# A Numerical Study on Curvilinear Free Surface Flows in Venturi Flumes

## Abstract

**:**

**J0101**

## 1. Introduction

## 2. Governing Equations

_{f}denotes the friction slope, calculated from Manning’s equation or the smooth boundary resistance law; $Q$ is the discharge; $A$ is the flow cross-sectional area; $\mathsf{\rho}$ is the density of the fluid; $p$ is the pressure; $\mathsf{\beta}$ refers to the Boussinesq coefficient; $g$ is gravitational acceleration; ${\mathsf{\beta}}_{L}{q}_{L}{U}_{L}$ is the contribution of an inflow of ${q}_{L}$ volume rate per unit length, with streamwise velocity component ${U}_{L}$, and $t$ is the time.

**-**order equation for steady rapidly-varied flow problems where the effects of the sidewalls and streamline vertical curvatures are significant. It includes terms due to sidewall curvature that originate from the vertical velocity profile as a result of the breadthwise contraction and/or expansion. This implies that a higher-order correction is incorporated in Equation (8) compared to the correction that was applied to Cheng et al.’s [19] equation for the effects of the dynamic pressure. In a special case of weakly-curved free surface flow in a prismatic channel, the contributions of the products of the derivatives are insignificant compared to the derivatives themselves. Using this approximation, Equation (8) reduces to an equation structurally similar to the classical Boussinesq-type momentum equation [28] that was developed based on the assumption of irrotational flow. Nonetheless, the method presented here demonstrates that this assumption is not really required to develop such a type of flow equation.

#### Boundary Conditions

#### Inflow Boundary Condition

_{0}is the bed slope.

#### Outflow Boundary Condition

## 3. Numerical Solution Procedure

## 4. Model Verifications

#### 4.1. Curvilinear Flow in Short-Throated Flumes

#### 4.1.1. Flow in a Parshall Flume

#### 4.1.2. Flow in a Horizontal Bed Flume with Rounded Transition

#### 4.2. Flow in a Channel with Curved Sidewall and Bottom Hump

## 5. Discharge Coefficients

#### 5.1. Method

#### 5.2. Validation of the Proposed Method

## 6. Discussion

## 7. Conclusions

## Acknowledgments

## Conflicts of Interest

## Abbreviations

2D | Two-dimensional |

3D | Three-dimensional |

BLD | Boundary layer displacement |

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**Figure 5.**Free surface profiles for flow in a Venturi flume with an upstream total energy head to throat length ratio of: (

**a**) 4.38; (

**b**) 3.80.

**Figure 6.**Pressure distributions: (

**a**) at the end of the converging section; (

**b**) at the beginning of the diverging section.

**Figure 7.**Horizontal velocity profiles at the upstream inlet section and at the upstream and downstream ends of the throat section.

**Figure 9.**Free surface profiles for curvilinear flows in a non-prismatic channel with a contraction ratio of 0.6.

**Figure 11.**Comparison of the results of the present model with Khafagi’s [37] experimental data and previous results: (

**a**) expansion ratio = 1:8; (

**b**) expansion ratio = 1:6.

H_{0}/R | ${\mathsf{\omega}}_{\mathbf{1}}$ | ${\mathsf{\omega}}_{\mathbf{2}}$ | ${\mathsf{\omega}}_{\mathbf{3}}$ | ${\mathsf{\omega}}_{\mathbf{4}}$ |
---|---|---|---|---|

0.25 | 0.628 | −0.378 | 2.744 | 1.562 |

0.31 | 0.624 | −0.387 | 2.278 | 1.316 |

0.35 | 0.624 | −0.382 | 2.021 | 1.129 |

0.40 | 0.617 | −0.393 | 1.812 | 1.062 |

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Zerihun, Y.T.
A Numerical Study on Curvilinear Free Surface Flows in Venturi Flumes. *Fluids* **2016**, *1*, 21.
https://doi.org/10.3390/fluids1030021

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Zerihun YT.
A Numerical Study on Curvilinear Free Surface Flows in Venturi Flumes. *Fluids*. 2016; 1(3):21.
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**Chicago/Turabian Style**

Zerihun, Yebegaeshet T.
2016. "A Numerical Study on Curvilinear Free Surface Flows in Venturi Flumes" *Fluids* 1, no. 3: 21.
https://doi.org/10.3390/fluids1030021