# Dip Phenomenon in High-Curved Turbulent Flows and Application of Entropy Theory

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## Abstract

**:**

## 1. Introduction

_{max}, and the mean flow velocity, u

_{m}, through a dimensionless function Φ(M), depending on the entropic parameter, M ([25,26,27,28,29]).

## 2. Materials and Methods

#### 2.1. Experimental Dataset

_{0}= 110° (see Figure 1).

_{50}= medium sediment diameter = 0.65 mm, σ

_{g}= geometric standard deviation = 1.34). The meandering channel is around 25 m long in order to accommodate two meander wavelengths; the upstream and downstream ends of the meandering channel are connected to two straight channels 3 m and 2 m long, respectively.

^{3}/s (channel-averaged flow depth h = 5.5 cm; B/h = 9.09 < 10); Run 2 was conducted with a flow discharge Q = 0.007 m

^{3}/s (channel-averaged flow depth h = 3.0 cm, B/h = 16.67 > 10). Herein, in accordance with other authors ([13,34]), B/h < 10 indicates “small” width-to-depth ratio and B/h > 10 indicates “large” width-to-depth ratio.

#### 2.2. Pertinent Aspects and Summary of Previous Results

- -
- Termini ([33]) conducted experiments under the hydraulic conditions that are considered in the present work in order to examine the effect of the continuously changing channel’s curvature on flow pattern. She verified that, because of the changing channel curvature, the flow accelerates near the outer bank from the beginning of the bend to the inflection section downstream and decelerates near the inner bank. The flow accelerated zone is more evident for B/h > 10 than for B/h < 10. Such a different behaviour is caused by the secondary circulation that develops more significantly in the case of B/h < 10 than in the case of B/h > 10, and that attenuates the effect of the convective acceleration close to the outer bank (see also in [31]).
- -
- Termini and Moramarco ([29]) investigated the effect of the downstream variation of the channel’s curvature on the applicability of the linear entropic relationship between the maximum velocity, u
_{max}, and the mean flow velocity, u_{m}, through a dimensionless parameter Φ(M). As result, they observed that the ratio u_{m/}u_{max}, and thus the value of the parameter Φ(M), varies along the bend. In particular, Termini and Moramarco ([29]) demonstrated that, in contrast to what observed in straight channels ([28]) where Φ(M) assumed a value that was almost constant and on average equal to 0.65, along the meandering bend the parameter Φ(M) increases passing from the inflection section to the apex section and decreases from the apex section to the inflection section downstream, assuming a value of around 0.8 close to the apex section and of around 0.6 at the inflection section. This behavior can be observed from Figure 2, where the values of the parameter Φ(M), as obtained by Termini and Moramarco ([29]), along the bend are reported.

## 3. Results

#### 3.1. Measured Velocity Profiles and Effect of the Aspect Ratio on the Velocity-Dip

#### 3.2. Application of the Entropic Model to Estimate the Velocity-Dip Phenomenon

_{max,v}is the maximum velocity along the each measurement vertical, D is the water depth, z is the distance of the velocity point from the bed, x is the distance of the considered vertical from the left bank, and M is the entropic parameter, which can be estimated by using the entropic relation ([25]):

_{max}= max[u

_{max,v}] which is located at z-axis. The velocity-dip is estimated by using the modified expression proposed by Yang et al. ([16]), written as ([24]):

_{p}+ δ(x) with δ = 1 + 1.3e

^{−x/D}

_{surf}, and the knowledge of Φ(M), the maximum velocity u

_{max,v}and the velocity-dip are computed, respectively, through Equations (2) and (4) by initializing a

_{p}= a

_{1}. Then, the velocity profiles are computed through Equation (1) and the maximum velocity u

_{max}= max[u

_{max,v}] is assessed. Once the velocity profiles are identified, the mean flow velocity, u

_{m}, can be assessed by applying the velocity area method ([38]) at the estimated velocity profiles; therefore, the ratio ${\Phi}_{com}\left({M}_{1}\right)=\frac{{u}_{m}\left({a}_{1}\right)}{{u}_{max}\left({a}_{1}\right)}$ can be computed by Equation (3). The iterative procedure ends when the optimal value of ${\Phi}_{com}\left({M}_{p}\right)$ is achieved in accordance with:

_{i,m}and δ

_{i,e}are the δ-values determined, respectively, from the measured and from the estimated profiles, N is the total number of measurements that are considered for each profile. Figure 6 reports the pair of values (δ

_{i,m}, δ

_{i,e}) and the line of perfect agreement (bisector line). The error bar is defined by the value of σ. As shown in Figure 6, with a few exceptions, the points arrange around the bisector line and the σ is quite low in comparison to the magnitude of the measured values.

## 4. Discussion and Concluding Remarks

## Author Contributions

## Conflicts of Interest

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**Figure 3.**Measured profiles at peculiar verticals of the examined sections: (

**a**) Run 1 (B/h < 10); (

**b**) Run 2 (B/h > 10); x/B indicates the relative distances from the left (outer) bank.

δav (%) = 53 | |||||||||

B/h < 10 | x/B | ||||||||

Section | 0.06 | 0.17 | 0.28 | 0.39 | 0.5 | 0.61 | 0.72 | 0.83 | 0.94 |

A | 57.36 | 61.24 | 52.94 | 52.85 | 52.94 | 53.40 | 53.85 | 47.47 | 56.52 |

B | 53.33 | 43.56 | 35.94 | 43.40 | 44.55 | 58.19 | 52.48 | 60.35 | 64.99 |

C | 39.62 | 47.26 | 51.81 | 52.94 | 57.20 | 57.69 | 61.83 | 61.17 | 60.78 |

δav (%) = 34 | |||||||||

B/h > 10 | x/B | ||||||||

Section | 0.06 | 0.17 | 0.28 | 0.39 | 0.5 | 0.61 | 0.72 | 0.83 | 0.94 |

A | 43.22 | 48.39 | 0.00 | 47.37 | 35.90 | 35.90 | 0.00 | 38.27 | 36.60 |

B | 50.00 | 35.86 | 20.97 | 37.69 | 35.48 | 26.42 | 37.50 | 36.62 | 44.44 |

C | 32.89 | 32.20 | 24.91 | 40.48 | 33.55 | 34.21 | 34.64 | 33.77 | 32.89 |

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Termini, D.; Moramarco, T.
Dip Phenomenon in High-Curved Turbulent Flows and Application of Entropy Theory. *Water* **2018**, *10*, 306.
https://doi.org/10.3390/w10030306

**AMA Style**

Termini D, Moramarco T.
Dip Phenomenon in High-Curved Turbulent Flows and Application of Entropy Theory. *Water*. 2018; 10(3):306.
https://doi.org/10.3390/w10030306

**Chicago/Turabian Style**

Termini, Donatella, and Tommaso Moramarco.
2018. "Dip Phenomenon in High-Curved Turbulent Flows and Application of Entropy Theory" *Water* 10, no. 3: 306.
https://doi.org/10.3390/w10030306