# Analytical Determination of Irrotational Flow Profiles in Open-Channel Transitions

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Irrotational Flow Modeling

#### 2.1. The x-ψ Method

_{b}(x), with x as the horizontal coordinate. The velocity components in Cartesian directions (x, z) are (u, w), and the total energy head of the flow is H. For irrotational and incompressible fluid flow, a stream function ψ defines the velocity components as [32,33]

^{2}+ w

^{2})

^{1/2}and the streamline inclination tanθ = w/u. In the complex potential plane (ϕ, ψ), lnV and θ are harmonic functions; they must satisfy the Laplace equation, namely

#### 2.2. Free Surface Boundary Condition

_{s}is the absolute free surface velocity, and E is the specific energy of the free surface streamline. Some former models have directly used a finite difference version of Equation (11), to be satisfied in a numerical setting [16,17,27]. The process is far from simple for flows with a critical point, as will be discussed below. Here, we will transform Equation (11) to produce a more suitable form of this challenging boundary condition.

_{c}, and

_{c}. Here

_{c}is interpreted as a critical flow depth, then Equations (17) and (18) define sub- and supercritical flow profiles, respectively. Note that the solution is analytical and exact: so far, no approximations are invoked, given that no assumptions were made on the nature of coefficient α. Therefore, the solution is exact and can be employed to determine the irrotational flow profiles in open channel transitions. If α = 1, the flow is hydrostatic, and Equations (17) and (18) represent alternate depths in the specific energy diagram of gradually varied open channel flow [31,39,40,41].

#### 2.3. Critical Point

## 3. Solution Method

_{min}(i). The maximum value of this set will satisfy this requirement at each free surface node, resulting in the new head estimation [44]

_{critical}, and a new estimation of the total energy head of the flow.

_{b}(i), the flow profile

_{critical}), and the branch

_{critical}), where

- Select a discharge q;
- Determine the critical point and total energy head H. Start assuming α(i) = 0 (initial hydrostatic flow profile);
- Compute the sub- and supercritical portions of the flow profile using Equations (28) and (29);
- For the new free surface position, solve the Laplace equation by SOR and compute a new set of coefficients α(i) at each node on the free surface;
- Compute the free surface pressure residuals. If their average is not below a tolerance, go back to 2 and start a new free surface iteration.

^{−12}is suggested. Otherwise, the computed velocity field may not be accurate enough, such as in flow over an ogee weir.

## 4. Test Cases

#### 4.1. Flow over Low Humps

^{2}/s.

^{−12}. The overrelaxation coefficient was set to 1.8. To increase stability, the new free surface position was taken as the average of the former and new estimations achieved in each iteration. Overall, after 20–25 iterations, the free surface was estimated with enough accuracy. During the solution, it was observed that small wiggles formed near the estimated critical point. On tracking the numerical solution in detail, this was produced by small oscillations of the surface velocity near the critical point. These surface velocity oscillations were produced by the Laplace solver in response to curvature discontinuities in the free surface boundary resulting when assembling Equations (28) and (29) at the critical point. An expansion of the velocity field near the free surface reveals that (u, w) are functions of the water depth derivatives h

_{x}, h

_{xx}, h

_{xxx}… [6,60]. This explains the observed behaviour during the solution procedure. A Savitzky–Golay smoothing filter was then applied to remove these oscillations at the critical point. Note from Figure 4 that the solution produced is in excellent agreement with the observations, resulting in an accurate transitional flow profile from sub- to supercritical flows. The average residual pressure on the free surface for this solution is 2 × 10

^{−4}m. The critical point is at x = 0.04 m, whereas the crest is at x = 0.02 m, i.e., there is a shift in the downstream direction, albeit small. The impact of filtering on the accuracy and reliability of results was double checked by comparing this test solution with that determined with the method by Castro-Orgaz [46], resulting in imperceptible differences. For high curvature effects, like those in the ogee crest (Section 4.3), the method by Castro-Orgaz [46] failed to produce a solution, whereas the current method produced a solution in conformity with experiments and CFD results, thereby ensuring that the filtering effects only stabilized without any impact on the physical accuracy of the solutions produced (see Section 4.3).

_{b}= 0.2exp[−0.5(x/0.24)

^{2}] (m) [49]. The unit discharge for this test is q = 0.1102 m

^{2}/s. The parameters in the numerical model were kept unaltered, and the resulting free surface and bed pressure head profiles are plotted in Figure 5. For this test, the irrotational flow solution is also in excellent agreement with observations.

#### 4.2. Flow Transition from Mild to Steep Slopes

_{c}= (q

^{2}/g)

^{1/3}. The energy head is thus known and of value H = (3/2)h

_{c}. Simulation with the irrotational flow model was accomplished using ∆x = 0.005 m and 20 streamlines to adequately capture the pressure drop in the transition zone. Further refinement of the mesh did not produce better results. For the 45° chute slope, the irrotational flow results presented in Figure 6 are in excellent agreement both for the free surface and piezometric pressure profiles for the three discharges tested. Note the non-hydrostatic bottom pressure on the chute slope. Simulations for the 60° slope presented in Figure 7 also reveal excellent free surface predictions. However, piezometric bed pressure profiles for 20 and 40 L/s show an appreciable distortion as compared to experiments. For 60 L/s, the prediction is in better agreement with the observations. The reason for the discrepancy at 20 and 40 L/s is unknown. It could be due to flow separation. However, no definite statement can be made in this regard, and it can only be said that the irrotational flow model prediction is in fair agreement with observations for the 60° chute flow test.

#### 4.3. Flow over the Spillway of a High Dam

_{D}, R = 0.2H

_{D}and R = 0.04H

_{D}) for the upstream quadrant [58,62,63], and the power equation

_{D}is the design head and z

_{b}is the bed profile elevation. The upstream face is vertical, and the chute angle is 45°. Other proposals for the upstream quadrant are a 2-circular arc profile and elliptical profile [64,65,66]. Knapp [67] and Hager [68,69] proposed continuous bottom profiles for the entire crest to avoid curvature discontinuities, such as those occurring at the apex and at the joins of the circular arcs on the standard USACE profile considered here.

_{D}= 0.5, 1, and 1.33 is considered in Figure 8, where free surface experimental data by USACE [63] is included. Simulated free surface and piezometric bottom pressure profiles are also plotted. The simulation with the irrotational flow model was accomplished using ∆x/H

_{D}= 0.02 and 20 streamlines for H/H

_{D}= 0.5. For the heads H/H

_{D}= 1 and 1.33, this mesh was adequate to simulate the flow profile, yet to obtain an accurate piezometric bottom pressure head estimation in the upstream quadrant, ∆x/H

_{D}= 0.01 and 80 streamlines had to be used. In the x-ψ mapping, streamlines cannot become vertical, given that their slope ∂y/∂x appears in the Laplacian Equation (8). It implies that, strictly, the upstream face angle of the spillway cannot be 90°. This face was thus simulated as a 2:1 slope (63.5°). Numerical simulations confirmed that it has a weak effect. It was found possible by numerical experimentation to increase the upstream face angle to almost 80°. However, this demanded too high a number of iterations in the SOR solver for an accurate solution. Results of Figure 8 confirm that the approximation adopted is accurate enough. The spillway discharge coefficient emerges as part of the computational solution, e.g.,

_{d}was fixed by a trial-and-error procedure observing if the resulting flow profile agreed with observations. Here, both C

_{d}and the flow profile are theoretical results obtained simultaneously without resorting to experimental data.

_{D}= 0.5, 1 and 1.33. Note that this is a generalization to 2D irrotational flows of open channel flow concepts used in gradually varied flow computations [32,40]. Second, Figure 9b,d,f present the initial hydrostatic flow profile considered and the final flow profile obtained. Other models for weir flow [16,27,46] demanded a highly accurate initial flow profile to ensure the convergence of a Newton–Raphson-type free surface iteration method. In our case, this was not a limitation, as confirmed by Figure 9. Note the significant displacement of the critical point from the ogee crest in all tests.

_{D}equal to 0.5, 1, and 1.33 with CFD results by Ho et al. [71], solving the Navier–Stokes equations using the industrial code in Figure 11. CFD models are an accepted tool to study spillway flow [72]. The irrotational model results are in excellent agreement with CFD results for the free surface, and in close agreement for bottom pressure head simulations.

_{D}equal to 0.5, 1, and 2 are compared with experimental data by Hager [69] in Figure 12, confirming the accuracy resolving the velocity field of the proposed ideal fluid flow model. The free surface and piezometric bottom pressure head measured by Hager [69] for H/H

_{D}= 1 and 2 are considered in Figure 13. The predictions of the irrotational flow model are included there, resulting again in an agreement with the observations. Table 1 contains the C

_{d}experimentally measured by Hager up to the high operational head H/H

_{D}≈ 3. Predicted values of C

_{d}using the irrotational flow solver are included, resulting in small deviations from measurements, in all cases within experimental uncertainty. This confirms that the present model reproduces flow over an ogee crest under a high operational head reasonably well.

## 5. Discussion of Results

_{d}coefficient is automatically determined in the solution procedure. There is no need to use a trial-and-error guessing method as was done by Cassidy [35] or Diersch et al. [22]. Others have used the experimentally obtained C

_{d}, e.g., Montes [16].

_{xx}when assembling the two portions. This impacts the boundary value problem posed by the Laplace equation, resulting in free surface velocity peaks near the critical point. Those peaks produce α peaks which are transmitted in the form of surface wiggles as water surface iterations progress. However, a simple smoothing filter eliminates the problem.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Notation

u | horizontal velocity |

w | vertical velocity |

ψ | stream function |

z | elevation |

x | horizontal coordinate |

V | modulus of velocity vector |

θ | streamline inclination |

p | pressure head |

ρ | water density |

g | gravity acceleration |

H | total energy head |

z_{s} | free surface elevation |

z_{b} | channel bed elevation |

E | specific energy |

V_{s} | free surface velocity |

q | discharge per unit channel width |

α | ratio of surface velocity to depth-averaged velocity |

h_{c} | critical depth of 2D irrotational flow |

X, a, b | auxiliar variables |

Γ | auxiliar function in specific energy |

i | mesh index |

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**Figure 1.**Irrotational open channel flows with a critical point (

**a**) flow over an ogee crest, (

**b**) transition from mild to steep slope (photos by W.H. Hager).

**Figure 4.**Flow over an asymmetrical bed profile: comparison of irrotational flow model with experiments [49].

**Figure 5.**Flow over a symmetrical bed profile: comparison of irrotational flow model with experiments [49].

**Figure 6.**(

**a**–

**c**) Flow in transition of horizontal channel to 45°: comparison of irrotational flow model with experiments [61]. The scaling used is h

_{c}= (q

^{2}/g)

^{1/3}.

**Figure 7.**(

**a**–

**c**) Flow in transition of horizontal channel to 60°: comparison of irrotational flow model with experiments [61]. The scaling used is h

_{c}= (q

^{2}/g)

^{1/3}.

**Figure 8.**Flow over an ogee crest for relative heads H/H

_{D}= (

**a**) 0.5, (

**b**) 1, and (

**c**) 1.33: comparison with USACE [63] experiments.

**Figure 9.**Flow over an ogee crest for relative heads H/H

_{D}= (

**a**) 0.5, (

**b**) 1, and (

**c**) 1.33: determination of the critical point and initial and final free surface profiles.

**Figure 10.**Flow over an ogee crest for relative heads H/H

_{D}= 0.5, 1, and 1.33: comparison of bottom pressure simulations with USACE [63] measurements.

**Figure 11.**Flow over an ogee crest for relative heads H/H

_{D}= 0.5, 1, and 1.33: comparison of (

**a**) free surface and (

**b**) bottom pressure simulations with CFD results [71].

**Figure 12.**Comparison of predicted irrotational crest velocity with experimental data [69].

**Figure 13.**Flow over an ogee crest for heads H/H

_{D}equal to 1 and 2: comparison of (

**a**) free surface and (

**b**) piezometric bottom pressure simulations with Hager’s [69] experiments.

**Table 1.**Comparison of predicted C

_{d}(H/H

_{D}) by the present irrotational flow model with experimental data for USACE 3-arc spillway by Hager [69]; H

_{D}= 0.1 m, chute angle 45°.

C_{d} | Error | ||
---|---|---|---|

H/H_{D} | Experimental | Irrotational | % |

0.5 | 0.6320 | 0.6406 | +1.36 |

1 | 0.7044 | 0.7015 | −0.41 |

1.38 | 0.7378 | 0.7328 | −0.68 |

1.88 | 0.7680 | 0.7574 | −1.38 |

2.46 | 0.7818 | 0.7784 | −0.43 |

3.06 | 0.7910 | 0.7917 | +0.09 |

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**MDPI and ACS Style**

Castro-Orgaz, O.; Hager, W.H.
Analytical Determination of Irrotational Flow Profiles in Open-Channel Transitions. *Water* **2023**, *15*, 4217.
https://doi.org/10.3390/w15244217

**AMA Style**

Castro-Orgaz O, Hager WH.
Analytical Determination of Irrotational Flow Profiles in Open-Channel Transitions. *Water*. 2023; 15(24):4217.
https://doi.org/10.3390/w15244217

**Chicago/Turabian Style**

Castro-Orgaz, Oscar, and Willi H. Hager.
2023. "Analytical Determination of Irrotational Flow Profiles in Open-Channel Transitions" *Water* 15, no. 24: 4217.
https://doi.org/10.3390/w15244217