# Numerical Analysis on Hydraulic Characteristics of U-shaped Channel of Various Trapezoidal Cross-Sections

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Model

^{2}and the flow rate, 0.038 m

^{3}/s, remained constant. And constant equivalent width (B

_{0}) is 0.4 m, which was defined in Equation (1).

**m**) is the ratio of the projection of the wall on the horizontal plane to the vertical plane, as shown in Figure 1d. In order to distinguish the slope coefficient (

**m**) from the unit meter (m), the slope coefficient (

**m**) was shown in bold. The radius-to-width ratio ($\lambda $) can be defined as follows:

_{0}is equivalent width.

_{1}and R

_{2}are the inner radius and outer radius of the bed in bend, respectively.

_{up}is the width of the surface and

**m**is slope coefficient.

_{0}) to the inlet water depth (H) of U-shaped channel, remained 1.97 in this study. The numerical simulations were performed for four radius-to-width ratios ($\lambda $) of 1.0, 1.5, 2.0 and 3.0 and seven slope coefficients (

**m**) of 0, 0.25, 0.5, 0.75, 1.0, 1.10 and 1.24. The test runs program were summarized in Table 1. The m

_{1}and m

_{2}of case9 were 0.5 and 0, respectively and the m

_{1}and m

_{2}of case10 were 0.5 and 0.5, respectively. For convenience, the slope coefficients (

**m**) of case9 and case10 were calculated by the definition (shown in Figure 1d) and the values were 1.10 and 1.24, respectively.

#### 2.1. Numerical Methods

^{+}values required in the turbulence model. The number of the single trapezoidal U-shaped channel were 20 × 30 × 300, whereas for the compound channel, were 25 × 35 × 300. The number of volume cells for ten tests investigated in the paper were listed in Table 1.

#### 2.2. Governing Equations

#### 2.3. Boundary Conditions and Assumptions

#### 2.4. Model Verification

_{ave}).

## 3. Results and Discussion

#### 3.1. Analysis of the Water Surface Transverse Slope in Bend Apex

#### 3.2. Analysis of the Longitudinal Velocity

_{ave}are the longitudinal velocity and the corresponding average in the cross-section. The vertical distributions of the longitudinal velocity calculated by Equation (22) were shown in Figure 9 and Figure 10.

_{ave}). The velocity locating close to the sidewall at 5%B

_{0}was extracted to represent the velocity at the sidewall, as the velocity at the sidewall is zero.

#### 3.3. Analysis of the Secondary Flow

**m**= 0 is consistent with the investigation of Ma [24], and Vaghefi et al. [39] also presented that the maximum intensity of secondary flow appears at approximately 60°~80° via various analysis methods, which demonstrated good agreement with the above conclusion.

#### 3.4. Analysis of the Shear Stress

_{1}is 0.19 [43], $\rho $ is corresponding density and ${u}^{\prime},{v}^{\prime}and{\omega}^{\prime}$ are the streamwise, spanwise and vertical fluctuating velocity, respectively.

#### 3.5. Analysis of the TKE

_{0}was extracted to represent the sidewall. The turbulent kinetic energy was nondimensionalized via the mean turbulent kinetic energy (TKE

_{ave}).

## 4. Conclusions

- (1)
- The transverse slope decreased with increasing radius-to-width ratios and increased with increasing slope coefficients. The effect of the radius-to-width ratio on the WTS in the bend apex was greater than that of the slope coefficient.
- (2)
- A new equation concerning the vertical distribution of the longitudinal velocity in trapezoidal cross-section channel was proposed. The influence of the slope coefficient on the distribution of the longitudinal velocity was higher than that of the radius-to-width ratio.
- (3)
- The slope coefficient had an effect on the structure of secondary flow in the bend. When the slope coefficient was large enough (m = 0.75 in the present study), obvious double vortexes were captured in the bend apex. The corresponding angle of the maximum intensity of secondary flow in the bend gradually moved to the upstream with increasing radius-to-width ratios. The intensity of secondary flow in the bend gradually decreased with the increase in slope coefficients.
- (4)
- The maximum of the shear stress of the bend was observed in the convex bank, which decreased gradually and the corresponding angle decreased with increasing radius-to-width ratios. As the slope coefficient increased, the maximum of the shear stress in the convex bank gradually increased.
- (5)
- The maximum TKE of the section in the bend decreased with increasing radius-to-width ratios and increased with increasing slope coefficients.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**U-shaped channel and investigated cross-sections. (

**a**) 3D view of the U-shaped channel; (

**b**) vertical view of the channel; (

**c**) geometry of the cross-section; (

**d**) Definition of the slope coefficient.

**Figure 3.**Discretization error bars computed using the GCI index for different grid densities, (

**a**) the concave bank, (

**b**)center line and (

**c**)convex bank of the bend apex.

**Figure 4.**Comparison of dimensionless longitudinal velocity of various cross-sections between the numerical simulation and experiment, (

**a**–

**c**) the concave bank, center line and convex bank of the inlet of the bend, respectively; (

**d**–

**f**) the concave bank, center line and convex bank of the bend apex, respectively; and (

**g**–

**i**) the concave bank, center line and convex bank of the outlet of the bend, respectively.

**Figure 5.**Comparison of the secondary flow between the numerical simulation and experiment, (

**a**) experiment [24], (

**b**) numerical simulation L = 6 m, (

**c**) numerical simulation L = 4 m.

**Figure 6.**Comparison of the shear stress between the numerical simulation and experiment, (

**a**) experiment [24], (

**b**) numerical simulation L = 6 m, (

**c**) numerical simulation L = 4 m.

**Figure 7.**The free surface level across the across-section in the bend apex (C90), (

**a**) the radius-to-width ratios ($\lambda $) of 1.0, 1.5, 2.0 and 3.0, (

**b**) the slope coefficients (

**m**) of 0, 0.25, 0.5, 0.75, 1.0, 1.10 and 1.24.

**Figure 8.**The WTS-BA (C90), (a) the radius-to-width ratios ($\lambda $) of 1.0, 1.5, 2.0 and 3.0; (

**b**) the slope coefficients (

**m**) of 0, 0.25, 0.5, 0.75, 1.0, 1.10 and 1.24.

**Figure 9.**Vertical distribution of the dimensionless longitudinal velocity of the bend in different positions with varying radius-to-width ratios, (

**a**–c) the concave bank, centerline and convex bank of the inlet of the bend, respectively; (

**d**–

**f**) the concave bank, centerline and convex bank of the bend apex, respectively; (

**g**–

**i**) the concave bank, centerline and convex bank of the outlet of the bend, respectively.

**Figure 10.**Vertical distribution of the dimensionless longitudinal velocity of the curved channel in different positions with varying slope coefficients, (

**a**–

**c**) the concave bank, center line and convex bank of the inlet of the bend, respectively; (

**d**–

**f**) the concave bank, center line and convex bank of the bend apex, respectively; (

**g**–

**i**) the concave bank, center line and convex bank of the outlet of the bend, respectively.

**Figure 11.**Structure of secondary flow of the cross-section in the bend apex for varying operating conditions, (

**a**–

**j**) case1–case10, respectively.

**Figure 12.**Graph of the dimensionless intensity of secondary flow for varying radius-to-width ratios and slope coefficients, (

**a**) radius-to-width ratio, (

**b**) slope coefficient.

**Figure 13.**Shear stress distribution of the riverbed and sidewall for varying operating conditions, (

**a**)–(

**j**) case1–case10, respectively.

**Figure 14.**Maximum shear stress for varying operating conditions, (

**a**) radius-to-width ratio, (

**b**) slope coefficient.

**Figure 15.**Corresponding angle of the maximum shear stress for varying operating conditions, (

**a**) radius-to-width ratio, (

**b**) slope coefficient.

**Figure 16.**Graphs of the dimensionless TKE with the radius-to-width ratios and slope coefficients in different positions of the bend, (

**a**,

**b**) the inlet of the bend; (

**c**,

**d**) the apex of the bend; (

**e**,

**f**) the outlet of the bend.

**Figure 17.**Relation graphs of the change extent of the dimensionless TKE with radius-to-width ratios, (

**a**) the difference between inlet and apex of the bend, (

**b**) the difference between apex and outlet of the bend.

**Figure 18.**Relation graphs of the change extent of the dimensionless TKE with slope coefficients, (

**a**) the difference between inlet and apex of the bend, (

**b**) the difference between apex and outlet of the bend.

**Table 1.**Summary of the test runs for the simulation, where

**m**is slope coefficient; $\lambda $is radius-to-width ratio; θ is the central angle of the bend; B and B

_{0}are the width of the bed and equivalent width; R, R

_{1}and R

_{2}are centerline radius, inner radius and outer radius of the bend, respectively; Q is the flow rate; Fr is Froude number; Re is Reynolds number; B

_{0}/H is the aspect ratio; and cells are the volume elements.

Series | m | $\mathit{\lambda}$ | $\mathit{\theta}$ | B | R | R_{1} | R_{2} | Q | B_{0} | Fr | Re | B_{0}/H | Cells |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

(°) | (m) | (m) | (m) | (m) | (L s^{−1}) | (m) | |||||||

case1 | 0.50 | 1.0 | 180 | 0.30 | 0.40 | 0.25 | 0.55 | 38 | 0.40 | 0.33 | $4.98\times {10}^{4}$ | 1.73 | 180,000 |

case2 | 0.50 | 1.5 | 180 | 0.30 | 0.60 | 0.45 | 0.75 | 38 | 0.40 | 0.33 | $4.98\times {10}^{4}$ | 1.73 | 180,000 |

case3 | 0.50 | 2.0 | 180 | 0.30 | 0.80 | 0.65 | 0.95 | 38 | 0.40 | 0.33 | $4.98\times {10}^{4}$ | 1.73 | 180,000 |

case4 | 0.50 | 3.0 | 180 | 0.30 | 1.20 | 1.05 | 1.35 | 38 | 0.40 | 0.33 | $4.98\times {10}^{4}$ | 1.73 | 180,000 |

case5 | 0.00 | 2.0 | 180 | 0.30 | 0.80 | 0.60 | 1.00 | 38 | 0.40 | 0.33 | $4.66\times {10}^{4}$ | 1.73 | 180,000 |

case6 | 0.25 | 2.0 | 180 | 0.35 | 0.80 | 0.625 | 0.975 | 38 | 0.40 | 0.33 | $4.88\times {10}^{4}$ | 1.73 | 180,000 |

case7 | 0.75 | 2.0 | 180 | 0.25 | 0.80 | 0.675 | 0.925 | 38 | 0.40 | 0.33 | $4.95\times {10}^{4}$ | 1.73 | 180,000 |

case8 | 1.00 | 2.0 | 180 | 0.20 | 0.80 | 0.70 | 0.90 | 38 | 0.40 | 0.33 | $4.85\times {10}^{4}$ | 1.73 | 180,000 |

case9 | 1.10 | 2.0 | 180 | 0.15 | 0.80 | 0.725 | 0.875 | 38 | 0.40 | 0.33 | $4.03\times {10}^{4}$ | 1.73 | 262,500 |

case10 | 1.24 | 2.0 | 180 | 0.15 | 0.80 | 0.725 | 0.875 | 38 | 0.40 | 0.33 | $4.15\times {10}^{4}$ | 1.73 | 262,500 |

**Table 2.**Summary of the maximums of relative error and absolute error in different location. The locations of (a–i) were introduced in Figure 4.

a | b | c | d | e | f | g | h | i | ||
---|---|---|---|---|---|---|---|---|---|---|

L = 6 m | Absolute error (m/s) | 0.03 | 0.05 | 0.06 | 0.04 | 0.06 | 0.05 | 0.01 | 0.06 | 0.07 |

Relative error (%) | 12 | 10.8 | 13.9 | 15.4 | 12 | 12.5 | 4.6 | 13.3 | 14.6 | |

L = 4 m | Absolute error (m/s) | 0.03 | 0.04 | 0.05 | 0.04 | 0.07 | 0.06 | 0.02 | 0.06 | 0.06 |

Relative error (%) | 12 | 8.7 | 11.6 | 15.4 | 14 | 15 | 7.6 | 13.3 | 12.2 |

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**MDPI and ACS Style**

Hu, R.; Zhang, J.
Numerical Analysis on Hydraulic Characteristics of U-shaped Channel of Various Trapezoidal Cross-Sections. *Water* **2018**, *10*, 1788.
https://doi.org/10.3390/w10121788

**AMA Style**

Hu R, Zhang J.
Numerical Analysis on Hydraulic Characteristics of U-shaped Channel of Various Trapezoidal Cross-Sections. *Water*. 2018; 10(12):1788.
https://doi.org/10.3390/w10121788

**Chicago/Turabian Style**

Hu, Ruichang, and Jianmin Zhang.
2018. "Numerical Analysis on Hydraulic Characteristics of U-shaped Channel of Various Trapezoidal Cross-Sections" *Water* 10, no. 12: 1788.
https://doi.org/10.3390/w10121788