# Numerical and Experimental Study of Classical Hydraulic Jump

^{*}

## Abstract

**:**

^{®}environment.

## 1. Introduction

^{6}. Moreover, the computational cost of such models is very high, because the computational time if a decent computing machine is very long. For practical, mainly civil engineering applications, shallow water modeling is much simpler to use, while the results can be acceptable. In the present work, we will show that one-dimensional modeling of a hydraulic jump is improved using the more accurate, unsteady, one-dimensional Boussinesq equations. They are solved numerically using the MacCormack and the Dissipative two-four finite difference schemes. The practicality of such a model used in applications like the design of stilling basins, has led us to choose this model for analysis. The maximum error of the flow depth is used as an iterative parameter, for achieving the steady state solution for the free surface profile and the location of a hydraulic jump, in a horizontal rectangular open channel for a wide range of Froude numbers. To apply the Boussinesq model at the downstream boundary node, the method of specified intervals was used for the evaluation of the unknown flow variables. The results are compared to experiments performed at the Laboratory of Applied Hydraulics, School of Civil Engineering, National Technical University of Athens, Greece, showing that such numerical schemes can accurately simulate the classical hydraulic jump.

## 2. Governing Equations

_{f}is the energy grade line slope, S

_{o}is the longitudinal bottom slope and g is the gravitational acceleration. E = E(x,t) is the Boussinesq term, which makes the difference if compared to the Saint Venant equations where the pressure distribution is assumed to be hydrostatic. The energy grade slope can be computed using the Manning formula in SI, u = (R

^{2/3}S

^{1/2})/n, where u is the mean over the wetted cross-section velocity, R is the hydraulic radius (cross sectional area over the wetted perimeter ratio), S = S

_{f}the energy grade slope and n the Manning friction coefficient as S

_{f}= ${\mathrm{n}}_{\mathrm{f}}^{2}$u

^{2}/R

^{4/3}. Alternatively, the Darcy–Weisbach formula could be used for computing S

_{f}, but we used the Manning formula in order to simplify calculations since the friction coefficient is kept constant for every step and the results are not affected.

## 3. Numerical Schemes

_{up}and h

_{do}respectively. A brief presentation of the numerical schemes for the discretization of Equation (1) follows.

#### 3.1. MacCormack Scheme

#### 3.2. Dissipative Two-Four Scheme

#### 3.3. Initial and Boundary Conditions

_{up}is known (obtained from the experiments downstream of the sluice gate) and the computation is performed with a Kutta-Merson method, beginning the calculations with the flow depth h

_{up}.

_{up}and h

_{do}at nodes i = 1 and i = n respectively. The mean velocity is known at i = 1 but it has to be computed at end node i = n. Apparently, the flow is assumed to be supercritical at node i = 1 and subcritical at node i = n.

#### 3.4. Courant Condition Artificial Viscosity

_{n}is the Courant number that must be less than or equal to 0.65 [24], and Δx is the constant spatial step as shown in Figure 1.

_{i}at the computational node i is calculated:

_{art}is a coefficient adjusting the amount of dissipation.

^{®}computational environment, while the input data for the numerical simulations are the channel geometry, the flow depths h

_{up}, h

_{do}and the flow rate Q.

## 4. Results

_{up}and h

_{do}respectively, and the Froude number of the flow, Fr, at the toe of the jump from the experimental measurements. The six different jump cases measured, were computed using same conditions (upstream and tailwater depths and the flow rate) and the numerical results are compared to experiments in the following paragraphs.

_{do}, alongside the dimensionless longitudinal distance of the channel, x/L, for each test case are shown in Figure 2a, Figure 3a, Figure 4a, Figure 5a, Figure 6a and Figure 7a. The measured and computed flow profiles by the three numerical algorithms are plotted together along the channel, and the significance of the Boussinesq terms (due to non-parallel streamlines) inside the region of the hydraulic jump are depicted sideways in Figure 2b, Figure 3b, Figure 4b, Figure 5b, Figure 6b and Figure 7b. Outside the jump these terms are almost zero (small enough), thus confirming the hydrostatic pressure distribution. From the numerical results, it must be noted that the flow depth increases from the vena contracta downstream of the sluice gate up to the jump, due to energy losses encountered in the governing equations. One may note that in test case 1, the dissipative behavior of the numerical results is probably due to the artificial dissipation added in the numerical scheme. In addition, from the Boussinesq term distribution along the channel, one may note that this term is not zero at the upstream end, therefore we would not expect a smooth transition in free surface from supercritical to subcritical flow. Furthermore, note that the Froude number is quite low and the hydraulic jump is characterized as weak [26], since 1.7 < Fr < 2.5, which means that it is a non-fully developed jump because of the weak surface roller with low energy loss. From these figures we conclude that the agreement between experiments and computations is satisfactory.

_{up}, where u

_{up}is the average velocity at x = 0, with the dimensionless longitudinal distance, x/L, for all three algorithms for test cases 1 and 6 respectively. It is evident that the MacCormack scheme overestimates the flow velocity at the upstream end of the channel, while the results from the other two methods are almost identical. Figure 9a,b depict the evolution of convergence criteria of the maximum absolute change in flow depth between two successive iterations, until a steady state is obtained for test case 2 using the Dissipative two-four scheme with Boussinesq terms, and test case 5 using the MacCormack scheme without Boussinesq terms respectively. Figure 10 and Figure 11 present the temporal evolution in ‘‘computational-pseudo time’’ of the free surface elevation for the Dissipative two-four scheme including the Boussinesq terms for test cases 3 and 4 respectively, both in dimensionless form. Note that the jump moves upstream until it is stabilized. Similar results have been produced for all other test cases.

^{−4}m or 5 × 10

^{−5}m, along with the closure of continuity equation for each test case and all numerical schemes are shown in Table 2, where the flow rate was computed from the depth and the average velocity over the cross section. It is evident that for all test cases the MacCormack algorithm gave the highest error in mass conservation, if compared to the other algorithms, due to the lower order of spatial accuracy.

## 5. Case Studies

^{3}/s. In Figure 12a–c we present the computed dimensionless flow depth, h/h

_{do}, versus the dimensionless distance x/L, for three Froude numbers at the vena contracta cross-section namely 2, 4 and 6, including or excluding the Boussinesq terms, using a dissipation parameter equal to 0.011. In Figure 12d we demonstrate the temporal evolution of the jump until it is stabilized, for Froude number equal to 4. In all three cases one may notice the formation of H3 and H2 types of free surface curves upstream and downstream of the jump respectively, due to energy loss. From Figure 12b,c it is evident that the hydraulic jump is much longer when computed using Boussinesq terms.

_{j}. We consider a 100 m long channel and 5 m wide (wide section, depth to width ratio smaller than one) with orthogonal cross section conveying 10 m

^{3}/s, with same dissipation factor equal to 0.011. In Figure 13a–c the computed dimensionless flow depth, h/h

_{do}, is plotted versus the dimensionless distance x/L, for three Froude numbers at the vena contracta namely 2, 4 and 6, with or without the Boussinesq terms incorporated. The flow depth at x/L = 0 is the boundary condition, and it is the same with and without Boussinesq terms, as indicated in Figure 12 and Figure 13. Near the supercritical boundary the depth deviates from that at x/L = 0, with deviation being greater when Boussinesq terms are not included. This is probably due to numerical instability from the boundary condition, also shown clearly in Figure 2b, Figure 3b, Figure 4b, Figure 5b, Figure 6b and Figure 7b, where Boussinesq terms are plotted vs x/L. The non-Boussinesq terms showed greater instability because the depth near the supercritical boundary deviates more than that with Boussinesq terms.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Experimental and Numerical Results for test case 1: (

**a**) Free surface profile; (

**b**) Boussinesq term versus distance.

**Figure 3.**Experimental and Numerical Results for test case 2: (

**a**) Free surface profile; (

**b**) Boussinesq term versus distance.

**Figure 4.**Experimental and Numerical Results for test case 3: (

**a**) Free surface profile; (

**b**) Boussinesq term versus distance.

**Figure 5.**Experimental and Numerical Results for test case 4: (

**a**) Free surface profile; (

**b**) Boussinesq term versus distance.

**Figure 6.**Experimental and Numerical Results for test case 5: (

**a**) Free surface profile; (

**b**) Boussinesq term versus distance.

**Figure 7.**Experimental and Numerical Results for test case 6: (

**a**) Free surface profile; (

**b**) Boussinesq term versus distance.

**Figure 8.**Variation of the mean velocity alongside the channel: (

**a**) for test case 1; (

**b**) for test case 6.

**Figure 9.**Evolution of the convergence criteria: (

**a**) for test case 2 with Dissipative scheme with Boussinesq terms; (

**b**) for test case 5 with MacCormack scheme without Boussinesq terms.

**Figure 10.**Temporal evolution of jump formation for test case 3 using the Dissipative two-four scheme including the Boussinesq terms.

**Figure 11.**Temporal evolution of jump formation for test case 4 using the Dissipative two-four scheme including the Boussinesq terms.

**Figure 12.**Dimensionless free surface elevation for 2 m wide channel: (

**a**) for Fr = 2; (

**b**) for Fr = 4; (

**c**) for Fr = 6; (

**d**) temporal evolution of the jump for Fr = 4.

**Figure 13.**Dimensionless free surface elevation for 5 m wide channel: (

**a**) for Fr = 2; (

**b**) for Fr = 4; (

**c**) for Fr = 6.

Test Case/Experiment | q (L/s/m) | Fr | h_{up} (m) | h_{do} (m) |
---|---|---|---|---|

1 | 24.28 | 2.44 | 0.0217 | 0.0682 |

2 | 54.32 | 3.06 | 0.0319 | 0.1335 |

3 | 24.28 | 3.40 | 0.0174 | 0.0735 |

4 | 28.72 | 4.03 | 0.0174 | 0.0788 |

5 | 28.72 | 4.48 | 0.0162 | 0.0906 |

6 | 21.72 | 5.38 | 0.0119 | 0.0895 |

Test Case | Numerical Scheme | Iterations | Maximum Mass Conservation Error (%) |
---|---|---|---|

1 | Dissipative with Boussinesq terms | 3099 | 0.39 |

Dissipative without Boussinesq terms | 3098 | 0.39 | |

MacCormack without Boussinesq terms | 3722 | 2.79 | |

2 | Dissipative with Boussinesq terms | 946 | 0.72 |

Dissipative without Boussinesq terms | 946 | 0.73 | |

MacCormack without Boussinesq terms | 1243 | 2.77 | |

3 | Dissipative with Boussinesq terms | 3375 | 0.55 |

Dissipative without Boussinesq terms | 3146 | 0.55 | |

MacCormack without Boussinesq terms | 3173 | 2.86 | |

4 | Dissipative with Boussinesq terms | 2296 | 0.78 |

Dissipative without Boussinesq terms | 1807 | 0.79 | |

MacCormack without Boussinesq terms | 2219 | 2.81 | |

5 | Dissipative with Boussinesq terms | 2218 | 0.93 |

Dissipative without Boussinesq terms | 2217 | 0.92 | |

MacCormack without Boussinesq terms | 2164 | 2.95 | |

6 | Dissipative with Boussinesq terms | 2874 | 1.08 |

Dissipative without Boussinesq terms | 2873 | 1.09 | |

MacCormack without Boussinesq terms | 2603 | 3.02 |

Fr | Scheme | Lj_{computed} (m) | Lj_{experimental} (m) | % Difference |
---|---|---|---|---|

2 | Dissipative with Boussinesq terms | 10.12 | 8.81 | 14.86 |

Dissipative without Boussinesq terms | 10.12 | 9.51 | 6.41 | |

4 | Dissipative with Boussinesq terms | 15.74 | 13.14 | 19.79 |

Dissipative without Boussinesq terms | 12.36 | 14.47 | −14.58 | |

6 | Dissipative with Boussinesq terms | 26.47 | 15.07 | 75.65 |

Dissipative without Boussinesq terms | 8.99 | 17.03 | −47.21 |

Fr | Scheme | Lj_{computed} (m) | Lj_{experimental} (m) | % Difference |
---|---|---|---|---|

2 | Dissipative with Boussinesq terms | 7.59 | 7.02 | 8.11 |

Dissipative without Boussinesq terms | 7.59 | 7.56 | 0.40 | |

4 | Dissipative with Boussinesq terms | 8.99 | 8.41 | 6.90 |

Dissipative without Boussinesq terms | 7.87 | 9.69 | −18.78 | |

6 | Dissipative with Boussinesq terms | 9.18 | 10.74 | −14.52 |

Dissipative without Boussinesq terms | 6.74 | 12.74 | −47.10 |

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**MDPI and ACS Style**

Retsinis, E.; Papanicolaou, P.
Numerical and Experimental Study of Classical Hydraulic Jump. *Water* **2020**, *12*, 1766.
https://doi.org/10.3390/w12061766

**AMA Style**

Retsinis E, Papanicolaou P.
Numerical and Experimental Study of Classical Hydraulic Jump. *Water*. 2020; 12(6):1766.
https://doi.org/10.3390/w12061766

**Chicago/Turabian Style**

Retsinis, Eugene, and Panayiotis Papanicolaou.
2020. "Numerical and Experimental Study of Classical Hydraulic Jump" *Water* 12, no. 6: 1766.
https://doi.org/10.3390/w12061766