# Application of Numerical and Experimental Modeling to Improve the Efficiency of Parshall Flumes: A Review of the State-of-the-Art

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Parshall Measuring Flume

#### 2.1. Types of Parshall Flume

#### 2.2. Rating Equations for Parshall Flume

## 3. Numerical Modeling in CFD

#### 3.1. Governing Equations

#### 3.2. Equation of the Free Surface

## 4. Literature Review

#### 4.1. Numerical Studies

#### 4.2. Experimental Studies

^{3}/s, they proposed a new empirical formula that provides the depth–discharge relationship. They reported that this equation is suitable for simple and small Parshall Flumes under free flow conditions. In their paper, four different sizes of Parshall flumes, having different throat widths were fabricated and tested in the laboratory under free-flow condition. The coefficient of discharge and exponent were determined and a single equation for the different flume sizes was developed. The relationship is simple and suitable to use for small Parshall Flumes.

## 5. Discussion

## 6. Conclusions

- The application of numerical modeling is considered to be a reliable technique to provide fast and accurate results with minimum cost in terms of time and money.
- The efficiencies of various Computational Fluid Dynamic (CFD) software programs have been well described by different scholars who demonstrated the accuracy of results of their numerical models. The reliability of these models made their application popular among researchers and engineers who have access to suitable computer processors. The freedom to alter, as needed, the geometry of the modeled hydraulic structure adds to the popularity of numerical modeling.
- Laboratory tests are useful in conjunction with the results of the numerical models, as the accuracy of the CFD models’ output can be determined via the data obtained physically on the same setup. It is important to calibrate the mathematical relationship that was used for the Parshall flume to provide the flowrate since the entrance condition and upstream flow type directly affect the precision of the height–discharge relationship. Nevertheless, choosing inappropriate equations and constants with respect to the type of flow and hydraulic jump, whether it is submerged or not, can trigger inaccurate flowrate results.
- Due to the accessibility of testing facilities to the majority of engineers and researchers, most of the time, building and testing methods for small scale models are not as expensive as for large projects. However, for larger projects, numerical models are typically less expensive than physical models. Additionally, the application of hydraulic structures, i.e., large scale Parshall flumes, requires the consideration of different scenarios to assess the tolerance of the structure under unforeseen harsh conditions. For example, choosing a Parshall flume as a measuring device for a wide-open channel requires extensive study on the hydrology of the region to understand the extreme weather conditions over the life span of the structure. The return period should be selected based on the codes and regulations that are available in various versions from different municipalities.
- Since having access to high performance processors is essential to run numerical models, in some articles, it was highlighted that to tackle the deficit of limited access to proper computer hardware, the proposed structures were usually designed with a symmetrical shape. Therefore, only one side of the symmetrical structure was modeled to deal with the limited computational resources. Although this method provides reasonable results, the reliability of this technique requires greater validation, particularly for LES and DES simulations, since the behavior of the fluid in motion is highly turbulent.
- Based on the review of numerous articles, for further study, the use of CFD models is recommended to perform numerical modeling on various sizes of Parshall flumes since there is a gap in the current state of application of the existing numerical modeling for this type of flume. A large number of turbulence models exist that have not yet been used to simulate the flow in Parshall flumes.
- Parshall flume is a flow measuring device that does not require energy to operate. The only vital energy to operate is taken from the water flow. As was explained earlier in this paper, the primary water level at throat section is used to calculate the discharge when there is free flow. The water level in the converging section determines the degree of submergence when submerged flow is experienced. These water level values are directly used to determine the flowrate using the rating equation explained in Section 2.2. Using the flow energy to convert the water level into discharge is highly useful in Parshall flumes. Therefore, the authors advocate that such systems are used to retrofit the open channels in agricultural section, wastewater treatment plants, and also encourage different industrial sectors to substitute the current electrical flowmeters with Parshall flumes whenever feasible and applicable.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Cone, V.M. The Venturi Flume; U.S. Government Printing Office: Washington, DC, USA, 1917.
- 20-Foot Concrete Parshall Flume with Radius Wing Walls. Available online: https://www.openchannelflow.com/assets/uploads/media/_large/20-foot-parshall-flume-curved-wing-walls.jpg (accessed on 12 January 2021).
- Fiberglass 6-Inch Parshall Flume with Gauge. Available online: https://www.openchannelflow.com/assets/uploads/media/_large/flume-parshall-6-inch-fiberglass.png (accessed on 12 January 2021).
- Parshall, R.L. The Parshall Measuring Flume; Colorado State College, Colorado Experiment Station: Fort Collins, CO, USA, 1936. [Google Scholar]
- Selecting Between a Weir and a Flume. 2022. Available online: https://www.openchannelflow.com/blog/selecting-a-primary-device-part-1-choosing-between-a-weir-and-a-flume (accessed on 29 December 2021).
- Parshall, R.L. The Improved Venturi Flume. Trans. Am. Soc. Civ. Eng.
**1928**, 89, 841–851. [Google Scholar] [CrossRef] - Heyrani, M.; Mohammadian, A.; Nistor, I. Numerical Simulation of Flow in Parshall Flume Using Selected Nonlinear Turbulence Models. Hydrology
**2021**, 8, 151. [Google Scholar] [CrossRef] - Heyrani, M.; Mohammadian, A.; Nistor, I.; Dursun, O.F. Numerical Modeling of Venturi Flume. Hydrology
**2021**, 8, 27. [Google Scholar] [CrossRef] - Alfonsi, G. Reynolds-Averaged Navier–Stokes Equations for Turbulence Modeling. Appl. Mech. Rev.
**2009**, 62, 040802. [Google Scholar] [CrossRef] - Imanian, H.; Mohammadian, A. Numerical Simulation of Flow over Ogee Crested Spillways under High Hydraulic Head Ratio. Eng. Appl. Comput. Fluid Mech.
**2019**, 13, 983–1000. [Google Scholar] [CrossRef][Green Version] - Khosronejad, A.; Herb, W.; Sotiropoulos, F.; Kang, S.; Yang, X. Assessment of Parshall Flumes for Discharge Measurement of Open-Channel Flows: A Comparative Numerical and Field Case Study. Measurement
**2020**, 167, 108292. [Google Scholar] [CrossRef] - Dursun, O.F. An Experimental Investigation of the Aeration Performance of Parshall Flume and Venturi Flumes. KSCE J. Civ. Eng.
**2016**, 20, 943–950. [Google Scholar] [CrossRef] - Shih, T.-H.; Liu, N.-S.; Chen, K.-H. A Non-Linear k-Epsilon Model for Turbulent Shear Flows. In Proceedings of the 34th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, Cleveland, OH, USA, 13 July 1998; p. 3983. [Google Scholar]
- Lien, F.S. Low-Reynolds-Number Eddy-Viscosity Modelling Based on Non-Linear Stress-Strain/Vorticity Relations. In Proceedings of the 3rd Symposium on Engineering Turbulence Modelling and Measurement, Heraklion, Greece, 27 May 1996. [Google Scholar]
- Davis, R.W.; Deutsch, S. A Numerical-Experimental Study of Parhall Flumes. J. Hydraul. Res.
**1980**, 18, 135–152. [Google Scholar] [CrossRef] - Xiao, Y.; Wang, W.; Hu, X.; Zhou, Y. Experimental and Numerical Research on Portable Short-Throat Flume in the Field. Flow Meas. Instrum.
**2016**, 47, 54–61. [Google Scholar] [CrossRef] - Wright, S.J.; Tullis, B.P.; Long, T.M. Recalibration of Parshall Flumes at Low Discharges. J. Irrig. Drain. Eng.
**1994**, 120, 348–362. [Google Scholar] [CrossRef] - Heiner, B.; Barfuss, S.L. Parshall Flume Discharge Corrections: Wall Staff Gauge and Centerline Measurements. J. Irrig. Drain. Eng.
**2011**, 137, 779–792. [Google Scholar] [CrossRef] - Savage, B.M.; Heiner, B.; Barfuss, S. Parshall Flume Discharge Correction Coefficients through Modelling. Proc. ICE Water Manag.
**2013**, 167, 279–287. [Google Scholar] [CrossRef] - Zerihun, Y.T. A Numerical Study on Curvilinear Free Surface Flows in Venturi Flumes. Fluids
**2016**, 1, 21. [Google Scholar] [CrossRef] - Sun, B.; Zhu, S.; Yang, L.; Liu, Q.; Zhang, C.; Zhang, J. ping Experimental and Numerical Investigation of Flow Measurement Mechanism and Hydraulic Performance on Curved Flume in Rectangular Channel. Arab. J. Sci. Eng.
**2020**. [Google Scholar] [CrossRef] - Hu, H.; Huang, J.; Qian, Z.; Huai, W.; Yu, G. Hydraulic Analysis of Parabolic Flume for Flow Measurement. Flow Meas. Instrum.
**2014**, 37, 54–64. [Google Scholar] [CrossRef] - Sun, B.; Yang, L.; Zhu, S.; Liu, Q.; Wang, C.; Zhang, C. Study on the Applicability of Four Flumes in Small Rectangular Channels. Flow Meas. Instrum.
**2021**, 80, 101967. [Google Scholar] [CrossRef] - Willeitner, R.P.; Barfuss, S.L.; Johnson, M.C. Using Numerical Modeling to Correct Flow Rates for Submerged Montana Flumes. J. Irrig. Drain. Eng.
**2013**, 139, 586–592. [Google Scholar] [CrossRef] - Ran, D.; Wang, W.; Hu, X. Three-Dimensional Numerical Simulation of Flow in Trapezoidal Cutthroat Flumes Based on FLOW-3D. Front. Agric. Sci. Eng.
**2018**, 5, 168–176. [Google Scholar] [CrossRef][Green Version] - Kim, S.-Y.; Lee, J.-H.; Hong, N.-K.; Lee, S.-O. Numerical Simulation for Determining Scale of Parshall Flume. Proc. Korea Water Resour. Assoc. Conf.
**2010**, 719–723. [Google Scholar] - Tekade, S.A.; Vasudeo, A.D.; Ghare, A.D.; Ingle, R.N. Measurement of Flow in Supercritical Flow Regime Using Cutthroat Flumes. Sadhana
**2016**, 41, 265–272. [Google Scholar] [CrossRef] - Wahl, T.L.; Replogle, J.A.; Wahlin, B.T.; Higgs, J.A. New Developments in Design and Application of Long-Throated Flumes. In Proceedings of the Joint Conference on Water Resource Engineering and Water Resources Planning and Management, Minneapolis, MN, USA, 30 July–2 August 2000. [Google Scholar]
- Howes, D.J.; Burt, C.M.; Sanders, B.F. Subcritical Contraction for Improved Open-Channel Flow Measurement Accuracy with an Upward-Looking ADVM. J. Irrig. Drain. Eng.
**2010**, 136, 617–626. [Google Scholar] [CrossRef][Green Version] - Tiwari, N.K.; Sihag, P. Prediction of Oxygen Transfer at Modified Parshall Flumes Using Regression Models. ISH J. Hydraul. Eng.
**2020**, 26, 209–220. [Google Scholar] [CrossRef] - Thornton, C.I.; Smith, B.A.; Abt, S.R.; Robeson, M.D. Supercritical Flow Measurement Using a Small Parshall Flume. J. Irrig. Drain. Eng.
**2009**, 135, 683–692. [Google Scholar] [CrossRef] - Cox, A.L.; Thornton, C.I.; Abt, S.R. Supercritical Flow Measurement Using a Large Parshall Flume. J. Irrig. Drain. Eng.
**2013**, 139, 655–662. [Google Scholar] [CrossRef] - Ribeiro, Á.S.; Sousa, J.A.; Simões, C.; Martins, L.L.; Dias, L.; Mendes, R.; Martins, C. Parshall Flumes Flow Rate Uncertainty Including Contributions of the Model Parameters and Correlation Effects. Meas. Sens.
**2021**, 18, 100108. [Google Scholar] [CrossRef] - Singh, J.; Mittal, S.K.; Tiwari, H.L. Discharge Relation for Small Parshall Flume in Free Flow Condition. Int. J. Res. Eng. Technol.
**2014**, 3, 317–321. [Google Scholar] - Kim, S.-D.; Lee, H.-J.; Oh, B.-D. Investigation on Application of Parshall Flume for Flow Measurement of Low-Flow Season in Korea. Meas. Sci. Rev.
**2010**, 10, 111. [Google Scholar] [CrossRef] - Willeitner, R.P.; Barfuss, S.L.; Johnson, M.C. Montana Flume Flow Corrections under Submerged Flow. J. Irrig. Drain. Eng.
**2012**, 138, 685–689. [Google Scholar] [CrossRef][Green Version] - Dufresne, M.; Vazquez, J. Head–Discharge Relationship of Venturi Flumes: From Long to Short Throats. J. Hydraul. Res.
**2013**, 51, 465–468. [Google Scholar] [CrossRef]

**Figure 1.**Parshall flume measuring structure, installed [2].

**Figure 2.**Parshall flume measuring structure, uninstalled [3].

**Figure 3.**Sketch of Parshall Flume basic design (

**a**) is top view and (

**b**) is cross sectional side view.

**Figure 4.**Mesh sensitivity analysis: top view and side view of the Parshall flume: (

**a**) contains 27,000 cells; (

**b**) 52,000 cells; (

**c**) 75,000 cells; (

**d**) 270,000 cells. The C setup was used in their simulation [7].

**Figure 5.**Basic Illustration of VoF method (Reprinted with permission from Ref. [11]. 2020 ELSEVIER).

**Figure 6.**Comparison between numerical data and experimental results [8].

**Figure 7.**The simulated velocity (

**a**) and simulated pressure pattern (

**b**) across the Parshall flume. The patterns match the physical behavior of actual Parshall flumes [7].

**Figure 8.**Computational grid system in the Side A flume. (

**a**) contains a triangular grid system (

**b**) demonstrates the rectangular grid system. (

**c**,

**d**) are three-dimensional schematics showing the superimposed grid system. (

**e**) magnifies the dashed section in (

**b**). (Reprinted with permission from Ref. [11]. 2020 ELSEVIER). ).

**Figure 9.**Consistency in the simulated data and physical results [20].

**Figure 10.**The results of flow patterns in different flumes; (

**a**) Cutthroat flume, (

**b**) airfoil-shaped flume, (

**c**) airfoil pillar-shaped flume, (

**d**) optimized airfoil-shaped flume [23].

**Figure 11.**Experimental setup: contraction ratio used on each flume [23].

**Figure 12.**Entire flume geometry [25].

Categories | Article # | CFD | Turbulence Model | Flume Type | Standard/Nonstandard | Free Surface | Wall Function | Submergence | Flowrate |
---|---|---|---|---|---|---|---|---|---|

Numerical | [7] | OpenFOAM | SQ/LC/V2-f | Parshall | Nonstandard | VOF method | Standard | Unsubmerged | 20 L/s |

[8] | OpenFOAM | 7 Turbulence Models | Parshall | Nonstandard | VOF method | Standard | Unsubmerged | 20–30 L/s | |

[15] | SOLA-FLUMP | UNSPECIFIED | Parshall | Nonstandard | UNSPECIFIED | UNSPECIFIED | Unsubmerged | UNSPECIFIED | |

[20] | 1D-flow Simulation model | UNSPECIFIED | Short-throat | Standard | UNSPECIFIED | UNSPECIFIED | UNSPECIFIED | UNSPECIFIED | |

[11] | VFS-Geophysics code | UNSPECIFIED | Parshall | Standard | VOF method | UNSPECIFIED | UNSPECIFIED | UNSPECIFIED | |

[21] | Ansys Fluent | Standard K-Epsilon | Curved | Nonstandard | UNSPECIFIED | UNSPECIFIED | Submerged | 5–27.19 L/s | |

[16] | FLOW-3D | RNG Kepsilon | Short-throat | UNSPECIFIED | UNSPECIFIED | UNSPECIFIED | UNSPECIFIED | UNSPECIFIED | |

[19] | FLOW-3D | RNG Kepsilon | Parshall | Nonstandard | VOF method | FAVOR method | UNSPECIFIED | UNSPECIFIED | |

[17] | UNSPECIFIED | UNSPECIFIED | Parshall | Standard | UNSPECIFIED | UNSPECIFIED | Submerged | UNSPECIFIED | |

[23] | Ansys Fluent | Standard KEpsilon | Cutthroat | Nonstandard | VOF method | UNSPECIFIED | UNSPECIFIED | 4.8–27.15 L/s | |

[25] | FLOW-3D | RNG Kepsilon | New design | Nonstandard | TruVOF method | FAVOR method | UNSPECIFIED | up to 75 L/s | |

[24] | FLOW-3D | RNG Kepsilon | Montana | Standard | VOF method | FAVOR method | Submerged | UNSPECIFIED | |

[26] | FLOW-3D | UNSPECIFIED | Parshall | Nonstandard | UNSPECIFIED | UNSPECIFIED | UNSPECIFIED | UNSPECIFIED | |

[22] | FLOW-3D | Standard KEpsilon | Parshall | Standard | VOF method | No-Slip | UNSPECIFIED | 40 L/s | |

[18] | FLOW-3D | UNSPECIFIED | Parshall | Nonstandard | Yes | Yes | UNSPECIFIED | UNSPECIFIED | |

[29] | FLOW-3D | UNSPECIFIED | Contraction flume | Nonstandard | UNSPECIFIED | UNSPECIFIED | UNSPECIFIED | UNSPECIFIED | |

[28] | FLOW-3D | UNSPECIFIED | Modified Parshall | Nonstandard | UNSPECIFIED | UNSPECIFIED | UNSPECIFIED | UNSPECIFIED | |

[30] | Regression Analysis | UNSPECIFIED | Contraction flume | Nonstandard | UNSPECIFIED | UNSPECIFIED | UNSPECIFIED | UNSPECIFIED |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Heyrani, M.; Mohammadian, A.; Nistor, I.; Dursun, O.F.
Application of Numerical and Experimental Modeling to Improve the Efficiency of Parshall Flumes: A Review of the State-of-the-Art. *Hydrology* **2022**, *9*, 26.
https://doi.org/10.3390/hydrology9020026

**AMA Style**

Heyrani M, Mohammadian A, Nistor I, Dursun OF.
Application of Numerical and Experimental Modeling to Improve the Efficiency of Parshall Flumes: A Review of the State-of-the-Art. *Hydrology*. 2022; 9(2):26.
https://doi.org/10.3390/hydrology9020026

**Chicago/Turabian Style**

Heyrani, Mehdi, Abdolmajid Mohammadian, Ioan Nistor, and Omerul Faruk Dursun.
2022. "Application of Numerical and Experimental Modeling to Improve the Efficiency of Parshall Flumes: A Review of the State-of-the-Art" *Hydrology* 9, no. 2: 26.
https://doi.org/10.3390/hydrology9020026