# Studying Inertia Effects in Open Channel Flow Using Saint-Venant Equations

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical and Numerical Approximations

#### 2.1. Theoretical Basis

^{2}], S

_{i}is source/sink flow rate [L

^{2}/t], Q is the flow rate of the river/stream/canal [L

^{3}/t], V is the river/stream/canal velocity [L/t], h is the water depth [L], g is the gravity [L/t

^{2}], S

_{0}is the slope of the bottom elevation, and S

_{f}is the friction loss.

^{#}is a function of the water depth (h) along the river/stream/canal direction x; and B = ∂A

^{#}/∂h is the top width of the cross-section [L]. The wave speed (c) and its transformed value (ω) are expressed as

^{−1/3}], R is hydraulic radius [L] to be assigned, and H is the water stage [L]. The cross-sectional discharge calculated by the average velocity multiplied by its areas is used to substitute Equation (6) into Equation (1) to obtain

_{0}= S

_{f}) [2]. Advective transport of cross sections is expressed as

#### 2.2. Numerical Approximations

_{i}, which is the coordinate of node i, at the new time level to x

^{*}

_{i}

_{1}and x

^{*}

_{i}

_{2}, which show the location of a fictitious particle tracked backward from x

_{i}along the first and second characteristics. The derivation is illustrated in detail in the work of Shih and Yeh (2018) [10]. Tracking time can be obtained via a backward track method along their first ($\mathsf{\Delta}{\tau}_{1}$) and second ($\mathsf{\Delta}{\tau}_{2}$) respective characteristics. To ignore the eddy diffusion effects, Equations (3) and (4) can be further expressed as follows:

_{i}and ω

_{i}are the values of V and ω at x

_{i}at a new time level, respectively; V

_{i}

_{1}

^{*}and ω

_{i}

_{1}

^{*}are the values of V and ω at point x

_{i}

_{1}

^{*}, respectively; (K

_{+})

_{i}and (S

_{+})

_{i}are the values of K

_{+}and S

_{+}, respectively, at node i at a new time level; (K

_{+})

_{i}

_{1}

^{*}and (S

_{+})

_{i}

_{1}

^{*}are the values of K

_{+}and S

_{+}, respectively, at node x

_{i}

_{1}

^{*}; V

_{i}

_{2}

^{*}and ω

_{i}

_{2}

^{*}are the values of V and ω at point x

_{i}

_{2}

^{*}, respectively; (K

_{−})

_{i}and (S

_{−})

_{i}are the values of K

_{−}and S

_{−}, respectively, at node i at a new time level; and (K)

_{i}

_{2}

^{*}and (S

_{−})

_{i}

_{2}

^{*}are the values of K

_{−}and S

_{−}, respectively, at node x

_{i}

_{2}

^{*}. The primary variables are calculated through interpolations at both their new and previous time levels.

_{1}

^{(i)}and k

_{2}

^{(i)}are the two nodes of the element in which the backward tracking from node i, along the first characteristic, stops; j

_{1}

^{(i)}and j

_{2}

^{(i)}are the two nodes of the element in which the backward tracking from node i, along the second characteristic, stops; and a

_{1(i)}, a

_{2(i)}, a

_{3(i)}, a

_{4(i)}, b

_{1(i)}, b

_{2(i)}, b

_{3(i)}, and b

_{4(i)}are the interpolation parameters associated with the backtracking of the i-th node, all within the range [0,1].

_{f}= S

_{0}.

## 3. Model Calibrations

#### 3.1. Calibration of the Fully Dynamic Wave Module

_{mean}) and to enlarge error discrepancy via weighting quadratic terms (root mean square error, RMSE) is performed. The variability on relations (Pearson’s coefficient, C

^{2}), goodness-of-fit efficiency (Nash–Sutcliffe efficiency, R

^{2}), and Theil’s prediction/inequality coefficient (U

^{2}) of paired simulations and analytics are performed.

_{p}= −0.0125). The indicators of the wave propagation trend, i.e., C

^{2}, R

^{2}, and U

^{2}, display the agreement between the CAMP1DF simulation and analytical data. For the fluvial transition problem of case 4, in which the subcritical to supercritical flow with a hydraulic jump, CAMP1DF also exhibits a significant agreement with the benchmark results and precisely captures the hydraulic jump. In short, the above DYW simulation results show a near perfect match to those of the analytical solutions in all the cases. The results indicate that the CAMP1DF model can accurately consider all types of transient fluvial problems between subcritical and supercritical flows in these benchmark problems.

#### 3.2. Calibration of Diffusion Wave Module

_{mean}and RMSE indicate the presence of very minor errors, within 1.0D-3 m, for the rest of the cases. Moreover, the error indicators of C

^{2}, R

^{2}, and U

^{2}all suggest that the values from the CAMP1DF DIW and WASH123D simulations are nearly identical. Thus, the CAMP1DF simulation is demonstrated to offer the same simulation trend as the WASH123D simulations for all four open-channel flow problems. Briefly, the DIW simulation module shows a nearly perfect match to the WASH123D simulations in all the cases, where the true values are assumed. The results indicate that the CAMP1DF model can accurately implement DIW routing results of transient flow between subcritical and supercritical flows in these benchmark problems.

#### 3.3. Calibration of Kinematic Wave Module

_{0}= S

_{f}. MacDonald et al. (1997) [30] examined two types of cross-sectional geometries: a rectangular channel in cases 1 and 2 and a trapezoidal channel in cases 3 and 4. Therefore, we used Manning’s equation, expressed as follows, solved by the Newton–Raphson method to estimate the analytic solutions.

_{0}is the hydraulic gradient.

_{mean}) or quadratic errors (RMSE) but also their variabilities and prediction trends (C

^{2}, R

^{2}, and U

^{2}) revealed nearly fitting estimations of the paired CAMP1DF KIW and Newton–Raphson approximation results. In short, in all the cases, the KIW simulation module shows a near perfect match to the numerical approximation results obtained by the Newton-Raphson method. The results demonstrate that the CAMP1DF model can implement an accurate KIW routing for the nonprismatic cross-section, nonuniform bed slope, and transient flow scenarios between subcritical and supercritical flows in the benchmark problems.

## 4. Result and Discussions

#### 4.1. Discussion on Prismatic Cases

_{mean}and RMSE indicate that the inertial effects are important to subcritical routing. The flow pattern related errors, i.e., C

^{2}and R

^{2}, suggest that the acceleration term plays an important role in this example. It is also noted that the simulation time of DYW is 1.3 times that of KIW and almost as the same as the DIW routing. Therefore, using DYW to solve this case can provide both efficiency and accuracy in achieving the solution.

_{mean}and RMSE of the KIW simulations. Therefore, the upstream boundary effects play an important role in this case. The error indicators of the flow pattern, i.e., C

^{2}, R

^{2}, and U

^{2}, yield similar trends for the two modules. The DYW flow clearly provides the most accurate simulations among the three approximations; however, it also spends 1.42 times the calculation time of the KIW flow and 1.23 times that of the DIW flow. In short, these three approaches can all achieve good simulations of the subcritical and supercritical transform problems because the proposed approximation utilizes sub-element tracking with monotonic shape-preserving interpolation, and the inertial effects are not obvious in this case.

_{mean}) at slightly lower water levels obtained relative to the analytics. Error indicators of C

^{2}and R

^{2}depict the agreement between the simulations and analytics for these two approaches. The simulations are confirmed to reveal a good flow pattern in this oscillating open-channel problem. However, the water depths reveal a large discrepancy in the KIW routing. Therefore, the inertia and gravity terms are important in this wave oscillation problem.

_{mean}and RMSE indicate the DYW has smaller errors among the three options. The KIW flow has a significant error in the downstream jet, whereas the DIW flow yields an inaccurate result around the hydraulic jump. Only the DYW flow reveals a near exact agreement at the location of the hydraulic jump. Error indicators of the mean errors and biases or variabilities all reveal slightly discrepancies between the simulations and analytical derivations. Therefore, the fluid condition of a subcritical/supercritical fluid transfer with a hydraulic jump confirms that only considering the inertial effect can yield accurate results.

#### 4.2. Discussion of Nonprismatic Cases

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Schematic diagram of river illustrates in (

**a**) top view, (

**b**) longitudinal view, and (

**c**) the hydraulic section.

**Figure 2.**Illustration of the respective characteristics’ lines in one-dimensional backward tracking.

**Figure 3.**Calibration result of the DYW model for (

**a**) benchmark 1, (

**b**) benchmark 2, (

**c**) benchmark 3, and (

**d**) benchmark 4 of MacDonald et al. (1997) [30].

**Figure 4.**Calibration result of the diffusion wave (DIW) of (

**a**) benchmark 1, (

**b**) benchmark 2, (

**c**) benchmark 3, and (

**d**) benchmark 4 of MacDonald et al. (1997) [30].

**Figure 5.**Calibration result of KIW of (

**a**) benchmark 1, (

**b**) benchmark 2, (

**c**) benchmark 3, and (

**d**) benchmark 4 of MacDonald et al. (1997) [30].

**Figure 6.**Comparison between the DYW, DIW, and KIW flows simulated by CAMP1DF of (

**a**) benchmark 1, (

**b**) benchmark 2, (

**c**) benchmark 3, and (

**d**) benchmark 4 of MacDonald et al. (1997) [30].

**Table 1.**Error calculations in the CAMP1DF fully dynamic wave (DYW) simulations of the benchmark problems relative to the analytical results by MacDonald et al. (1997) [30]. RMSE: root mean square error; ε

_{mean}: mean error; U

^{2}: Theil’s prediction/inequality coefficient; C

^{2}: Pearson´s coefficient; R

^{2}: Nash–Sutcliffe efficiency; ε

_{p}: peak error.

Case 1 | Case 2 | Case 3 | Case 4 | |
---|---|---|---|---|

ε_{mean} | −0.0006 | 0.0000 | −0.0046 | −0.0052 |

ε_{p} | 0.0012 | 0.0001 | −0.0125 | −0.0004 |

RMSE | 0.0009 | 0.0001 | 0.0123 | 0.0185 |

C^{2} | 1.0000 | 1.0000 | 0.9995 | 0.9955 |

R^{2} | 1.0000 | 1.0000 | 0.9952 | 0.9949 |

U^{2} | 0.0000 | 0.0000 | 0.0001 | 0.0003 |

Case 1 | Case 2 | Case 3 | Case 4 | |
---|---|---|---|---|

ε_{mean} | 0.0000 | 0.0000 | 0.0033 | −0.0009 |

ε_{p} | 0.0000 | 0.0000 | −0.0018 | 0.0004 |

RMSE | 0.0000 | 0.0001 | 0.0051 | 0.0011 |

C^{2} | 1.0000 | 1.0000 | 0.9994 | 1.0000 |

R^{2} | 1.0000 | 1.0000 | 0.9990 | 1.0000 |

U^{2} | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

**Table 3.**Error calculations for the CAMP1DF kinematic wave (KIW) simulations of the benchmark problems and Newton–Raphson approximations.

Case 1 | Case 2 | Case 3 | Case 4 | |
---|---|---|---|---|

ε_{mean} | 0.0006 | −0.0003 | 0.0098 | 0.0003 |

ε_{p} | −0.0005 | 0.0004 | −0.0136 | −0.0008 |

RMSE | 0.0008 | 0.0004 | 0.0115 | 0.0011 |

C^{2} | 1.0000 | 1.0000 | 0.9997 | 1.0000 |

R^{2} | 1.0000 | 1.0000 | 0.9988 | 1.0000 |

U^{2} | 0.0000 | 0.0000 | 0.0001 | 0.0000 |

**Table 4.**Error calculations obtained from the CAMP1DF DYW/DIW/KIW simulations and analytical results by MacDonald et al. (1997) [30].

Case 1 | Case 2 | Case 3 | Case 4 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

DYW | DIW | KIW | DYW | DIW | KIW | DYW | DIW | KIW | DYW | DIW | KIW | |

ε_{mean} | −0.0006 | −0.0028 | 0.0025 | 0.0000 | −0.0168 | 0.0059 | −0.0046 | −0.0102 | 0.0837 | −0.0052 | 0.0135 | −0.1023 |

ε_{p} | 0.0012 | 0.0168 | −0.0279 | 0.0001 | 0.0151 | −0.0099 | −0.0125 | 0.0243 | 0.4328 | −0.0004 | 0.0000 | 0.1686 |

RMSE | 0.0009 | 0.0202 | 0.0306 | 0.0001 | 0.0203 | 0.0108 | 0.0123 | 0.0275 | 0.2580 | 0.0185 | 0.0569 | 0.1449 |

C^{2} | 1.0000 | 0.9770 | 0.9478 | 1.0000 | 0.9956 | 0.9991 | 0.9995 | 0.9873 | 0.4689 | 0.9955 | 0.9558 | 0.8912 |

R^{2} | 1.0000 | 0.9751 | 0.9426 | 1.0000 | 0.9753 | 0.9930 | 0.9952 | 0.9758 | −1.1311 | 0.9949 | 0.9518 | 0.6870 |

U^{2} | 0.0000 | 0.0005 | 0.0011 | 0.0000 | 0.0007 | 0.0002 | 0.0001 | 0.0006 | 0.0424 | 0.0003 | 0.0029 | 0.0242 |

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**MDPI and ACS Style**

Shih, D.-S.; Yeh, G.-T.
Studying Inertia Effects in Open Channel Flow Using Saint-Venant Equations. *Water* **2018**, *10*, 1652.
https://doi.org/10.3390/w10111652

**AMA Style**

Shih D-S, Yeh G-T.
Studying Inertia Effects in Open Channel Flow Using Saint-Venant Equations. *Water*. 2018; 10(11):1652.
https://doi.org/10.3390/w10111652

**Chicago/Turabian Style**

Shih, Dong-Sin, and Gour-Tsyh Yeh.
2018. "Studying Inertia Effects in Open Channel Flow Using Saint-Venant Equations" *Water* 10, no. 11: 1652.
https://doi.org/10.3390/w10111652