damage protection in the initial reach of the plain stilling basin. - The increased extreme pressures below stepped chutes within X
_{J}≤ 0.13 are not expected to represent a danger in terms of cavitation. In spite of this, the increased bottom pressure coefficients below stepped chutes should be considered in the plain stilling basin slab design.

_{1}≤ 0.37 and 10.9 ≤ F

_{1}≤ 15) and skimming stepped (0.41 ≤ C

_{1}≤ 0.55, 6.2 ≤ F

_{1}≤ 8.2 and 2.70 ≤ h

_{c}/s ≤ 7.94) chute inflows.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Chanson, H. Hydraulics of Stepped Chutes and Spillways; August Aimé Balkema: Rotterdam, The Netherlands, 2002. [Google Scholar]
- Chanson, H.; Bung, D.B.; Matos, J. Stepped spillways and cascades. In Energy Dissipation in Hydraulic Structures; CRC Press: Leiden, The Netherlands, 2015; pp. 45–64. [Google Scholar]
- Hager, W.H.; Schleiss, A.J.; Boes, R.M.; Pfister, M. Stepped chute. In Hydraulic Engineering of Dams; CRC Press: Leiden, The Netherlands, 2021; pp. 277–319. [Google Scholar]
- Frizell, K.W.; Frizell, K.H. Guidelines for Hydraulic Design of Stepped Spillways; Hydraulic Laboratory Report HL-2015-06; U.S. Bureau of Reclamation: Denver, CO, USA, 2015.
- Stojnic, I.; Pfister, M.; Matos, J.; Schleiss, A.J. Effect of 30-Degree Sloping Smooth and Stepped Chute Approach Flow on the Performance of a Classical Stilling Basin. J. Hydraul. Eng.
**2021**, 147, 04020097. [Google Scholar] [CrossRef] - Bung, D.B.; Sun, Q.; Meireles, I.; Viseu, T.; Matos, J.S. USBR type III stilling basin performance for steep stepped spillways. In Proceedings of the 4th International. IAHR Symposium on Hydraulic Structures, Porto, Portugal, 9–11 February 2012. [Google Scholar]
- Frizell, K.W.; Svoboda, C.D. Performance of Type III Stilling Basins–Stepped Spillway Studies; U.S. Bureau of Reclamation: Denver, CO, USA, 2012.
- Frizell, K.W.; Svoboda, C.D.; Matos, J. Performance of type III stilling basins for stepped spillways. In Proceedings of the 2nd International Seminar on Dam Protection Against Overtopping, Fort Collins, CO, USA, 7–9 September 2016. [Google Scholar]
- Meireles, I.; Matos, J.; Silva Afonso, A. Flow characteristics along a USBR type III stilling basin downstream of steep stepped spillways. In Proceedings of the 3rd International Junior Researcher and Engineer Workshop on Hydraulic Structures, Brisbane, Australia, 2–4 May 2010. [Google Scholar]
- Novakoski, C.K.; Conterato, E.; Marques, M.; Teixeira, E.D.; Lima, G.A.; Mees, A. Macro-turbulent characteristics of pressures in hydraulic jump formed downstream of a stepped spillway. RBRH
**2017**, 22. [Google Scholar] [CrossRef][Green Version] - Novakoski, C.K.; Hampe, R.F.; Conterato, E.; Marques, M.G.; Teixeira, E.D. Longitudinal distribution of extreme pressures in a hydraulic jump downstream of a stepped spillway. RBRH
**2017**, 22. [Google Scholar] [CrossRef][Green Version] - Stojnic, I.; Pfister, M.; Matos, J.; Schleiss, A.J. Air-water flow in a plain stilling basin below smooth and stepped chutes. J. Hydraul. Res.
**2022**. [Google Scholar] [CrossRef] - Schwalt, M.; Hager, W.H. Die Strahlbox. Schweiz. Ing. Archit.
**1992**, 110, 547–549. (In German) [Google Scholar] [CrossRef] - Stojnic, I. Stilling basin performance downstream of stepped chutes. Ph.D. Thesis, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland, 2020. [Google Scholar]
- Wood, I. Uniform region of self-aerated flow. J. Hydraul. Eng.
**1983**, 109, 447–461. [Google Scholar] [CrossRef] - Wood, I. Air entrainment in free-surface flows. In IAHR Hydraulic Structures Design Manuals 4: Hydraulic Design Considerations, 1st ed.; Wood, I., Ed.; CRC Press/Balkema: Rotterdam, The Netherlands, 1991; pp. 1–152. [Google Scholar]
- Chanson, H.; Toombes, L. Air–water flows down stepped chutes: Turbulence and flow structure observations. Int. J. Multiph. Flow
**2002**, 28, 1737–1761. [Google Scholar] [CrossRef][Green Version] - Takahashi, M.; Ohtsu, I. Aerated flow characteristics of skimming flow over stepped chutes. J. Hydraul. Res.
**2012**, 50, 427–434. [Google Scholar] [CrossRef] - Matos, J.; Meireles, I. Hydraulics of stepped weirs and dams spillways: Engineering challenges, labyrinths of research. In Proceedings of the 5th IAHR International Symposium on Hydraulic Structures (ISHS 2014), Brisbane, Australia, 25–27 June 2014. [Google Scholar]
- Peterka, A.J. Hydraulic Design of Stilling Basins and Energy Dissipators; U.S. Department of the Interior: Denver, CO, USA, 2008.
- Fiorotto, V.; Rinaldo, A. Turbulent pressure fluctuations under hydraulic jumps. J. Hydraul. Res.
**1992**, 30, 499–520. [Google Scholar] [CrossRef] - Lopardo, R.A.; De Lio, J.C.; Vernet, G.F. Physical modelling on cavitation tendency for macroturbulence of hydraulic jump. In Proceedings of the International Conference on the Hydraulic Modelling of Civil Engineering Structure, Conventry, UK, 22–24 September 1982. [Google Scholar]
- Toso, J.W.; Bowers, C.E. Extreme Pressures in Hydraulic-Jump Stilling Basins. J. Hydraul. Eng.
**1988**, 114, 829–843. [Google Scholar] [CrossRef] - Peterka, A.J. The effect of entrained air on cavitation pitting. In Proceedings: Minnesota International Hydraulic Convention, Proceedings of the 1953 Minnesota International Hydraulics Convention, Minneapolis, MN, USA, 1–4 September 1953; American Society of Civil Engineers: New York, NY, USA, 1953. [Google Scholar]
- Rasmussen, R.E.H. Some experiments on cavitation erosion in water mixed with air. In Proceedings International Symposium on Cavitation in Hydrodynamics; National Physical Laboratory: London, UK, 1956. [Google Scholar]
- Russell, S.O.; Sheehan, G.J. Effect of entrained air on cavitation damage. Can. J. Civ. Eng.
**1974**, 1, 97–107. [Google Scholar] [CrossRef]

**Figure 1.**Definition sketch with instrumentation, notations and nomenclature. FOP—Fiber optical probe; PPT—Pitot-Prandtl tube; US—Ultrasonic displacement meter; APS—Automatic positioning system.

**Figure 2.**Photo of the spillway model with Q = 0.14 m

^{3}/s, the 50° smooth chute in the background and the plain stilling basin at the front. Flow direction from left to right.

**Figure 3.**(

**a**) Air concentration profiles at the inflow section for the 50° smooth chute (Runs 16–24, Table 1) and comparison with (—) advective diffusion model [17] with C

_{1 =}0.16, 0.29 and 0.36, and (

**b**) Dimensionless velocity V/V

_{90}profiles at the inflow section for the 50° smooth chute (Runs 16–24, Table 1) and comparison with (—) Equation (2).

**Figure 4.**(

**a**) Air concentration profiles at the inflow section for 50° stepped chute tests (Runs 25–30, Table 1), comparison with (―) advective diffusion model [17] with C

_{1}= 0.55 and 0.46, and (

**b**) Dimensionless velocity V/V

_{90}profiles at the inflow section for 50° stepped chute test runs (Runs 25–30, Table 1), comparison with (—) Equation (2).

**Figure 6.**(

**a**) Streamwise mean flow depths η along the stilling basin, and (

**b**) Sequent depth ratio h

_{2}/h

_{1}as a function of the approach Froude number F

_{1}; (—) Equation (3); [Runs 1–9: 30° smooth chute, Runs 10–15: 30° stepped chute, Runs 16–24: 50° smooth chute, Runs 25–30: 50° stepped chute].

**Figure 7.**Dimensionless flow depths Z along the jump roller for 30° and 50°: (

**a**) smooth and (

**b**) stepped chute inflows; (—) Equation (4); [Runs 1–9: 30° smooth chute, Runs 10–15: 30° stepped chute, Runs 16–24: 50° smooth chute, Runs 25–30: 50° stepped chute].

**Figure 8.**Roller lengths obtained from ultrasonic displacement meter measurement L

_{R,η}and visual observation L

_{R,D}plotted against F

_{1}as: (

**a**) L

_{R}/h

_{2}and (

**b**) L

_{R}/h

_{1}; (− −) Equation (5); [Runs 1–9: 30° smooth chute, Runs 10–15: 30° stepped chute, Runs 16–24: 50° smooth chute, Runs 25–30: 50° stepped chute].

**Figure 9.**(

**a**) Streamwise flow depth fluctuations η’ along the stilling basin (Table 1) and (

**b**) Dimensionless jump lengths L

_{J,η’}obtained from flow depth fluctuations η’ as a function of the inflow Froude number F

_{1}, compared to jump length prediction of [20]; [Runs 1–9: 30° smooth chute, Runs 10–15: 30° stepped chute, Runs 16–24: 50° smooth chute, Runs 25–30: 50° stepped chute].

**Figure 10.**Streamwise development of the flow depth fluctuation coefficient C

_{H}’ versus the normalized streamwise coordinate x/L

_{J,η’}for 30° and 50° smooth and stepped chute tests; (− −) Equation (6), (—) Equation (7); [Runs 1–9: 30° smooth chute, Runs 10–15: 30° stepped chute, Runs 16–24: 50° smooth chute, Runs 25–30: 50° stepped chute].

**Figure 11.**Streamwise pressure distribution (Run 22, smooth chutes, Table 1) of: (

**a**) extreme maximum p

_{max}, 99.9% probability p

_{99.9}, mean p

_{m}, 0.1% probability p

_{0.1}, and extreme minimum pressure p

_{min}, and (

**b**) fluctuating pressure p’, skewness S and excess kurtosis K.

**Figure 12.**Dimensionless jump lengths L

_{J}/h

_{2}downstream of 30° and 50° smooth and stepped chutes from pressure (L

_{J,p}

_{′}and L

_{J,SK}) and flow depth (L

_{J,η′}/h

_{2}) measurements, plotted against the inflow Froude number F

_{1}; (− −) [20]; [Runs 1–9: 30° smooth chute, Runs 10–15: 30° stepped chute, Runs 16–24: 50° smooth chute, Runs 25–30: 50° stepped chute].

**Figure 13.**Streamwise distribution of: (

**a**) mean pressure coefficient P

_{m}, (

**b**) pressure fluctuation coefficient C

_{P}’, (

**c**) maximum pressure coefficient C

_{P}

^{max}, (

**d**) 99th percentile coefficient C

_{P}

^{99.9}, (

**e**) minimum pressure coefficient C

_{P}

^{min}, (

**f**) 0.1th percentile coefficient C

_{P}

^{0.1}, (

**g**) skewness S, and (

**h**) excess kurtosis K; (—) Equations (10) and (11); (− −) Equations (6)–(9) and (12); [Runs 16–24: 50° smooth chute; Runs 25–30: 50° stepped chute].

**Figure 14.**Streamwise distribution of mean pressure coefficients P

_{m}downstream of 30° and 50°: (

**a**) smooth chutes, and (

**b**) stepped chutes; (—) Equation (10); (− −) Equations (8) and (9); [Runs 1–9: 30° smooth chute, Runs 10–15: 30° stepped chute, Runs 16–24: 50° smooth chute, Runs 25–30: 50° stepped chute].

**Figure 15.**Mean pressure coefficients C

_{P}

^{def}against inflow Froude number F

_{1}; [Runs 1–9: 30° smooth chute, Runs 10–15: 30° stepped chute, Runs 16–24: 50° smooth chute, Runs 25–30: 50° stepped chute].

**Figure 16.**Streamwise development of pressure fluctuation coefficient C

_{P}’ for 30° and 50°: (

**a**) smooth chute, and (

**b**) stepped chute; (—) Equation (11); (− −) Equations (12) and (13); [Runs 1–9: 30° smooth chute, Runs 10–15: 30° stepped chute, Runs 16–24: 50° smooth chute, Runs 25–30: 50° stepped chute].

**Figure 17.**Pressure fluctuation coefficients C

_{P}’ at the flow deflection point and the jump toe against inflow Froude number F

_{1}; [Runs 1–9: 30° smooth chute, Runs 10–15: 30° stepped chute, Runs 16–24: 50° smooth chute, Runs 25–30: 50° stepped chute].

**Figure 18.**Streamwise development of bottom pressure fluctuations p′ for Run 24 (50°, smooth chute) and Run 30 (50°, stepped chute chute).

**Figure 19.**Streamwise development of extreme pressure coefficients: (

**a**) C

_{P}

^{max}for 30° and 50° smooth chutes, (

**b**) C

_{P}

^{max}for 30° and 50° stepped chutes, (

**c**) C

_{P}

^{min}for 30° and 50° smooth chutes, and (

**d**) C

_{P}

^{min}for 30° and 50° stepped chutes; (—) Equation (11); (− −) Equations (12) and (13); [Runs 1–9: 30° smooth chute, Runs 10–15: 30° stepped chute, Runs 16–24: 50° smooth chute, Runs 25–30: 50° stepped chute].

**Figure 20.**Extreme pressure coefficients: (

**a**) C

_{P}

^{max}and (

**b**) C

_{P}

^{min}at the flow deflection point and jump toe; [Runs 1–9: 30° smooth chute, Runs 10–15: 30° stepped chute, Runs 16–24: 50° smooth chute, Runs 25–30: 50° stepped chute].

**Figure 21.**Streamwise development of bottom air concentration C

_{b}; (—) Equation (13); (− −) Equations (14) and (15); [Runs 1–9: 30° smooth chute, Runs 10–15: 30° stepped chute, Runs 16–24: 50° smooth chute, Runs 25–30: 50° stepped chute].

**Table 1.**Test program. SM = smooth chute; R = roughened with grid; PA = roughened with grid and pre-aeration; ST = stepped chute; s = step size.

Chute Configuration | Test Run | φ (°) | q (m ^{2}/s) | h_{c}/s(-) | C_{1}(-) | h_{1}(m) | V_{1}(m/s) | α (-) | F_{1} (-) | R_{1} × 10^{5}(-) | W_{1}(-) |
---|---|---|---|---|---|---|---|---|---|---|---|

SM + PA | 1 | 30 | 0.198 | / | 0.32 | 0.031 | 6.47 | 1.09 | 11.8 | 1.98 | 133 |

2 | 30 | 0.277 | / | 0.32 | 0.038 | 7.22 | 1.08 | 11.8 | 2.77 | 166 | |

3 | 30 | 0.358 | / | 0.32 | 0.046 | 7.79 | 1.09 | 11.6 | 3.58 | 196 | |

SM + R | 4 | 30 | 0.198 | / | 0.25 | 0.032 | 6.21 | 1.09 | 11.1 | 1.98 | 130 |

5 | 30 | 0.277 | / | 0.26 | 0.040 | 6.86 | 1.08 | 10.9 | 2.77 | 162 | |

6 | 30 | 0.356 | / | 0.26 | 0.047 | 7.51 | 1.08 | 11.0 | 3.56 | 192 | |

SM | 7 | 30 | 0.198 | / | 0.16 | 0.029 | 6.95 | 1.08 | 13.1 | 1.98 | 137 |

8 | 30 | 0.278 | / | 0.15 | 0.036 | 7.68 | 1.08 | 12.9 | 2.78 | 171 | |

9 | 30 | 0.356 | / | 0.15 | 0.042 | 6.38 | 1.08 | 13.0 | 3.56 | 202 | |

ST s = 0.06 m | 10 | 30 | 0.204 | 2.70 | 0.42 | 0.048 | 4.24 | 1.18 | 6.2 | 2.04 | 109 |

11 | 30 | 0.284 | 3.36 | 0.42 | 0.058 | 4.91 | 1.18 | 6.5 | 2.84 | 138 | |

12 | 30 | 0.362 | 3.95 | 0.41 | 0.068 | 5.35 | 1.18 | 6.6 | 3.62 | 163 | |

ST s = 0.03 m | 13 | 30 | 0.204 | 5.40 | 0.41 | 0.047 | 4.34 | 1.19 | 6.4 | 2.04 | 110 |

14 | 30 | 0.282 | 6.70 | 0.41 | 0.056 | 5.07 | 1.19 | 6.9 | 2.82 | 140 | |

15 | 30 | 0.364 | 7.94 | 0.41 | 0.066 | 5.54 | 1.18 | 6.9 | 3.64 | 166 | |

SM + PA | 16 | 50 | 0.198 | / | 0.37 | 0.028 | 7.16 | 1.08 | 13.8 | 1.98 | 140 |

17 | 50 | 0.280 | / | 0.36 | 0.036 | 7.88 | 1.07 | 13.4 | 2.80 | 174 | |

18 | 50 | 0.358 | / | 0.35 | 0.043 | 8.38 | 1.07 | 13.0 | 3.58 | 203 | |

SM + R | 19 | 50 | 0.199 | / | 0.28 | 0.029 | 6.88 | 1.08 | 12.9 | 1.99 | 137 |

20 | 50 | 0.278 | / | 0.29 | 0.038 | 7.41 | 1.07 | 12.2 | 2.78 | 168 | |

21 | 50 | 0.358 | / | 0.28 | 0.044 | 8.07 | 1.08 | 12.2 | 3.58 | 199 | |

SM | 22 | 50 | 0.199 | / | 0.16 | 0.026 | 7.61 | 1.08 | 15.0 | 1.99 | 144 |

23 | 50 | 0.279 | / | 0.16 | 0.033 | 8.38 | 1.09 | 14.7 | 2.79 | 179 | |

24 | 50 | 0.356 | / | 0.15 | 0.040 | 8.99 | 1.08 | 14.4 | 3.56 | 210 | |

ST s = 0.06 m | 25 | 50 | 0.205 | 2.71 | 0.55 | 0.041 | 4.95 | 1.18 | 7.8 | 2.05 | 118 |

26 | 50 | 0.284 | 3.36 | 0.53 | 0.051 | 5.52 | 1.19 | 7.8 | 2.84 | 147 | |

27 | 50 | 0.364 | 3.97 | 0.50 | 0.061 | 5.96 | 1.19 | 7.7 | 3.64 | 173 | |

ST s = 0.03 m | 28 | 50 | 0.205 | 5.51 | 0.50 | 0.040 | 5.10 | 1.18 | 8.1 | 2.05 | 120 |

29 | 50 | 0.284 | 6.73 | 0.48 | 0.050 | 5.71 | 1.19 | 8.2 | 2.84 | 149 | |

30 | 50 | 0.364 | 7.94 | 0.46 | 0.060 | 6.08 | 1.19 | 7.9 | 3.64 | 174 |

Equation | Coefficient | C_{P}′ (-) | C_{P}^{max} (-) | C_{P}^{99.9} (-) | C_{P}^{min} (-) | C_{P}^{0.1} (-) |
---|---|---|---|---|---|---|

(11) | d | 1.006 | 1.035 | 1.020 | 1.020 | 1.015 |

e | 3.00 | 3.41 | 3.31 | 2.52 | 2.46 | |

f | 0.31 | 3.41 | 1.60 | 1.80 | 0.79 | |

j | 0.006 | 0.060 | 0.037 | 0.010 | 0.010 | |

R^{2} | 0.99 | 0.97 | 0.98 | 0.91 | 0.97 | |

(12) | k | 1.04 | 1.36 | 1.19 | 1.24 | 1.12 |

l | 19 | 28 | 21 | 24 | 18 | |

m | 6 | 154 | 31 | 28 | 10 | |

n | 0.000 | 0.020 | 0.070 | −0.050 | −0.007 | |

R^{2} | 0.98 | 0.96 | 0.97 | 0.86 | 0.96 | |

(13) | o | −0.266 | −2.400 | −1.650 | −0.560 | −0.375 |

r | 0.076 | 0.604 | 0.362 | 0.278 | 0.175 | |

R^{2} | 0.90 | 0.75 | 0.90 | 0.42 | 0.60 |

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

© 2022 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**

Stojnic, I.; Pfister, M.; Matos, J.; Schleiss, A.J. Plain Stilling Basin Performance below 30° and 50° Inclined Smooth and Stepped Chutes. *Water* **2022**, *14*, 3976.
https://doi.org/10.3390/w14233976

**AMA Style**

Stojnic I, Pfister M, Matos J, Schleiss AJ. Plain Stilling Basin Performance below 30° and 50° Inclined Smooth and Stepped Chutes. *Water*. 2022; 14(23):3976.
https://doi.org/10.3390/w14233976

**Chicago/Turabian Style**

Stojnic, Ivan, Michael Pfister, Jorge Matos, and Anton J. Schleiss. 2022. "Plain Stilling Basin Performance below 30° and 50° Inclined Smooth and Stepped Chutes" *Water* 14, no. 23: 3976.
https://doi.org/10.3390/w14233976