# Flow Patterns in an Open Channel Confluence with Increasingly Dominant Tributary Inflow

^{*}

## Abstract

**:**

## 1. Introduction

**Figure 1.**Conceptual model of an open channel confluence (after Best [1]).

- What happens at extremely low discharge ratios when the tributary flow impinges on the opposing wall? How does this influence the small inflow coming from the upstream channel?
- Are known trends regarding flow patterns in the function of the discharge ratio confirmed at limiting small discharge ratios?
- Does the small discharge ratio influence the position of the mixing interface and possible helicoidal cells?
- To what extent does the small discharge ratio influence possible intermittent flow features?

## 2. Laboratory Experiments for Numerical Model Validation

_{d}is kept constant at a value of 40 L/s. In addition, the downstream water level is constant, and hence the downstream Froude number Fr

_{d}is constant as well, equalling 0.05. Although the Froude number is small compared to other laboratory experiments (see overview in [12]), the current value is in the typical range for lowland rivers [19]. As expected for such low subcritical flows, measurements by means of ultrasonic water level sensors show that the variation in the water level over the confluence is negligible. The Reynolds number based on the hydraulic radius equals 9.8 × 10

^{4}.

_{d}(= 0.104 m·s

^{−1}), respectively, are utilized. The (downstream) water depth h (= 0.415 m) is also used as a length scale in the vertical direction, when appropriate.

## 3. Numerical Model Simulations

#### 3.1. Model Description

^{+}and z

^{+}are the wall coordinates; k is the von Karman constant and the parameter B accounts for the wall roughness [28]:

^{−3}W/U

_{d}). After about 7.5 flow through times (= 760 s = 80 W/U

_{d}), collection of flow data started (t = 0), over an additional 16.5 flow through times (= 1640 s = 175 W/U

_{d}).

#### 3.2. Verification of the Simulations

^{+}is in an acceptable range for the use of a wall function. It is found that z

^{+}is generally sufficiently high (20 to 45) to allow the use of a wall function (requiring $20\le {z}^{+}\le 500$ [22,23,24]). However, in regions of very low velocity, e.g., still flow in the upstream channel in the case where q = 0, the use of a wall function introduces some inaccuracies [27], due to ${z}^{+}$ being lower than 20. However, to keep inaccuracies as small as possible, the adopted wall function includes the buffer layer and the laminar sublayer.

**Figure 3.**(

**a**) Percentage of turbulent kinetic energy resolved in the simulations; (

**b**) Spectral density of the turbulent kinetic energy at the point with coordinates (x = −1.33 W, y = 0.75 W, z = 0.95 h) at q = 0.25. The frequency associated with the Kolmogorov scale is indicated by a dotted line.

#### 3.3. Validation of the Simulations

**Figure 4.**Comparison of experimental measurements on the left and numerical results on the right, for the measured cross-sections. The color scale indicates the time-averaged longitudinal velocity, which is interpolated between the measured verticals for the ADV data, and arrows represent lateral and vertical velocity. The dotted line indicates $\overline{u}$ = 0, i.e., separates upstream and downstream flow. In the left figures, the measured locations are indicated with grey lines, and the arrows are located in the sweet spots. (

**a**,

**b**) q = 0.25, x = −1.33 W; (

**c**,

**d**) q = 0.25, x = −2 W; (

**e**,

**f**) q = 0.05, x = −1.33 W; (

**g**,

**h**) q = 0.05, x = −2 W.

**Figure 5.**Comparison of experimental measurements of the surface velocity (PTV) and the numerical predictions. (

**a**) q = 0.25, experimental; (

**b**) q = 0.25, LES; (

**c**) q = 0.05, experimental; (

**d**) q = 0.05, LES.

**Figure 6.**Time-averaged flow patterns near the surface (z = 0.9 h). Arrows indicate the longitudinal and lateral velocity component, while the color represents the vertical velocity component. The location of the mixing interface and the extent of the separation zone are indicated by a line. (

**a**) q = 0.25; (

**b**) q = 0.05; (

**c**) q = 0.

## 4. Results

_{t}is (only) 3 times larger than the upstream channel incoming flow rate Q

_{m}(or U

_{t}Q

_{t}/U

_{m}Q

_{m}= 9). The changes in flow features at small discharge ratios will be presented in Section 4.2 for the cases with q = 0.05 (or Q

_{t}/Q

_{m}= 19 and U

_{t}Q

_{t}/U

_{m}Q

_{m}= 361) and q = 0 (or Q

_{t}/Q

_{m}→ ∞ and U

_{t}Q

_{t}/U

_{m}Q

_{m}→ ∞).

#### 4.1. Reference Case with Moderate Discharge Ratio (q = 0.25)

#### 4.2. Cases with a Small Discharge Ratio (q = 0.05 and q = 0)

#### 4.2.1. Confluence Area and Upstream Channel

**Figure 7.**Cross section at x/W = 0, looking downstream. Color scale represents the time-averaged downstream velocity $\overline{u}$. Arrows represent lateral and vertical velocity. The dotted line indicates $\overline{u}$ = 0, i.e., separates upstream and downstream flow; (

**a**) q = 0.25; (

**b**) q = 0.05; (

**c**) q = 0.

_{u}through the section x/W = 0 (i.e., the last section of the upstream channel) is split into two parts: Q

_{u}= Q

_{u,out}+ Q

_{u,in}. The “outflow” discharge Q

_{u,out}represents the surface integral of all flow velocities oriented from the upstream channel to the confluence, whereas Q

_{u,in}denotes the surface integral of all flow velocities oriented from the confluence to the upstream channel.

**Table 1.**Composition of the total discharge Q

_{u}through the section x/W = 0, for different discharge ratios q.

q (−) | Q_{u} (m³·s^{−1}) | Q_{u,in} (m³·s^{−1}) | Q_{u,out} (m³·s^{−1}) |
---|---|---|---|

0.25 | 0.010 | 0.000 | 0.010 |

0.05 | 0.002 | 0.000 | 0.002 |

0 | 0.000 | 0.002 | 0.002 |

#### 4.2.2. Mixing Layer

#### 4.2.3. Separation Zone

**Figure 8.**Turbulent kinetic energy (TKE) near the surface (z = 0.9 h). The location of the mixing interface, the velocity gradient between impinging and non-impinging flow and the extent of the separation zone are indicated by a line. (

**a**) q = 0.25; (

**b**) q = 0.05; (

**c**) q = 0.

width/ W (−) | length/ W (-) | |||
---|---|---|---|---|

q (-) | Present Work | Best & Reid [34] | Present Work | Best & Reid [34] |

0.25 | 0.50 | 0.45 | 1.40 | 2.29 |

0.05 | 0.55 | 0.50 | 1.25 | 2.53 |

0 | 0.55 | 0.51 | 1.35 | 2.59 |

#### 4.2.4. Contracted Flow and Flow Recovery Areas

**Figure 9.**Cross section at x = −1.33 W, looking downstream. Color scale represents the streamwise-oriented vorticity of the mean flow. Arrows represent time-averaged lateral and vertical velocity. The dotted line indicates $\overline{u}=0$, i.e., separates upstream and downstream flow; (

**a**) q = 0.25; (

**b**) q = 0.05; (

**c**) q = 0.

**Figure 10.**Momentum coefficient $\beta $ of mean flow, characterizing the non-uniformity of the velocity distribution at different sections, as a function of the discharge ratio.

#### 4.2.5. Turbulent Kinetic Energy

#### 4.2.6. Intermittent Flow Patterns and Implications for Time-Averaged Results

**Figure 11.**Time series of lateral velocity v at the point with coordinates (x = −1.33 W, y = 0.75 W, z = 0.95 h), in the experiment (ADV) and in the LES. (

**a**) q = 0.25; (

**b**) q = 0.05; (

**c**) q = 0.

**Figure 12.**Probability density functions of the time series of lateral velocity $v$ shown in Figure 11, both for the experiment (ADV) and for the LES.

**Figure 13.**Instantaneous streamlines when q = 0.05, starting at x = −0.20W in the tributary channel, at (

**a**) t = 540 s; (

**b**) t = 740 s. Color scale indicates the vertical height at which the streamline is located. Red arrows indicate flow direction.

**Figure 14.**Instantaneous velocity in the section x = −1.33 W when q = 0.05, looking downstream, at (

**a**) t = 540 s; (

**b**) t = 740 s. Color scale indicates longitudinal velocity u, while arrows show the lateral and vertical velocity.

## 5. Discussion

## 6. Conclusions

- New features in the flow patterns induced by the impinging of the tributary flow on the opposing bank. In the upstream channel, a recirculating eddy develops, imposing rather significant changes on the incoming velocity. By changing the size of the stagnation zone, this also influences the mixing layer. In the downstream channel, the impinging flow causes stronger helicoidal cells, upwelling near the right bank and associated higher levels of TKE.
- Confirmation of some of the known trends in confluence literature, such as increased three-dimensionality of the flow, and increasing dimensions of the separation zone. However, the new flow features can be regarded as deviations from the known trends.
- Changes in the mixing layer, as the upstream channel inflow is influenced by the recirculating eddy. In addition, a new shear interface develops between the upwelling flow caused by impinging, and the non-impinging part of the tributary inflow follows a curved path to the downstream channel. The impinging flow enforces stronger helicoidal cells, though in the end, these do not result in faster flow recovery.
- Upwelling events of much stronger upwelling flow, having an intermittent character. They seem to be linked to the height at which the tributary impinges on the opposing wall, thus they are associated with the small discharge ratios.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Schindfessel, L.; Creëlle, S.; De Mulder, T.
Flow Patterns in an Open Channel Confluence with Increasingly Dominant Tributary Inflow. *Water* **2015**, *7*, 4724-4751.
https://doi.org/10.3390/w7094724

**AMA Style**

Schindfessel L, Creëlle S, De Mulder T.
Flow Patterns in an Open Channel Confluence with Increasingly Dominant Tributary Inflow. *Water*. 2015; 7(9):4724-4751.
https://doi.org/10.3390/w7094724

**Chicago/Turabian Style**

Schindfessel, Laurent, Stéphan Creëlle, and Tom De Mulder.
2015. "Flow Patterns in an Open Channel Confluence with Increasingly Dominant Tributary Inflow" *Water* 7, no. 9: 4724-4751.
https://doi.org/10.3390/w7094724