# An Experimental Study of Hydraulic Jump in a Gradually Expanding Rectangular Stilling Basin with Roughened Bed

^{1}

^{2}

^{*}

## Abstract

**:**

_{1}/b

_{2}= 0.4, 0.6, 0.8, 1) and the inflow Froude numbers (6 < Fr

_{1}< 12) were compared with each other and with the corresponding values measured for the classical hydraulic jump. The results showed that the tailwater depth required to form a hydraulic jump and also the roller length of the hydraulic jump and the length of the hydraulic jump on a gradual expansion basin with the rough bed were appreciably smaller than that of the corresponding hydraulic jumps in a rectangular basin with smooth and rough bed. With the experimental data, empirical formulae were developed to express the hydraulic jump characteristics relating to roughness elements height and divergence ratio of wall. Also, the applicability of some empirical relationships for estimating the roller length was tested.

## 1. Introduction

_{2}) required to form a hydraulic jump on rough bed could be appreciably smaller than the corresponding sequent depth on the smooth bed (${y}_{2}^{*}$) [18]. Rajaratnam [17] found that the hydraulic jump on rough beds were significantly shorter than the classical hydraulic jump on smooth bed [18]. Hughes and Flack [19] measured hydraulic jump characteristics over several artificially roughened beds in a horizontal rectangular flume with smooth side walls. Their observations showed that boundary roughness reduces both the sequent depth and the length of a hydraulic jump and that the observed reductions were related to both the inflow Froude number and the degree of roughness. Ead and Rajaratnam [18] analyzed the hydraulic jump on a corrugated bed and identified substantial axial velocity profile similitude at different cross sections of the hydraulic jump. The effect of the tailwater level on flow conditions at the hydraulic jump were studied by Mossa et al. [20]. The experimental results showed that, design stilling basins with abrupt drops may be used to stabilize the position of the jump. Pagliara et al. [21] conducted many laboratory tests in order to propose empirical relationships to predict the main hydraulic jump characteristics on a rough uniform and non-homogeneous granular bed material. Oscillating characteristics and cyclic mechanisms of the hydraulic jump was studied by Mossa [22]. The results showed that the vortex roll-up process was linked to fluctuations of the longitudinal location of the jump toe.

_{1}and A

_{2}are the flow cross-sectional area, upstream and downstream of the hydraulic jump, respectively. B′ is a characteristics free-surface width (B

_{1}< B′ < B

_{2}), B

_{1}and B

_{2}are the upstream and downstream free-surface widths, respectively. Here the inflow Froude number Fr

_{1}is $F{r}_{1}={v}_{1}/\sqrt{g\frac{{A}_{1}}{{B}_{1}}}$ and B is another free-surface width maybe defined as:

_{1}is the inflow depth of the hydraulic jump and d

_{2}is the flow depth downstream of the hydraulic jump. For a rectangular channel, Equation (2) becomes the Bélanger equation. To date a large number of stilling basin types has been proposed. The usual stilling basin is characterized by a (gradual or abrupt) enlargement of the channel width in order to raise its efficiency. Hager [24] studied the characteristics of the hydraulic jump in non-prismatic rectangular channels. The results showed that the hydraulic jump in a diverging channel needs lower tailwater flow depth for identical flow conditions. Also, he obtained that the length characteristics were nearly independent of the width ratio and had the same order of magnitude as in prismatic channels. Lawson and Phillips [25] used the momentum analysis of Koloseus and Ahmad (1969), Arbhabhirama and Abella [26] and France (1981) adopted to obtain a suitable circular hydraulic jump equation. The results showed that the length of the circular hydraulic jump was found to be considerably less than that of a corresponding rectangular hydraulic jump. The head loss in a circular hydraulic jump was greater than in a rectangular jump for the same inflow Froude number.

_{p}

_{1}, F

_{p}

_{2}, M

_{1}and M

_{2}are the pressure forces and the momentum upstream and downstream of the hydraulic jump, respectively, ${F}_{\tau}$ is the integrated bed shear stress on the horizontal plane and F

_{e}is pressure force on the expanding wall [29]. If the pressure force on the expansion walls is negligible, Equation (4) can be written as [29]:

## 2. Materials and Methods

#### 2.1. Experimental Set-Up and Instrumentation

_{1}), the sequent depth of flow (y

_{2}) and the fluctuating of free surface elevations above the hydraulic jump were measured using ultrasonic sensors Data Logic US30 with operation range of 10–100 cm and an accuracy of ±0.1 mm that were mounted above the channel. Following Hager [10], the length of the hydraulic jump (L

_{j}) was measured between the toe of the hydraulic jump and the location where the gradually varied flow starts. The length of the hydraulic jump was measured by a fabric ruler with a reading accuracy of ±1 mm installed along the channel.

_{1}= 1.58 m for all experiments. To investigate the effect of roughness, discontinuous dissipative elements of lozenge shape (the shape of roughness elements was chosen according to the study of Bejestan and Neisi [33]) with two heights (r = 0.014 m and 0.028 m) were installed on the horizontal bed (Figure 3). The crest of these elements were at the same level as the upstream bed. In this way, the dissipative elements would not be protruding into the flow [18]. Thus, the elements were not directly subjected to the incoming jet.

_{1}/b

_{2}) of 0.4, 0.6, 0.8, and 1 (where b

_{1}and b

_{2}are the widths of the stilling basin in upstream and downstream of the hydraulic jump, respectively). For each set of experiments, each model was tightly installed in the flume. For each experiment, the inflow depth (y

_{1}) and the sequent depth (y

_{2}), the length of the hydraulic Jump (L

_{j}), the roller length of the hydraulic jump (L

_{r}) and the width of channel where the hydraulic jump ends (b) were measured. Table 2 lists the experiments carried out in the present study.

#### 2.2. Dimensional Analysis

_{2}) and length of the hydraulic jump (L

_{j}), can be expressed as:

_{1}is the inflow depth, V

_{1}is the average velocity at the beginning of the hydraulic jump, g is the gravitational acceleration, $\nu $ is the kinematic viscosity of water, $\rho $ is the density of water, r is the height of dissipative elements, b

_{1}and b

_{2}are the widths of stilling basin upstream and downstream of the hydraulic jump, respectively. Based on the principle of dimensional analysis, using the Pi theorem by selecting y

_{1}, V

_{1}and $\rho $ as three repeated variables, the functional relationships can be expressed in dimensionless form as:

_{1}is the inflow Froude number at the beginning of the hydraulic jump and Re

_{1}is the inflow Reynolds number. For large values of Re

_{1}(as in this study), viscous effects can be neglected (Rajaratnam [17], Hager and Bremen [9]). Therefore the sequent depth ratio and relative length of the hydraulic jump are obtained as follows:

## 3. Results and Discussion

#### 3.1. Sequent Depth Ratio

_{2}/y

_{1}), considering the Equation (10), is dependent on the inflow Froude number (Fr

_{1}), the relative height of roughness elements (r/y

_{1}) and the divergence ratio of walls (B = b

_{1}/b

_{2}). To evaluate the effect of dissipative elements on the sequent depth ratio of the hydraulic jump, in Figure 4 the values of y

_{2}/y

_{1}are plotted versus the inflow Froude number. Also in Figure 4, the sequent depth ratios are compared with Equation (2) proposed by Chanson and Carvalho [1] for the hydraulic jump in an irregular channel.

_{2}/y

_{1}), decreased as the height of the roughness elements increased. Visual observations indicated that the flow separation and recirculation vortex were formed between the roughness elements. By the increasing dissipative elements height, a recirculation vortex was developed, reducing the sequent depth ratio. Also, The results showed that the sequent depth ratio for the hydraulic jump on gradually expanding channel was smaller than that of the corresponding classic hydraulic jump on a rectangular channel.

_{2}/y

_{1}) with Fr

_{1}and two height of the roughness elements and four divergence ratio of the channel walls could be described with the following regression-based equation with a coefficient of determination (R

^{2}) equal to 0.95.

_{2}/y

_{1}values in the present study, Ead and Rajaratnam [18] and CarolloFerro [31], and the values calculated from Equation (12) are presented in the Figure 5. The results indicated that the computed data were close to the line of agreement and there is a ±10% difference with the corresponding measured values. Also, the computed data for y

_{2}/y

_{1}by Equation (12) shows a good agreement with previous investigations and only 6 of 190 ratios y

_{2}/y

_{1}fall out the error band of ±10%.

_{1}= 1.33.

#### 3.2. Relative Roller Length of the Hydraulic Jump

_{r}) was actually a better length characteristic than the hydraulic jump length because it was easy to observe and was properly defined for steady flow conditions [35]. The roller length (L

_{r}), is the horizontal distance between the toe section with the flow depth y

_{1}and the roller end. For the hydraulic jump in smooth and rough horizontal channels, using the available experimental data CarolloFerro [31] proposed the following equation for the roller length of the hydraulic jump:

_{0}and b

_{0}are numerical coefficients depending on bed roughness. In this investigation, the experimental data of the roller length were used to test the applicability of Equations (13)–(15).

_{r}/y

_{1}, y

_{1}/y

_{2}) and (L

_{r}/y

_{1}, y

_{2}/y

_{1}− 1) can be obtained independent of the roughness height and divergence ratio but according to Figure 8, a single relationship cannot be established using the pairs (L

_{r}/y

_{1}, Fr

_{1}− 1), as it was shown by CarolloFerro [31].

_{0}and b

_{0}are listed in Table 4. The estimates of a, a

_{0}and b

_{0}obtained by Carollo and Ferro (2004) using the measurements of Hughes and Flack [19] and HagerBremen [36] and also the estimates of a, a

_{0}and b

_{0}obtained by CarolloFerro [31] are listed in Table 4.

_{0}depends on the roughness height and divergence ratio whereas a and a

_{0}can be assumed constant. Using experimental data from this study and fitting Equations (13) and (14) to them, gives a = 4.745 and a

_{0}= 2.309. So Equations (13) and (14) can be written as it follows:

^{2}) was equal to 0.84. This equation shows that the relative roller length linearly increased with the increasing ratio of (y

_{2}/y

_{1}− 1).

^{2}) was equal to 0.86. This equation shows that the relative roller length exponentially decreased with the increasing ratio of (y

_{1}/y

_{2}).

_{r}/y

_{1}in the present study, Ead and Rajaratnam [18] and CarolloFerro [31] and the values from Equations (16) and (17) are plotted in the Figure 9. As shown in Figure 9, the computed data by Equations (16) and (17) showed a ±25% and ±22% difference with the corresponding measured values, respectively. Also, the computed ratios of L

_{r}/y

_{1}by Equations (16) and (17) for the experimental data of Ead and Rajaratnam [18] and CarolloFerro [31] are in the range of ±25%.

_{0}, the divergence ratio, and the relative roughness, the relationship between them, could be described by the following regression-based equation with a coefficient of determination (R

^{2}) equal to 0.89.

_{r}/y

_{1}in the present study, Ead and Rajaratnam [18] and CarolloFerro [31] and those calculated using the Equation (19) are plotted in Figure 10. According to this Figure, the computed data by the Equation (19) showed a ±26% difference with the corresponding measured values.

#### 3.3. Relative Length of the Hydraulic Jump

_{j}/y

_{1}) for different divergence ratio, are plotted versus the inflow Froude number in Figure 11. The results showed that the roughness element caused a significant reduction of the hydraulic jump length. Also, Figure 11 showed that the roughness with height 0.028 m was more effective than the other height of dissipative elements to reduce the relative length of the hydraulic jump (L

_{j}/y

_{1}). In these Figure, the relative lengths of a classical hydraulic jump based on the following equations proposed by USBR [37] and Hager [10], respectively, were also compared.

_{j}/y

_{1}) obtained in the present study with those obtained by Hager [10] and USBR [37] are compared in Figure 12 and Figure 13.

_{j}/y

_{1}), the inflow Froude number (Fr

_{1}), divergence ratio (B) and the relative height of roughness (r/y

_{1}) can be described by the Equation (22) with a coefficient of determination (R

^{2}) equal to 0.89. Then the values of relative length ratio for all the experiments were computed using the Equation (22) and compared with corresponding measured values in Figure 14. It was obtained from the Figure 14 that the result produced by the proposed relationship showed ±19% difference with the corresponding measured values.

#### 3.4. Energy Dissipation

_{L}) is equal to the difference between the specific energy upstream and downstream of the hydraulic jump, and it can be defined as below:

_{1}and E

_{2}are the energy height at the toe and the end of the hydraulic jump, respectively. The energy loss in the diverging hydraulic jump was calculated from the specific energy and continuity equation as it follows:

_{L}and E

_{L}/E

_{1}are the energy loss and the relative loss of energy, respectively. The measured relative energy losses (E

_{L}/E

_{1}) due to the hydraulic jump were calculated for different divergence ratio with two heights of bed roughness elements and the relative energy loss values were plotted versus the inflow Froude number in Figure 15.

_{L}/E

_{1}) values corresponding to rough beds are greater than those for the smooth bed. This difference increases with the increasing inflow Froude number (Fr

_{1}). The trend of the measured values for each model in these Figures shows that for the same inflow Froude number the relative energy loss increases with the increasing height of roughness elements. Also, Figure 15 shows that for the hydraulic jump in all expanding stilling basin, the relative energy loss is greater than those for the rectangular section.

_{L}/E

_{1}) for different divergence ratios were compared with each other. The Figures indicated that the relative energy loss increased with the decreasing divergence ratio and the channel with divergence ratio of B = 0.4 is more effective to dissipate the energy of the hydraulic jump. This is probably due to the existence of lateral force and high turbulence and rolling flow along the longitudinal section.

_{L}/E

_{1}), the inflow Froude number (Fr

_{1}), the relative height of roughness (r/y

_{1}) and the divergence ratio of channel walls (B) the following equation was obtained by non-linear regression as:

^{2}) was equal to 0.98.

#### 3.5. Bed Shear Stresses

_{1}) in Figure 17.

## 4. Conclusions

_{2}/y

_{1}), the relative length of the hydraulic jump (L

_{j}/y

_{1}) and the relative loss of energy (E

_{L}/E

_{1}), respectively. The Equation (12) shows that the sequent depth ratio increased with the increasing inflow Froude number and the divergence ratio. Also, the sequent depth ratio decreased as the roughness elements ratio increased. The relative length of the hydraulic jump for a given inflow Froude number, decreased as the roughness elements ratio increased. A comparing between the measured values of E

_{L}/E

_{1}and those calculated by Equation (26) showed that, Equation (26) allows an estimation for calculating the relative energy loss in expanding basin with roughened bed. Also, three Equations (16), (17) and (19) for calculating the roller length of the hydraulic jump were developed. The roller length ratio decreased by the increasing roughness elements ratio and increased as the divergence ratio increased.

_{0}can be considered constant whereas the coefficient b

_{0}is related to divergence ratio and relative roughness height. In conclusion, this paper investigated the effects of the height of roughness elements and divergence ratio of walls on the main characteristics of the hydraulic jump. Table 5 lists the equations derived in this study.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Notation

a | The coefficient of Equation (13) |

a_{0} | The coefficient of Equation (14) |

B | Divergence ratio |

b_{0} | The coefficient of Equation (15) |

b_{1} | Width of the stilling basin in upstream |

b_{2} | Width of the stilling basin in downstream |

d_{1} | Inflow depth of the hydraulic jump |

d_{2} | Flow depth in downstream of the hydraulic jump |

E_{1} | Specific energy upstream the hydraulic jump |

E_{2} | Specific energy downstream the hydraulic jump |

E_{L} | Energy loss in the hydraulic jump |

Fr_{1} | Inflow Froude number, where $F{r}_{1}={v}_{1}/\sqrt{g{y}_{1}}$ |

g | Gravitational acceleration |

Re_{1} | Inflow Reynolds number |

r | Height of roughness elements |

L_{j} | Length of the hydraulic jump |

L_{r} | Roller length of the hydraulic jump |

L_{j}/y_{1} | Relative length of the hydraulic jump |

L_{r}/y_{1} | Relative roller length of the hydraulic jump |

q | Discharge per unit width |

E_{L}/E_{1} | Relative energy loss |

y_{1} | Inflow depth of the hydraulic jump |

y_{2} | Sequent depth of the hydraulic jump |

y_{2}^{*} | Sequent depth of the classic hydraulic jump |

M_{1} | Momentum flux at the beginning of the hydraulic jump |

M_{2} | Momentum flux at the end of the hydraulic jump |

P_{1} | Hydrostatic force at the section upstream of the hydraulic jump |

P_{2} | Hydrostatic force at the section downstream of the hydraulic jump |

R^{2} | Coefficient of determination |

Fp_{1} | Pressure force upstream the hydraulic jump |

Fp_{2} | Pressure force downstream the hydraulic jump |

F_{e} | Pressure force on the expanding wall |

${F}_{\tau}$ | Integrated bed shear stress over the hydraulic jump length |

$\epsilon $ | Shear force coefficient |

$\rho $ | Mass density of water |

$\upsilon $ | Kinematic viscosity of water |

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**Figure 1.**Basic sketch of momentum equation for the hydraulic jump on rough bed of channel with expanding walls.

**Figure 3.**Definition sketch for dissipative elements with lozenge shape with divergence ratio of B = 0.4. (

**a**) Height of roughness (r) = 1.4 cm. (

**b**) Height of roughness (r) = 2.8 cm.

**Figure 4.**Variation of the sequent depth ratio vs. the inflow Froude number for two relative height of roughness elements (r/y

_{1}= 0.67 and r/y

_{1}= 1.33) and different divergence ratio (B = 1, 0.8, 0.6, 0.4).

**Figure 5.**Comparison between all the y

_{2}/y

_{1}values measured and those calculated by Equation (12).

**Figure 7.**Relationship between the ratio of L

_{r}/y

_{1}and (

**a**) the ratio of (y

_{1}/y

_{2}), (

**b**) the difference ((y

_{2}/y

_{1}) − 1) for all experimental data.

**Figure 9.**Comparison between the measured values of the ratio L

_{r}/y

_{1}and (

**a**) results from Equation (16), (

**b**) results from Equation (17).

**Figure 10.**Comparison between the measured values of the ratio L

_{r}/y

_{1}and those calculated by Equation (19).

**Figure 11.**Variation of the relative length of the hydraulic jump (L

_{j}/y

_{1}) vs. the inflow Froude number for different divergence ratio (B = 1, 0.8, 0.6 and 0.4) and two relative height of dissipative elements (r/y

_{1}= 0.67 and 1.33).

**Figure 12.**Variation of the relative length of the hydraulic jump (L

_{j}/y

_{1}) vs. the inflow Froude number with different divergence ratio on a smooth bed.

**Figure 13.**Variation of the relative length of the hydraulic jump (L

_{j}/y

_{1}) vs. the inflow Froude number with different divergence ratio on a rough bed.

**Figure 14.**Comparison between the L

_{j}/y

_{1}values measured in this investigation and those calculated by Equation (22).

**Figure 15.**Variation of the relative energy loss (${E}_{L}/{E}_{1}$) vs. the inflow Froude number for different divergence ratio (B = 1, 0.8, 0.6 and 0.4) and two relative heights of dissipative elements (r/y

_{1}= 0.67 and 1.33).

**Figure 16.**(

**a**) Variation of the relative energy loss (E

_{L}/E

_{1}) vs. the inflow Froude number for different divergence ratios on the smooth and rough bed (r/y

_{1}= 1.33). (

**b**) Comparison the effect of divergence ratio on relative energy loss on the smooth and rough bed (r/y

_{1}= 0.67 and 1.33).

**Figure 17.**Variation of the shear force coefficient ($\epsilon $) vs. the inflow Froude number for different divergence ratio (B = 1, 0.8, 0.6 and 0.4) and two relative heights of dissipative elements (r/y

_{1}= 0.67 and 1.33).

Reference | Cross Section and Change Bed | Flume Dimension (m) | Roughness (mm) | Range of Inflow Froude Number | Investigated Flow Properties |
---|---|---|---|---|---|

Hughes and Flack [19] | - Horizontal bed - Rectangular channel - Smooth and rough bed | FL = 2.13 FW = 0.3 | - Two striproughness beds (RH: 3.2 and 6.4) - Three densely packed gravel beds (RH: 4.3–11.3) | 3.44–8.04 2.34–10.5 | - Discharge - Upstream depth - Tailwater depth - Jump length - Roller length - Velocity |

Wanoschek and Hager [23] | - Horizontal Bed - Prismatic symmetrical trapezoidal - Side slope: 45° (m = 1) | FL = 8 FW = 0.22–1.60 FH = 0.7 | - Smooth Bed | 5.45–13 | - Discharge - Upstream depth - Tailwater depth - Jump length - Velocity |

Ead and Rajaratnam [18] | - Horizontal bed - Rectangular channel - Smooth and rough bed | FL = 7.6 FW = 0.44 FH = 0.6 | - Corrugated aluminum Sheets (RH = 13 and 22) | 4–10 | - Discharge - Upstream depth - Tailwater depth - Jump length - Roller length - Velocity - Water surface profile |

Carollo et al. [31] | - Horizontal bed - Rectangular channel - Smooth and rough bed | FL = 1.4 FW = 0.6 FH = 0.6 | - Crashed gravel particles (d _{50}: 4.6, 8.2, 14.6, 23.9, 32) | 1.1–9.9 | - Upstream depth - Tailwater depth - Jump length - Roller length - Velocity |

Izadjoo and Shafai Bejestan [32] | - Horizontal bed - Two rectangular channel - Smooth and rough bed | FL = 1.2, 9 FW = 0.25, 0.5 FH = 0.4 | - Wooden baffle with trapezoidal cross section (RH: 13, 26) | 6–12 | - Upstream depth - Tailwater depth - Jump length - Velocity |

Omid Esmaeeli Varaki [28] | - Horizontal bed - Gradually expanding channel - Trapezoidal cross section - Side slope (m): 0.5:1, 1:1, 1.5:1 - Divergence angle (ϴ): 3°, 5°, 7°, 9° | FL = 10 FW = 0.5 | - Smooth Bed | 2.99–9.83 | - Upstream depth - Tailwater depth - Jump length |

Bejestan and Neisi [33] | - Horizontal bed - Rectangular channel - Smooth and rough bed | FL = 7.5 FW = 0.35 FH = 0.5 | - Lozenge shape rough element (RH: 16) | 4.5–12 | - Discharge - Upstream depth - Tailwater depth - Jump length - Roller length - Velocity |

Carollo Ferro [3] | - Horizontal bed - Rectangular channel - Smooth and rough bed | FL = 4.9 FW = 0.3 FH = 0.24 | - Crushed gravel particles cemented (d_{50}: 5.6, 9.9, 15.3, 19) | 1.7–6.9 | - Discharge - Upstream depth - Tailwater depth |

Pagliara Lotti [21] | - Horizontal bed - Rectangular channel - Smooth and rough bed | FL = 6 FW = 0.35 FH = 0.5 | - Homogeneous and non-homogeneous sediments, gravel (d _{50}: 6.26–45.6)- Rough bed withboulders, metallic | 2.2–12.2 | - Discharge - Upstream depth - Tailwater depth - Jump length - Roller length - Velocity - Water surface profile |

Wang and Chanson [34] | - Horizontal bed - Rectangular channel | FL = 3.2 FW = 0.5 FH = 0.41 | - Smooth Bed | 3.8–7.5 | - Discharge - Upstream depth - Tailwater depth - Jump length - Roller length - Velocity - Air concentrations - Bubbles frequency - interfacial velocity - turbulence intensity |

Experiments | b_{1} (m) | b (m) | b_{2} (m) | q (m^{2}/s) | r (m) | Fr_{1} | Re_{1}*10^{6} | y_{1} (m) | V_{1} (m/s) | y_{2} (m) | L_{j} (m) |
---|---|---|---|---|---|---|---|---|---|---|---|

1 | 0.5 | 0.5 | 0.5 | 0.0700–0.1169 | Smooth bed | 7.4–12.4 | 0.065–0.109 | 0.021 | 3.368–5.622 | 0.174–0.298 | 1.05–2.06 |

2 | 0.5 | 0.5 | 0.5 | 0.0701–0.1174 | 0.014 | 7.4–12.4 | 0.065–0.108 | 0.021 | 3.338–5.592 | 0.156–0.237 | 0.9–1.75 |

3 | 0.5 | 0.5 | 0.5 | 0.0705–0.1173 | 0.028 | 7.3–12.3 | 0.064–0.108 | 0.021 | 3.322–5.584 | 0.151–0.233 | 0.8–1.65 |

4 | 0.4 | 0.44–0.50 | 0.5 | 0.0619–0.1071 | Smooth bed | 6.4–11.3 | 0.056–0.097 | 0.021 | 2.946–5.098 | 0.147–0.249 | 0.8–2.3 |

5 | 0.4 | 0.433–0.493 | 0.5 | 0.0619–0.1071 | 0.014 | 6.4–11.3 | 0.056–0.097 | 0.021 | 2.948–5.101 | 0.127–0.219 | 0.65–1.85 |

6 | 0.4 | 0.428-0.48 | 0.5 | 0.0607–0.1076 | 0.028 | 6.4–11.3 | 0.055–0.097 | 0.021 | 2.889–5.125 | 0.115–0.2 | 0.57–1.6 |

7 | 0.3 | 0.393–0.50 | 0.5 | 0.0601–0.1076 | Smooth bed | 6.3–11.3 | 0.053–0.094 | 0.021 | 2.864–5.123 | 0.136–0.235 | 0.95–2.2 |

8 | 0.3 | 0.385–0.461 | 0.5 | 0.0615–0.1070 | 0.014 | 6.4–11.2 | 0.054–0.094 | 0.021 | 2.928–5.097 | 0.13–0.207 | 0.87–1.65 |

9 | 0.3 | 0.378–0.451 | 0.5 | 0.0610–0.1074 | 0.028 | 6.4–11.3 | 0.053–0.094 | 0.021 | 2.906–5.115 | 0.126–0.196 | 0.8–1.55 |

10 | 0.2 | 0.324–0.432 | 0.5 | 0.0612–0.1090 | Smooth bed | 6.3–11.3 | 0.050–0.090 | 0.021 | 2.916–5.192 | 0.136–0.221 | 0.85–1.6 |

11 | 0.2 | 0.302–0.396 | 0.5 | 0.0611–0.1084 | 0.014 | 6.4–11.2 | 0.050–0.089 | 0.021 | 2.910–5.160 | 0.129–0.193 | 0.7–1.35 |

12 | 0.2 | 0.284–0.367 | 0.5 | 0.0603–0.1070 | 0.028 | 6.4–11.3 | 0.050–0.088 | 0.021 | 2.872–5.097 | 0.119–0.183 | 0.58–1.15 |

_{1}and b are the widths of the stilling basin upstream and downstream of the hydraulic jump, b

_{2}is the width of expanded channel, q: Discharge per unit width, r: Height of the dissipative elements, Fr

_{1}: Inflow Froude number, Re

_{1}: Inflow Reynolds number, y

_{1}: Inflow depth, V

_{1}: Average velocity of upstream, y

_{2}: Sequent depth, L

_{j}: Length of the hydraulic jump.

Experiments | D% |
---|---|

B = 1, r/y_{1} = 0.67 | 16.38 |

B = 1, r/y_{1} = 1.33 | 17.88 |

B = 0.8, r/y_{1} = 0.67 | 18.69 |

B = 0.8, r/y_{1} = 1.33 | 23.56 |

B = 0.6, r/y_{1} = 0.67 | 19.59 |

B = 0.6, r/y_{1} = 1.33 | 23.67 |

B = 0.4, r/y_{1} = 0.67 | 22.92 |

B = 0.4, r/y_{1} = 1.33 | 27.28 |

**Table 4.**Values of coefficients of a, a

_{0}and b

_{0}calculated from all experimental data and obtained by previous studies.

Experimental Investigation | r (cm) | B | a | a_{0} | b_{0} |
---|---|---|---|---|---|

Present Study | 0 | 1 | 5.24 | 2.44 | 6.02 |

CarolloFerro [31] | 0 | 1 | 4.12 | 2.04 | 5.73 |

Hughes and Flack [19] | 0 | 1 | 5.06 | 2.42 | 6.58 |

Present Study | 0 | 0.8 | 5.14 | 2.48 | 5.59 |

Present Study | 0 | 0.6 | 5.10 | 2.48 | 5.2 |

Present Study | 0 | 0.4 | 5.07 | 2.48 | 4.84 |

Hughes and Flack [19] | 0.32 | 1 | 4.53 | 2.25 | 5.9 |

CarolloFerro [31] | 0.46 | 1 | 4.26 | 2.15 | 5.02 |

Hughes and Flack [19] | 0.49 | 1 | 4.66 | 2.33 | 6.00 |

Hughes and Flack [19] | 0.61 | 1 | 4.06 | 2.00 | 4.92 |

Hughes and Flack [19] | 0.64 | 1 | 4.43 | 2.22 | 5.44 |

CarolloFerro [31] | 0.82 | 1 | 3.92 | 1.98 | 4.67 |

Hughes and Flack [19] | 1.04 | 1 | 4.07 | 2.01 | 4.79 |

Present Study | 1.4 | 1 | 4.49 | 2.17 | 4.21 |

Present Study | 1.4 | 0.8 | 4.85 | 2.39 | 4.39 |

Present Study | 1.4 | 0.6 | 4.89 | 2.41 | 4.34 |

Present Study | 1.4 | 0.4 | 4.40 | 2.20 | 3.67 |

CarolloFerro [31] | 1.46 | 1 | 3.86 | 1.94 | 4.16 |

Present Study | 2.8 | 1 | 4.08 | 1.98 | 3.724 |

Present Study | 2.8 | 0.8 | 4.33 | 2.15 | 3.662 |

Present Study | 2.8 | 0.6 | 4.55 | 2.27 | 3.789 |

Present Study | 2.8 | 0.4 | 3.93 | 2.00 | 3.13 |

Number of Equation | Equation Form | R^{2} | Note |
---|---|---|---|

12 | $\frac{{y}_{2}}{{y}_{1}}=0.832\left(F{r}_{1}\right)+1.998\left(B\right)-1.250\left(\frac{r}{{y}_{1}}\right)+0.432$ | 0.95 | This equation can be used for calculating sequent depth ratio in the gradually roughened stilling basin. |

16 | $\frac{{L}_{r}}{{y}_{1}}=4.745\left(\frac{{y}_{2}}{{y}_{1}}-1\right)$ | 0.84 | This equation can be used for calculating relative roller length in the gradually roughened stilling basin. |

17 | $\frac{{L}_{r}}{{y}_{1}}=2.309{\left(\frac{{y}_{1}}{{y}_{2}}\right)}^{-1.272}$ | 0.86 | This equation can be used for calculating relative roller length in the gradually roughened stilling basin. |

18 | ${b}_{0}=1.416\mathrm{exp}\left(-79.655\frac{r}{{y}_{1}}\right)-3.99{(B)}^{2}+6.535(B)+1.442$ | 0.89 | This equation can be used for calculating the coefficient of b_{0} in Equation (12). |

19 | $\frac{{L}_{r}}{{y}_{1}}=\left\{1.416\mathrm{exp}\left(-79.655\frac{r}{{y}_{1}}\right)-3.99{(B)}^{2}+6.535(B)+1.442\right\}\left(F{r}_{1}-1\right)$ | 0.89 | This equation can be used for calculating relative roller length in the gradually roughened stilling basin. |

22 | $\frac{{L}_{j}}{{y}_{1}}=8.924\left(F{r}_{1}\right)+11.473\left(B\right)-12.390\left(\frac{r}{{y}_{1}}\right)-21.541$ | 0.89 | This equation can be used for calculating the relative length of the hydraulic jump in the gradually roughened stilling basin. |

26 | $\frac{{E}_{L}}{{E}_{1}}=0.250Ln\left(F{r}_{1}\right)-0.024{\left(B\right)}^{2}-0.023\left(B\right)+0.026\left(\frac{r}{{y}_{1}}\right)+0.244$ | 0.97 | This equation can be used for calculating the relative loss of energy in the gradually roughened stilling basin. |

© 2017 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Hassanpour, N.; Hosseinzadeh Dalir, A.; Farsadizadeh, D.; Gualtieri, C.
An Experimental Study of Hydraulic Jump in a Gradually Expanding Rectangular Stilling Basin with Roughened Bed. *Water* **2017**, *9*, 945.
https://doi.org/10.3390/w9120945

**AMA Style**

Hassanpour N, Hosseinzadeh Dalir A, Farsadizadeh D, Gualtieri C.
An Experimental Study of Hydraulic Jump in a Gradually Expanding Rectangular Stilling Basin with Roughened Bed. *Water*. 2017; 9(12):945.
https://doi.org/10.3390/w9120945

**Chicago/Turabian Style**

Hassanpour, Nasrin, Ali Hosseinzadeh Dalir, Davod Farsadizadeh, and Carlo Gualtieri.
2017. "An Experimental Study of Hydraulic Jump in a Gradually Expanding Rectangular Stilling Basin with Roughened Bed" *Water* 9, no. 12: 945.
https://doi.org/10.3390/w9120945