# The Modified One-Dimensional Hydrodynamic Model Based on the Extended Chezy Formula

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Flume Experiments and Data

^{3}/h), and the discharge in the No. 2 experiments (2-1, 2-2, 2-3, 2-4, 2-5) was in the range of 60–100 (m

^{3}/h). During the unsteady flow experiments, both the cross-sectional averaged velocity V and the flow depth D were recorded every 5 s and the observation precision was evaluated using a posteriori error inspection method. According to the principle of conservation of mass, variables at each cross section should be compatible with following water balance equation:

_{in}), B is the width of the flume, L is the length of the part ahead of the cross section, D(t) is the flow depth series, V(t) is the average velocity series, $\overline{D}$(T) is the average stage of the part ahead of the cross section at the end of experiment, and $\overline{D}$(0) is average stage of the part ahead of the cross section at the beginning of the experiment. According to Equation (4), the ∑Q

_{in}can be calculated based on the time series of D(t) and V(t), referred to as calculated accumulative discharge. Moreover, ∑Q

_{in}was measured using the pipe ultrasonic flow meter as well. Therefore, the relative error of ∑Q

_{in}in each experimental flood event can be obtained, as shown in Table 1.

## 3. Methods

#### 3.1. Extended Chezy Formula

_{c}is the friction slope computed with the Chezy formula, V is the cross-sectional averaged velocity, R is the hydraulic radius, and C is the Chezy coefficient. The relationships among Chezy’s C, Manning’s n, and the Fanning friction factor f can be written as follows:

_{1}is a function of the factors which affect both steady uniform and unsteady frictions, and F

_{2}contains the factors which only affect unsteady friction, which depends on time and space derivatives of hydraulic parameters, such as ∂D/∂x, ∂D/∂t, ∂V/∂x, and ∂V/∂t. Based on dimensional analysis, a modified friction model for unsteady flow has been proposed by Bao et al. [24]. The modified model states the following:

_{m}is the friction slope calculated by the modified friction model, in which the steady uniform component of the modified model is the same as the Chezy formula. Moreover, it considers the effect of flow non-uniformity and flow unsteadiness by additional components related to ∂D/∂x, ∂D/∂t, and ∂V/∂t. Thus, the modified friction model can be regarded as an extension of the Chezy formula, namely, the extended Chezy formula. Since it is not deduced from the momentum equation of the Saint-Venant equations, the extended Chezy formula can be recognized as a substitute for the Chezy formula or Manning formula to modify the friction calculation of a 1D hydrodynamic model.

#### 3.2. Modified SVN Model

_{0}is the bed slope, S

_{f}is the friction slope, and g is the gravitational acceleration. In this paper, the extended Chezy formula is used to compute the friction term in the momentum equation of the Saint-Venant equations. That is, the S

_{f}is computed using Equation (9). Therefore, a new version of a 1D hydrodynamic model is proposed, which is referred to as the modified SVN model.

_{1j}, pb

_{1j}, pk

_{1j}, pd

_{1j}, pe

_{1j}, pa

_{2j}, pb

_{2j}, pk

_{2j}, pd

_{2j}, and pe

_{2j}are coefficients. They are determined by the current state of the hydraulic parameters and the four parameters involved in the extended Chezy formula, (pa

_{1j}… pe

_{1j}, pa

_{2j}… pe

_{2j}) = f (q, V

_{j}, D

_{j}, V

_{j}

_{+1}, D

_{j}

_{+1}, c

_{0}, c

_{1}, c

_{2}, c

_{3}). In addition, ∆V

_{j}, ∆D

_{j}, ∆V

_{j}

_{+1}, and ∆D

_{j}

_{+1}are the increment of the next time relative to the current moment.

#### 3.3. Original SVN Model

^{2}= c

_{0}, and then obtain the classic 1D hydrodynamic model. In this paper, it is referred to as the original SVN model. Similarly, the numerical solution of the original SVN model can be derived using the Preissmann differencing method to scatter its governing equations. Obviously, the discretized equation of the continuity equation is the same as Equation (15). On the other hand, the discretized equation of the momentum equation is different from Equation (16) due to the different friction formula, which can be expressed as follows:

_{3j}, pb

_{3j}, pk

_{3j}, pd

_{3j}, and pe

_{3j}are coefficients. Their calculation formulas are the same as pa

_{2j}, pb

_{2j}, pk

_{2j}, pd

_{2j}, and pe

_{2j}in the case of c

_{1}= c

_{2}= c

_{3}= 0.

#### 3.4. A Genetic Algorithm (GA)-Based Parameter Calibration Method

_{2,i}and D

_{2,i}are the observed cross-sectional averaged velocity and flow depth at cross section #2 during the jth experiment, whereas ${V}_{2,i}^{\prime}$ and ${D}_{2,i}^{\prime}$ are the computed ones; and α

_{j}is a weighing coefficient between the cross-sectional averaged velocity and flow depth, which is determined by the relative magnitude of the cross-sectional averaged velocity and flow depth [37]. It is given by Equation (20). c

_{k}is the kth parameter to be calibrated and ${c}_{k}^{a}$ and ${c}_{k}^{b}$ are the lower bound and the upper bound, respectively. The frictional slope of the modified SVN model is computed using the extended Chezy formula and four parameters must be calibrated (p = 4), whereas the friction slope of the original SVN model is computed using the Chezy formula and only one parameter must be calibrated (p = 1):

_{2,i}and D

_{2,i}are the observed cross-sectional averaged velocity and flow depth at cross section #2.

- Generating the initial community: at the beginning of the algorithm, a population of size N is randomly generated and enters the loop.
- Calculating the fitness of parent: after decoding the genetic code of each individual, the parameters are obtained. Then, flood events are simulated using the modified or original SVN model with the parameters to get the value of the objective function. Finally, the mapped method is adopted to calculate the fitness according to the objective function result.
- Generating offspring by crossing and mutation operation: genes are selected by using the roulette method in which the selection probability is proportional to the fitness of parent individuals. In addition, a certain size of offspring population is generated by crossing and mutation, where the simplex crossover operator is used to perform the crossing operation, with the crossover probability being P
_{x}, and the mutation operator is used based on the discrete mutation operator, with the mutation probability being P_{m}. The ratio of the offspring population to parent population is GGAP, where GGAP = 0.95. Hence, the number of the offspring population is GGAP × N. - Competition restructuring operation between parent and offspring: according to the principle of survival of the fittest, GGAP × N low fitness parent individuals will be eliminated by offspring ones, that is, 5% excellent individuals of parent and all of offspring will be recombined to form a new population.
- Judging whether the termination conditions are satisfied: if the hereditary algebra is greater than the maximum number of generations M, choose the best individual decoding output in the current population as the optimal approximation solution of the original problem; otherwise, substitute the current population as a parent into the next round of cycle operation, and return to step (2).

#### 3.5. Efficiency Criteria

_{c}(i) and X

_{o}(i) represent the ith calculated value and the observed value, respectively, $\overline{{X}_{\mathrm{o}}}$ is the average observed value, and n is the length of X series. NSE indicates how well the observed values match the simulated ones and can range from −∞ to 1; larger NSE values indicate a better model performance. PX

_{c}is the calculated peak value, PX

_{o}is the observed peak value, and RPE > 0 indicates that the calculated peak value is greater than the observed one, and vice versa. PT

_{c}is the time of calculated peak occurrence, PT

_{o}is the time of observed peak occurrence, RTE is the relative time-at-peak error, and RTE > 0 indicates that the calculated peak lags behind the observed peak, and vice versa.

## 4. Results and Discussion

#### 4.1. Model Calibration

_{0}are given after using the Chezy formula. However, the lower and upper bounds of other empirical parameters in the extended Chezy formula could not be estimated due to a lack of similar applications. In this paper, they were determined using the trial and error method.

#### 4.2. Performance of the Original SVN Model

_{0}= 0.0013 (the objective function value = 0.0196), c

_{0}= 0.0018 (the objective function value = 0.0441), and c

_{0}= 0.0008 (the objective function value = 0.0448). Then, the performances of the original SVN model with different parameters were compared (see Figure 7). According to Figure 7, it can be concluded that although c

_{0}= 0.0013 could achieve the best overall model performance according to the objective function value, it was not the best parameter for each situation. For example, c

_{0}= 0.0013 was good for flow depth simulating, whereas c

_{0}= 0.0018 was better for the sectional averaged velocity simulation. Moreover, c

_{0}= 0.0013 was a better choice for the Nos. 2-1 and 2-2 flood events, whereas for the Nos. 1-1 and 1-2 flood events, c

_{0}= 0.0018 was the preferred parameter because it could improve the performance of the sectional averaged velocity simulation without hurting the agreement between observed and simulated flow depths. Hence, not only the flow depth simulation and the sectional averaged velocity simulation had preferred different parameters, but also different flood events had preferred different parameters. In such a case, it is very difficult to obtain an ideal parameter which can support good performance for both the flow depth simulation and the sectional averaged velocity simulation and obtain a sound agreement for every flood event.

#### 4.3. Performance of the Modified SVN Model

## 5. Conclusions

_{0}= 0.0013 can achieve the best overall model performance, it is not the best parameter for each situation.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Fitting curves of velocity at fixed points versus the cross-sectional averaged velocity (V−V* relationships): (

**a**) the V–V* relationship at cross section #1; (

**b**) the V–V* relationship at cross section #2; (

**c**) the V–V* relationship at cross section #3.

**Figure 4.**The relationships between cross-sectional averaged velocity and the flow depth at: (

**a**) cross section #1, (

**b**) cross section #2, and (

**c**) cross section #3 in flood event No. 1-1.

**Figure 7.**Simulation results of the original SVN model with different parameter values: (

**a1**) flow depth hydrograph and (

**a2**) sectional averaged velocity hydrograph for No. 1-1 flood event; (

**b1**) flow depth hydrograph and (

**b2**) sectional averaged velocity hydrograph for No. 1-2 flood event; (

**c1**) flow depth hydrograph and (

**c2**) sectional averaged velocity hydrograph for No. 2-1 flood event; and (

**d1**) flow depth hydrograph and (

**d2**) sectional averaged velocity hydrograph for No. 2-2 flood event.

**Figure 8.**Simulation results of the modified and original SVN models during the four flood events: (

**a1**) flow depth hydrograph and (

**a2**) sectional averaged velocity hydrograph for No. 1-1 flood event; (

**b1**) flow depth hydrograph and (

**b2**) sectional averaged velocity hydrograph for No. 1-2 flood event; (

**c1**) flow depth hydrograph and (

**c2**) sectional averaged velocity hydrograph for No. 2-1 flood event; and (d1) flow depth hydrograph and (

**d2**) sectional averaged velocity hydrograph for No. 2-2 flood event.

**Table 1.**Relative error between the observed and calculated accumulative discharges at each cross section in man-made flood events.

No. | #1 | #2 | #3 | No. | #1 | #2 | #3 |
---|---|---|---|---|---|---|---|

1-1 | 0.1% | 1.4% | 1.4% | 2-1 | 0.4% | −0.5% | 1.9% |

1-2 | 1.2% | 0.5% | 1.2% | 2-2 | 0.3% | 1.6% | 0.1% |

1-3 | −2.1% | −3.1% | −1.0% | 2-3 | 0.0% | 0.8% | −1.0% |

1-4 | −2.2% | −1.2% | −1.1% | 2-4 | −0.3% | −0.4% | 1.0% |

1-5 | −1.3% | −0.8% | −2.7% | 2-5 | −0.3% | 0.6% | 1.3% |

Parameters of GA | The Original SVN Model | The Modified SVN Model |
---|---|---|

L | 20 | 80 |

N | 40 | 40 |

Px | 0.7 | 0.7 |

Pm | 0.01 | 0.01 |

GGAP | 0.95 | 0.95 |

M | 20 | 80 |

Empirical Parameter | The Original SVN Model | The Modified SVN Model | ||||
---|---|---|---|---|---|---|

Value | Lower Bound | Upper Bound | Value | Lower Bound | Upper Bound | |

c_{0} | 0.00130 | 0 | 1 | 0.00977 | 0 | 1 |

c_{1} | - | - | - | 3.42 | 3 | 4 |

c_{2} | - | - | - | 1.12 | 1 | 2 |

c_{3} | - | - | - | −0.724 | −1 | 0 |

E(c_{0},…,c_{k}) | 0.0196 | - | - | 0.0042 | - | - |

**Table 4.**The Nash–Sutcliffe model efficiency coefficients (NSEs), relative peak value errors (RPEs), and relative time-at-peak errors (RTEs) of the original SVN model.

No. | Flow Depth | Sectional Averaged Velocity | ||||
---|---|---|---|---|---|---|

NSE | RPE (%) | RTE (%) | NSE | RPE (%) | RTE (%) | |

1-1 | 0.86 | −2.2 | 4.7 | 0.62 | −3.6 | −9.3 |

1-2 | 0.88 | −3.8 | 2.3 | 0.44 | −7.2 | 0.0 |

1-3 | 0.88 | −3.5 | 8.3 | 0.49 | −5.2 | −16.7 |

1-4 | 0.89 | −3.4 | 4.3 | 0.47 | −4.1 | −8.5 |

1-5 | 0.84 | −4.5 | 4.7 | 0.63 | −3.8 | −9.3 |

2-1 | 0.80 | −4.2 | 11.1 | 0.62 | −2.6 | −2.8 |

2-2 | 0.63 | −5.7 | 2.2 | 0.46 | −3.6 | 4.4 |

2-3 | 0.65 | −4.0 | 7.5 | 0.42 | −3.8 | 5.0 |

2-4 | 0.77 | −5.1 | 4.5 | 0.68 | −1.6 | 0.0 |

2-5 | 0.86 | −4.0 | 6.4 | 0.40 | −4.2 | 8.5 |

AAV | 0.81 | 4.0 | 5.6 | 0.52 | 4.0 | 6.5 |

No. | Flow Depth | Sectional Averaged Velocity | ||||
---|---|---|---|---|---|---|

NSE | RPE (%) | RTE (%) | NSE | RPE (%) | RTE (%) | |

1-1 | 0.97 | 0.6 | 0.0 | 0.89 | 0.78 | 4.7 |

1-2 | 0.99 | −1.7 | 0.0 | 0.84 | −1.59 | 4.5 |

1-3 | 0.99 | −0.1 | 6.3 | 0.89 | −0.53 | 2.1 |

1-4 | 0.99 | −0.5 | 0.0 | 0.86 | −0.73 | −6.4 |

1-5 | 0.98 | −0.6 | 4.7 | 0.85 | 0.40 | 0.0 |

2-1 | 0.99 | 0.0 | 2.8 | 0.73 | 0.54 | 8.3 |

2-2 | 0.91 | −4.0 | 0.0 | 0.93 | −0.01 | 8.9 |

2-3 | 0.93 | −1.7 | 0.0 | 0.86 | −0.13 | 7.5 |

2-4 | 0.88 | −4.3 | −2.3 | 0.72 | 2.69 | 4.5 |

2-5 | 0.95 | −2.8 | 0.0 | 0.88 | −0.31 | −2.1 |

AAV | 0.96 | 1.65 | 1.60 | 0.84 | 0.77 | 4.91 |

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## Share and Cite

**MDPI and ACS Style**

Zhou, J.; Bao, W.; Li, Y.; Cheng, L.; Bao, M.
The Modified One-Dimensional Hydrodynamic Model Based on the Extended Chezy Formula. *Water* **2018**, *10*, 1743.
https://doi.org/10.3390/w10121743

**AMA Style**

Zhou J, Bao W, Li Y, Cheng L, Bao M.
The Modified One-Dimensional Hydrodynamic Model Based on the Extended Chezy Formula. *Water*. 2018; 10(12):1743.
https://doi.org/10.3390/w10121743

**Chicago/Turabian Style**

Zhou, Junwei, Weimin Bao, Yu Li, Li Cheng, and Muxi Bao.
2018. "The Modified One-Dimensional Hydrodynamic Model Based on the Extended Chezy Formula" *Water* 10, no. 12: 1743.
https://doi.org/10.3390/w10121743