# Experimental Analysis on the Use of Counterflow Jets as a System for the Stabilization of the Spatial Hydraulic Jump

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{1}/(g·h

_{1})

^{0.5}, where v

_{1}and h

_{1}are the velocity and the depth of the approaching flow and g is the gravitational acceleration).

- Froude number of the approaching flow (Fr) and the corresponding reference tailwater levels identified in the first phase (hs);
- Distance of the jet system from the expansion section in the flume (P): 0.4, 0.6, and 0.8 m;
- Jet density, i.e., number of open nozzles in water jet injection system (N): 5, 7, and 9 (Figure 2);
- Jet flow rate (Q
_{j}): 8.1 and 7.2 L/s.

_{b}, which is a modified version of the Boussinesq momentum correction coefficient for near-bed velocity:

_{b}(x) is the near-bed longitudinal velocity across the section, and v

_{mb}is the average value of v

_{b}(x).

_{b}·v

_{mb}

^{2}provides information on the dynamic force of the flow (per unit height at the measurement depth) and then on its scouring potential, which is important when assessing the effectiveness of a stilling basin downstream from a hydraulic structure. β

_{b}and β

_{b}·v

_{mb}

^{2}were computed for both the reference and the 54 “with device” conditions, thus allowing for a comparative analysis of the performances of the different tested configurations of the dissipator.

_{1}) and the control section at 0.5 m downstream from the device (E

_{2}) was carried out based on the experimental measurements. In calculating the efficiency (ΔE = (E

_{1}− E

_{2})/E

_{1}), the kinetic energy correction coefficient was neglected for the upstream section (1), under the hypothesis of uniform velocity distribution, while for the downstream one (2), the near-bed coefficient α

_{b}was used in lieu of the global Coriolis value (α), assuming it as representative of the whole cross-section. α

_{b}was computed similarly to β

_{b}based on the longitudinal velocities measured at the elevation of 0.5 cm from the channel bed:

_{m}is the mean flow velocity.

_{b}and β

_{b}, for describing the uniformity of the flow features instead of using the global coefficients, α and β. Indeed, although both the parameters (α and α

_{b}; β and β

_{b}) are flow uniformity coefficients, the ones denoted with lowercase “b” consider only the velocity field in the proximity of the channel bed instead of that for the whole cross-sectional area.

_{p}′ [29], which compares the standard deviation of the pressure head σ with the inflow velocity head, was calculated from the experimental measurements:

_{1}is the approaching flow velocity and g is the gravitational acceleration.

_{m}and both negative and positive fluctuations from the mean, ∆P

^{+}and ∆P

^{−}, were expressed as well [29]:

## 3. Results and Discussion

_{b}values are considerably greater than unity in all the measurement sections, reaching the maximum values at 2.5 m downstream from the expansion section, and still being over 2 at the end of the flume. The comparison of β

_{b}and β

_{b}·v

_{mb}

^{2}obtained with the installed device to the ones shown in Figure 3 then provides direct information on the effectiveness of the counterflow jets in stabilizing the hydraulic jump occurring in the expanding channel.

_{b}and β

_{b}·v

_{mb}

^{2}registered for all the experimental runs at the measurement section located 0.5 m downstream from the dissipator (i.e., 0.9, 1.1, and 1.3 m from the abrupt expansion for P equal to 0.4, 0.6, and 0.8 m, respectively). In this section, the flow depth was always slightly higher than the one at the end of the flume (as expected in a horizontal channel in subcritical condition) and the results for β

_{b}(ranging from 1.01 to 1.46, with a mean of 1.15) and β

_{b}·v

_{mb}

^{2}(the maximum registered value was equal to 0.35 m

^{2}/s

^{2}) reveal a good effectiveness of the system in improving the flow features in the tailwater channel, allowing the transition to subcritical flow in a stilling basin of limited length, even for the worst-performing configurations. At the end of the flume (i.e., at the measurement section at 8 m from the expansion), the flow was found to be always uniform, with β

_{b}values constantly smaller than 1.03.

_{b}and β

_{b}·v

_{mb}

^{2}found in the tests with the larger values of P.

_{mb}has been found to increase with the jet density (i.e., higher number of open nozzles, N), suggesting that the flow tends to be diverted towards the free surface when the jets are less uniformly distributed across the channel. This phenomenon could be a consequence of the increase in total momentum and kinetic energy per unit of Q

_{j}associated with a smaller number of nozzles (i.e., higher velocity of the jets), resulting in a higher local pressure upstream of the device. Regarding the influence of the jet discharge Q

_{j}, this was found to negatively affect β

_{b}, except for Fr = 7.4, while the values of β

_{b}·v

_{mb}

^{2}were not surprisingly higher for increasing Q

_{j}, because of the larger flow rate under the same downstream boundary condition.

- Q
_{j}= 7.2 L/s, N = 7, P = 0.4 m; - Q
_{j}= 8.1 L/s, N = 9, P = 0.4 m; - Q
_{j}= 7.2 L/s, N = 5, P = 0.6 m.

_{b}= α or neglecting it in the computation of the dissipated energy, observing that the use of α

_{b}instead of α causes an average error of 0.15%, with a maximum value of 1.16%, while neglecting it (i.e., α = 1) leads to an average error of 0.38%, thus confirming the suitability of approximating α with α

_{b}. Moreover, in these experiments, the values of α and β were found to be linearly dependent, according to the equation $\alpha =3.01\beta -2.01$ (R

^{2}= 0.9975), in line with the observations of Mohanty et al. [31].

_{p}roughly resembles the water surface profile, while the other plots in Figure 8 corroborate the qualitative description of the flow features provided in Figure 4.

_{p}

^{±}values of about 0.5 in the expansion area. C

_{p}′ was found to increase when moving from hs to 0.75 hs, because of the additional turbulent structures induced by the jets.

_{p}′ at y/P = 2.87 (C

_{p}′ ~ 0.1) were a bit larger than those reported in the literature for hydraulic jumps [29,32,33], suggesting that, in spite of the quite uniform flow velocity field registered (Figure 7), bed pressure oscillations are not completely contained within the basin, mainly as a consequence of the reported water surface fluctuations in the area downstream from the device [34].

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**View of the system of counterflow jets installed in the flume, (

**left**); and different tested arrangements of the jet density (i.e., varying number of open nozzles), (

**right**).

**Figure 3.**Values of β

_{b}and β

_{b}·v

_{mb}

^{2}in the measurement sections along the flume for the reference S-jump conditions.

**Figure 4.**Typical flow patterns induced by the counterflow jets: (

**a**) plan view, (

**b**) side view, and (

**c**) frontal view.

**Figure 5.**Performance of the dissipator as a function of P, N, and Qj, for the different tested Fr:β

_{b}and β

_{b}·v

_{mb}

^{2}registered 0.5 m downstream from the dissipator’s end.

**Figure 6.**3D velocity flow fields measured in the channel downstream from the dissipator, under variable tailwater conditions (as a portion of hs): Configuration 1 (Qj = 7.2 L/s, N = 7, P = 0.4 cm).

**Figure 7.**Momentum and kinetic energy correction coefficients (β and α) calculated downstream from the dissipator, under variable tailwater conditions: Configuration 1—Qj = 7.2 L/s, N = 7, P = 0.4 cm (

**a**); Configuration 2—Qj = 8.1 L/s, N = 9, P = 0.4 cm (

**b**); Configuration 3—Qj = 7.2 L/s, N = 5, P = 0.6 cm (

**c**).

**Figure 8.**Bed pressure coefficients along the flume for varying tailwater levels (Configuration 1—Fr = 7.4): standard deviation of pressure fluctuations Cp′, mean pressures Cp, and extreme positive and negative pressure deviations from the mean, Cp+ and Cp−.

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**MDPI and ACS Style**

Sharoonizadeh, S.; Ahadiyan, J.; Scorzini, A.R.; Di Bacco, M.; Sajjadi, M.; Moghadam, M.F. Experimental Analysis on the Use of Counterflow Jets as a System for the Stabilization of the Spatial Hydraulic Jump. *Water* **2021**, *13*, 2572.
https://doi.org/10.3390/w13182572

**AMA Style**

Sharoonizadeh S, Ahadiyan J, Scorzini AR, Di Bacco M, Sajjadi M, Moghadam MF. Experimental Analysis on the Use of Counterflow Jets as a System for the Stabilization of the Spatial Hydraulic Jump. *Water*. 2021; 13(18):2572.
https://doi.org/10.3390/w13182572

**Chicago/Turabian Style**

Sharoonizadeh, Shokoofeh, Javad Ahadiyan, Anna Rita Scorzini, Mario Di Bacco, Mohsen Sajjadi, and Manoochehr Fathi Moghadam. 2021. "Experimental Analysis on the Use of Counterflow Jets as a System for the Stabilization of the Spatial Hydraulic Jump" *Water* 13, no. 18: 2572.
https://doi.org/10.3390/w13182572