^{1}

^{*}

^{1}

^{1}

^{1}

^{2}

^{2}

^{1}

^{1}

Open channel junctions are encountered in urban water treatment plants, irrigation and drainage canals, and natural river systems. Junctions are very important in municipal sewerage systems and river engineering. Adequate theoretical description of flow through an open channel junction is difficult because numerous variables are to be considered. Equations of junction models are based on mass and momentum or mass and energy conservation. The objective of this study is to compare two junction models for subcritical flows. In channel branches, we solve numerically the Saint-Venant hyperbolic system by combining Preissmann scheme and double sweep method. We validate our results with HEC-RAS using Nash and Sutcliffe efficiency. In junction models, equality of water stage and complete energy conservation equation from HEC-RAS are compared. Outcome of the research clearly indicates that the complete conservation energy model is more suitable in flow through junction than equality of water stage model in serious situations.

Channel junctions are area of separation or meeting of natural or artificial flow networks. They are often encountered in open channel networks of drainage systems and river systems [

The objective of the present work is to show that the energy based junction model is generally more suitable than the equality of water stage junction model and should be applied in serious situations. We so compare the equality of water stage model, to the mass and energy model for a junction system. We calculate the external boundaries of the junction by solving Saint Venant’s equations using Preissmann scheme for discretization and double sweep method for numerical solution along the channels of the network. Our calculations are validated by HEC-RAS software. Inputs of the network are simple and complex hydrographs. We then apply equality of water stage model to the junction. We then use HEC-RAS mass and energy based junction model for the same network with the same conditions. The two junction models are compared at the output of the network.

A junction system is composed by three channels at least: two upstream channels and a downstream channel (

The one-dimensional Saint-Venant equations are used to model transient open- channel flow for an incompressible homogeneous fluid. Saint-Venant model is a nonlinear hyperbolic system of two equations based on mass conservation and momentum conservation laws. The first (1a) is the continuity equation, and the second (1b) is the momentum equation.

where Q = discharge; A = cross-sectional; g = acceleration due to gravity; Z =

elevation of the water surface, S_{f} = energy slope; t = temporal coordinate and x = longitudinal coordinate. Complete Saint-Venant’s system is very popular among hydraulic engineers and hydrologists but have no analytical solutions so numerical solutions have been developed. Many numerical methods have been proposed for discretization: implicit or explicit [

For a given space and time function

where j is the space index, n the time index and

Introducing Equations (2), (3) and (4) in Equations (1a) and (1b), we obtain two nonlinear algebraic equations:

The system is then linearized around an equilibrium steady-state by Taylor series expansion using Equations (7), (8) and (9):

The following finite difference linear system is finally obtained:

where

In this paper, we use the double sweep method as applied by Preissmann and Cunge. The number of elementary operations necessary to solve the system by this method is proportional to the number of points N while standard methods of matrix inversion is proportional to

The rating curve

We eliminate

Equation (12) is written again as follows:

where:

Substitution of Equation (11) into Equation (10a) leads to:

Then replacing

Rearranging Equation (18) finally leads to:

where:

Equation (19) represents the recurrence relation at point

We start the calculations from the upstream by setting the hydrograph Q(t) for

At the downstream boundary (

To validate our results, we need to compare them with a known package. Many packages can solve unsteady flow with Saint-Venant’s equations with appropriate formulation to take into account the complexity of free surface flow. The HEC-RAS software solves the one dimensional unsteady flow equations by writing Saint-Venant equations in a general formulation for 1D flow in flood plain as follow [

Subscripts (c) is associated to channel and (f) to floodplain,

In our application the channels are rectangular and there is no flow over the banks. Thus the system of equations used by Hec-Ras becomes the same as the system of Equations (1a), (1b) described previously.

Numerical solution program of Saint-Venant used in HEC-RAS is based on the U.S. Army Corp of Engineer’s (USACE) model Unsteady Network Model. This program solves the mass conservation and momentum conservation equations with an implicit linearized system of equations using Preissman’s second order box scheme. The simultaneous system of equations generated for each time step (and iterations within a time step) are stored with a skyline matrix scheme and reduced with a direct solver developed specifically for unsteady river hydraulics by Dr. Robert Barkau. The state variables for the numerical scheme are flow and stage, which are computed and stored at each cross section. The hydraulic resistance is based on the friction slope from the empirical Manning’s equation, with several ways of modifying the roughness. Roughness can be characterized with Manning’s (n) or roughness height’s (k) (William E. F. 2003).

Numerical models junction in open channel networks are based on the mass conservation associated either to the energy conservation or to the momentum conservation [

Stream junctions can be modeled by two methods within Hec-Ras: energy conservation or momentum conservation. The energy based method (EBM) has been used here. The main assumptions of this method are [

where:

Subcritical flow calculations are performed up to the most upstream section of branch 3. The water surface at branch 1 is calculated by performing a balance of energy from station 3.0 to station 4.0. Friction losses are based on the length from station 4.0 to 3.0 and the average friction slope between the two sections (

The water surface for the downstream end of branch 2 is calculated in the same manner.

Energy losses and differences in velocity head are difficult to evaluate, so that the interior boundary conditions may simply diminish to the equality of water surface elevations (Equation (28)) associated to the macroscopic version of continuity equation (Equation (29)), as in many software such as the One Dimensional Hydrodynamic Model Environment Canada 1988; Mike 11 model Danish Hydraulic Institute 1999; and Chaudhry 1993. The equations of the model are written as follows:

The network is represented by three identical rectangular branches related by a junction (

The geometric and hydraulic characteristics of the system are given in

・ Upstream boundary conditions

At the input of the upstream branches constituting the junction, we chose a simple Henderson sinusoidal hydrograph (HS, Equation (30)), a two complex Henderson hydrographs, one with two equal peaks (HC2, Equation (31)), and another with three decreasing peaks (HC3D, Equation (32)). The corresponding hydrographs are shown in Figures 5-7.

Branch | Length (m) | Width (m) | K_{s} | Slope I |
---|---|---|---|---|

B1 | 150,000 | 120 | 50 | 0.0001 |

B2 | 150,000 | 120 | 50 | 0.0001 |

B3 | 150,000 | 120 | 50 | 0.0001 |

Θ | ||
---|---|---|

7500 | 900 | 0.999 |

with

・ Downstream boundary conditions.

For downstream boundary conditions we have chosen a steady-state calibration curve:

・ Initial condition is set by using a uniform flow with

Hydrographs calculated according to our program are compared to those given by HEC RAS package. Two kind criteria are used for comparison: local criteria (Relative Peak Error or RPE, Equation (34); Relative Volume Error or RVE, Equation (35)) and global statistical criteria (Nash-Sutcliffe coefficient, Equation (36)).

where

There are two main characteristics of flow motion in channels: translation and attenuation. In translation, shape of the hydrograph is maintained along the channel while attenuation involves the reduction of the peak flow and the change of the shape of the hydrograph. Translation is dominant in steep straight channel, and attenuation is channel with storage. The downstream hydrographs are compared here.

We first validate our program in a single branch and then introduce the junction model.

・ Flow in a single branch (B1): model validation

We first compared downstream hydrographs of branch 1 that we have calculated to that computed with HEC-RAS. Corresponding results are presented in

According to

close to unity. This shows that our program reproduces well the flow in the channel compared to Hec-Ras model for simple and complex upstream hydrographs (

・ Flow through the whole network with Junction models

We have then compared the hydrographs downstream the junction computed with the water surface equality method (EWS) to that of Hec-Ras junction method (EBM).

Hydrograph | RPE | RVE | Nash | |
---|---|---|---|---|

Double sweep/Hec-Ras | HS | 0.058 | −0.003 | 0.93 |

Double sweep/Hec-Ras | HC2 | 0.046 | −0.003 | 0.95 |

Double sweep/Hec-Ras | HC3D | 0.059 | −0.001 | 0.95 |

Hydrograph | RPE | RVE | Nash | |
---|---|---|---|---|

Double sweep + EWS/Hec-Ras + EBM | HS | −0.064 | 0.056 | 0.64 |

Double sweep + EWS/Hec-Ras + EBM | HC2 | −0.302 | 0.028 | 0.52 |

Double sweep + EWS/Hec-Ras + EBM | HC3D | −0.314 | −0.002 | 0.69 |

junction obtained with the equality of the water surface (EWS) and that obtained by the energy based model (EBM): EWS junction model overestimates the peak flow and decreases the falling limb’s times of hydrographs when compared to EBM based model. We can see in

According to

Junction in river network is represented by two kinds of model: mass and energy conservation and mass and momentum. Mass and energy conservation based models are easier to implement because they avoid solving numerically nonlinear equations. When flow is subcritical, mass and energy conservation model can be approximated by equality of water surface model as used in many packages. In this paper, we compared the equality of water surface (EWS) and the energy based model (EBM) for junction model. We solved numerically Saint- Venant equations using the four points PREISSMANN implicit scheme for discretization and the double sweep method in the channels of the junction and validated successfully the results with HEC RAS in the same conditions. Then we compared two mass and energy conservation based junction models: the equality of water surface based model (EWS) and the energy based model (EBM) use in HEC RAS software. Comparison of the patterns of the hydrographs downstream of the junction shows that EBM reproduces better dispersion and diffusion encountered in the natural flood wave propagation in river or channels. Analyzing the equations governing the two junction models, it appears that this can be due to the fact that EWS based model neglects kinetic energy and friction losses. In conclusion, although much easier to implement, the junction model based on the equality of water surface is less suitable in channel network.

Kane, S., Sambou, S., Leye, I., Diedhiou, R., Tamba, S., Cisse, M.T., Ndione, D.M. and Sane, M.L. (2017) Modeling of Unsteady Flow through Junction in Rectangular Channels: Impact of Model Junction in the Downstream Channel Hydrograph. Computational Water, Energy, and Environmental Engineering, 6, 304-319. https://doi.org/10.4236/cweee.2017.63020