# Turbulent Flow through Random Vegetation on a Rough Bed

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Laboratory Experiments and Methodology

^{−1}, where B was the flume width. The longitudinal bottom slope of the flume, S, was fixed at 1.5‰ using a hydraulic jack.

_{c}= 0.40 m and d = 0.02 m, respectively. The stems were implanted into a 1.96-m long, 0.485-m wide and 0.015-m thick Plexiglas panel fixed to the channel bottom. A total of 68 cylinders were randomly arranged in the flume (Figure 2a). All the experiments were carried out in conditions of emergent vegetation. The frontal area per volume was a = nd = 1.4 m

^{−1}, where n = 71 m

^{−2}was the number of cylinders per bed area, while the solid volume fraction occupied by the canopy els per bed area, while the solid volume fraction occupied by the canopy elements was ϕ = πad/4 = nπd

^{2}4 = 0.02, which is consistent with typical laboratory studies with vegetation [17,26,27]. Following Nepf [28], this vegetation distribution can be classified as dense. Focus was given to the evolution of the turbulence characteristics in a study area selected around a single stem (Figure 2b).

_{50}= 1.53 mm), fine gravel (d

_{50}= 6.49 mm) and coarse gravel (d

_{50}= 17.98 mm), respectively (Figure 3). The grain size distributions were relatively uniform, i.e., as reported by Dey and Sarkar [29], with a geometric standard deviation σ

_{g}= (d

_{84}/d

_{16})

^{0.5}< 1.5, where d

_{16}and d

_{84}are the sediment sizes for which 16% and 84% by weight of sediment is finer, respectively. At the beginning of each run, the flume was filled in with the sediments, which were successively screeded to make the longitudinal bed slope equal to that of the flume bottom.

_{*}), the critical velocity for the inception of sediment motion (U

_{c}), the mean water temperature (T) measured with the Acoustic Doppler Velocimeter (ADV) integrated thermometer (having an accuracy of ±0.1 °C), the water kinematic viscosity (ν), computed as a function of the water temperature [32], the flow Froude number Fr [=U/(gh)

^{0.5}], the flow Reynolds number Re (=Uh/ν), the shear Reynolds number Re

_{*}(=u

_{*}ε/ν, where ε is the Nikuradse equivalent sand roughness, equal to about 2d

_{50}) and the Reynolds number of the vegetation stems Re

_{d}(=Ud/ν). In accordance with Manes et al. [33] and Dey and Das [34], the shear velocity used to scale the flow statistics was determined as u

_{*}=(τ

_{*}/ρ)

^{0.5}, where τ

_{*}is the total stress acting at the roughness tops. This can be obtained by extending linearly the distribution of the turbulent shear stress captured 50 cm upstream of the vegetation pattern (i.e., in correspondence with the undisturbed flow condition) from the region above the roughness elements to their tops. Thus, the shear velocity was evaluated at the sediment crest level as ${\left(-\overline{{u}^{\prime}{w}^{\prime}}\right)}^{0.5}$, where u′ and w′ are the fluctuations of the temporal velocity signal in the streamwise and vertical directions, respectively, and the symbol $\overline{\xb7}$ indicates the time averaging operation. The critical velocity for the inception of sediment motion U

_{c}was established 50 cm upstream of the vegetation array through the well-known Neill formula [35], as follows:

_{c}), which was also verified from the direct observation of the flow.

_{50}increases and, consequently, to a decrease in $\widehat{u}$. The distributions of the dimensionless turbulent shear stresses ${\widehat{\tau}}_{uw}$ ($=-\overline{u\u2019w\u2019}/{u}_{*}^{2}$) and of the dimensionless viscous shear stresses ${\widehat{\tau}}_{\nu}$ [$=\nu \left(\mathrm{d}\overline{u}/\mathrm{d}z\right)/{u}_{*}^{2}$] along $\widehat{z}$ are represented in Figure 4b,c, respectively. In particular, above the roughness surface, the prevalence of the Reynolds shear stresses can be noted, while the viscous shear stresses are practically negligible as $\widehat{z}$ increases. The viscous shear stresses achieve their maximum values near the grain crests for each experimental run. Conversely, the turbulent shear stresses reach the peak above the crest level and then they reduce as the vertical distance increases.

## 3. Results and Discussion

#### 3.1. Time-Averaged Flow

_{s}= 12 cm). The time-averaged flow is accelerated if $\left({\widehat{u}}_{UP}-\widehat{u}\right)<0$ (blue values in the colormaps) and decelerated if $\left({\widehat{u}}_{UP}-\widehat{u}\right)>0$ (red values in the colormaps), where $\widehat{u}$ is the dimensionless time-averaged streamwise velocity.

_{50}decreases, this zone becomes strongly accelerated. Conversely, as d

_{50}increases, in the near-bed layer, the flow field results to be influenced by the bed roughness and much more by the vegetation, with a minor acceleration intensity.

_{s}= 12 cm). In Figure 7, it is evident that the presence of vegetation has a very visible effect on the flow field: the flow is accelerated in all the runs with higher streamwise velocities than in the undisturbed flow profiles along the whole water depth. At each measurement location, the vegetation causes the velocity profile to maintain a constant value (for z > h

_{l}). Moving toward the bed (z < h

_{l}), the influence of vegetation decreases, and the flow field becomes more accelerated, owing to the presence of the bed.

#### 3.2. TKE and Normal Stresses

#### 3.3. Energy Spectra

_{s}/N, and N is the number of samples equal to 30,000 for an acquisition time of 300 s). The energy spectra were determined by employing the discrete fast Fourier transform of the autocorrelation function.

## 4. Conclusions

_{s}. The longitudinal TKE distribution revealed high values in the near-bed flow zone. This suggests that the velocity oscillations get excited by the rough bed, producing an increase of the turbulence level in the vicinity of the sediments. Mowing toward the free surface this effect disappears, inducing a decrease in the TKE.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**(

**a**) Sketch of the cylinder array in the laboratory flume (dimensions are in cm); (

**b**) the measurement verticals within the study area. Here, B

_{s}and L

_{s}are the width and length of the study area, respectively.

**Figure 3.**Sediments used in the experimental runs: (

**a**) d

_{50}= 1.53 mm; (

**b**) d

_{50}= 6.49 mm; (

**c**) d

_{50}= 17.98 mm.

**Figure 4.**Vertical profiles of (

**a**) dimensionless time-averaged velocity, (

**b**) dimensionless Reynolds shear stress and (

**c**) dimensionless viscous shear stress for the undisturbed flow condition (50 cm upstream to the vegetation array) in Run 1, Run 2 and Run 3.

**Figure 5.**Contours of the dimensionless time-averaged accelerated $\left({\widehat{u}}_{UP}-\widehat{u}\right)<0$and decelerated $\left({\widehat{u}}_{UP}-\widehat{u}\right)>0$ flow field in the upstream plane section (from LE to RE) for (

**a**) Run 1, (

**b**) Run 2 and (

**c**) Run 3. The black dashed lines show the cylinder position.

**Figure 6.**Contours of the dimensionless time-averaged accelerated $\left({\widehat{u}}_{UP}-\widehat{u}\right)<0$and decelerated $\left({\widehat{u}}_{UP}-\widehat{u}\right)>0$ flow field in the downstream plane section (from LV to RV) for (

**a**) Run 1, (

**b**) Run 2 and (

**c**) Run 3. The black dashed lines show the cylinder position.

**Figure 7.**Contours of the dimensionless time-averaged accelerated $\left({\widehat{u}}_{UP}-\widehat{u}\right)<0$and decelerated $\left({\widehat{u}}_{UP}-\widehat{u}\right)>0$ flow field in the longitudinal plane section (from RE to RV) for (

**a**) Run 1, (

**b**) Run 2 and (

**c**) Run 3. The black dashed lines show the cylinder position.

**Figure 8.**Contours of the dimensionless TKE in the upstream plane section (from LE to RE) for (

**a**) Run 1, (

**b**) Run 2 and (

**c**) Run 3. The black dashed lines show the cylinder position.

**Figure 9.**Contours of the dimensionless normal stresses (

**a**) $\overline{{u}^{\prime}{u}^{\prime}}/{u}_{*}^{2}$, (

**b**) $\overline{{v}^{\prime}{v}^{\prime}}/{u}_{*}^{2}$ and (

**c**) $\overline{{w}^{\prime}{w}^{\prime}}/{u}_{*}^{2}$ in the upstream plane section (from LE to RE) for (

**a**) Run 1, (

**b**) Run 2 and (

**c**) Run 3. The black dashed lines show the cylinder position.

**Figure 10.**Contours of the dimensionless TKE in the downstream plane section (from LV to RV) for (

**a**) Run 1, (

**b**) Run 2 and (

**c**) Run 3. The black dashed lines show the cylinder position.

**Figure 11.**Contours of the dimensionless normal stresses (

**a**) $\overline{{u}^{\prime}{u}^{\prime}}/{u}_{*}^{2}$, (

**b**) $\overline{{v}^{\prime}{v}^{\prime}}/{u}_{*}^{2}$ and (

**c**) $\overline{{w}^{\prime}{w}^{\prime}}/{u}_{*}^{2}$ in the downstream plane section (from LV to RV) for (

**a**) Run 1, (

**b**) Run 2 and (

**c**) Run 3. The black dashed lines show the cylinder position.

**Figure 12.**Contours of the dimensionless TKE in the longitudinal plane section (from RE to RV) for (

**a**) Run 1, (

**b**) Run 2 and (

**c**) Run 3. The black dashed lines show the cylinder position.

**Figure 13.**Energy spectra for (

**a**) Run 1, (

**b**) Run 2 and (

**c**) Run 3 of streamwise (blue lines), spanwise (red lines), and vertical (green lines) velocity fluctuations at three different levels $\widehat{z}$ of the vertical W.

Parameter (Units) | Run 1 | Run 2 | Run 3 |
---|---|---|---|

d_{50} (mm) | 1.53 | 6.49 | 17.98 |

h (m) | 0.12 | 0.12 | 0.12 |

Q (l/s) | 19.73 | 19.73 | 19.73 |

U (m/s) | 0.34 | 0.34 | 0.34 |

u_{*} (m/s) | 0.021 | 0.022 | 0.028 |

U_{c} (m/s) | 0.39 | 0.69 | 1.04 |

S (‰) | 1.50 | 1.50 | 1.50 |

T (°C) | 16.67 | 18.06 | 18.70 |

ν (m ^{2}/s) | 1.09 × 10^{−6} | 1.05 × 10^{−6} | 1.03 × 10^{−6} |

Fr | 0.31 | 0.31 | 0.31 |

Re | 37,431 | 38,857 | 39,612 |

Re_{*} | 59 | 272 | 978 |

Re_{d} | 6239 | 6192 | 6602 |

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**MDPI and ACS Style**

Coscarella, F.; Penna, N.; Ferrante, A.P.; Gualtieri, P.; Gaudio, R. Turbulent Flow through Random Vegetation on a Rough Bed. *Water* **2021**, *13*, 2564.
https://doi.org/10.3390/w13182564

**AMA Style**

Coscarella F, Penna N, Ferrante AP, Gualtieri P, Gaudio R. Turbulent Flow through Random Vegetation on a Rough Bed. *Water*. 2021; 13(18):2564.
https://doi.org/10.3390/w13182564

**Chicago/Turabian Style**

Coscarella, Francesco, Nadia Penna, Aldo Pedro Ferrante, Paola Gualtieri, and Roberto Gaudio. 2021. "Turbulent Flow through Random Vegetation on a Rough Bed" *Water* 13, no. 18: 2564.
https://doi.org/10.3390/w13182564