# Characteristics of Large-Scale Coherent Structures on Irregularly Arranged Rough-Bed Open-Channel Flows

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{50}) was observed in the present irregularly arranged rough-bed OCF, which is in contrast to that reported for regular rough-bed OCF. Furthermore, the relationship between the peak wavelength of VLSMs and the aspect ratio did not strictly follow a linear increase, in contrast to that documented in the literature.

## 1. Introduction

## 2. Methodology

#### 2.1. Facilities and Experiments

^{2}were placed at the flume entrance to flatten the flow and eliminate the potential large-scale fluctuations generated by the circulating system. Seven ultrasonic water-level gauges were arranged along the flume to monitor the water depth.

^{3}and a median particle size of 10 mm were used as the tracer particles.

_{m}up to over 10,000 water depths h, to obtain convergent spectral results of VLSMs.

#### 2.2. Power Spectrum

## 3. Results and Discussion

#### 3.1. Determination of the Theoretical Zero Point of Irregularly Arranged Rough-Bed

_{50}= 1.5 cm), and the rough-bed thickness was approximately 4d

_{50}(Figure 2a). Figure 2b shows the reconstruction results of the bed topography and provides a preliminary demonstration of the random distribution feature of the rough-bed; where the gray pattern in the lower left corner of the figure is a cubic concrete block (7 × 7 × 7 cm

^{3}) used for calibrating the rough-bed height.

#### 3.2. Profiles of Typical Statistics

_{t}4110 test condition, the dimensionless Reynolds shear stress per unit density attains a peak near y/H = 0.1 followed by a sharp decrease. The dimensionless Reynolds shear stress under H055Re

_{t}1780 and H075Re

_{t}2760 test conditions break over at y/H = 0.2. Compared with the other two groups of test conditions, H100Re

_{t}4110 shows that the sudden change in the Reynolds shear stress is closer to the bed surface, indicating that the turbulent structure tends to develop toward the bed surface under the condition of a large water depth. This linear variation can be verified by the net-force (wall-normal gradient of the Reynolds shear stress) profile (Figure 4b). In Figure 4b, the net force is roughly constant at the outer region, indicating a linear variation of Reynolds shear stress in the corresponding region. The variation of the wall-normal gradient of Reynolds shear stress near the bed surface is related to the aspect ratio (B/H). The wall-normal variation gradient of the viscosity effect under H100Re

_{t}4110 is small due to the smaller aspect ratio, and the net force distribution presents a larger slope.

#### 3.3. Presence of LSMs and VLSMs and Their Scales

_{τ}> 1700).

_{t}1780 in the present study (corresponding to irregularly arranged rough-bed OCF) are all approximately 10, the wavelengths of VLSMs in different scenarios exhibit the following trend: smooth-bed OCF > regular rough-bed OCF > irregularly arranged rough-bed OCF. This observation contrasts with Cameron et al.’s speculation that the wavelength of VLSMs exhibits a dependence on the aspect ratio, based on which they proposed a linear relationship between the maximum wavelength of VLSMs along the wall-normal and aspect ratio. This conflict can be more clearly observed in Figure 8 and will be discussed later. Another interesting feature observed in Figure 7 is that the empirical curve obtained by Monty et al. roughly represents the upper envelope of the results of the present irregularly arranged rough-bed OCF.

#### 3.4. Strength of VLSMs

_{uuVLSMs}and g

_{uvVLSMs}for brevity, respectively. For the cumulative energy fraction of the streamwise velocity fluctuations, the Reynolds number and water depth changes are not considered, as shown in the figure. More than 65.8% of the energy contribution is from VLSMs, and more than 38.9% of the energy is contained in wavelengths longer than 10H. Moreover, more than 47.5% of the Reynolds shear stress contribution is from VLSMs, and approximately 22–25% of the Reynolds shear stress is contained in wavelengths longer than 10H.

_{uuVLSMs}and g

_{uvVLSMs}are shown in Figure 10. The contributions of VLSMs to turbulent kinetic energy in the irregularly arranged rough-bed OCF are greater than those in the smooth-bed OCF (gray solid sign). Another apparent discovery is that the contribution fraction of VLSMs to the turbulent kinetic energy shows a similar trend (i.e., increasing first in the inner region and then decreasing along with the water depth) for all three Reynolds number sections as well as that in smooth-bed OCF. Here, g

_{uvVLSMs}increase monotonically between 0 < y/H < 0.5. At y/H = 0.5, g

_{uvVLSMs}reach approximately 0.5–0.6 and then gradually decrease. Similar to that of the smooth-bed OCF, VLSMs in the present irregularly arranged rough-bed OCF also play major roles in contributing to the turbulent energy and Reynolds shear stress. Comparisons between the results of the present irregularly arranged rough OCF and those of the smooth-bed OCF in the previous study by Duan et al. demonstrated that the contributions of VLSMs in the present irregularly arranged rough OCF are higher. As the turbulence intensity and Reynolds shear stress of the present irregularly arranged rough OCF are not lower than those of the smooth-bed OCF, the higher contributions of VLSMs to the turbulent kinetic energy and the Reynolds shear stress indicate that the VLSMs in the present irregularly arranged rough-bed OCF are stronger.

## 4. Conclusions

- (1)
- The difference in the measurement locations (pebble gap and top) had a considerably small influence on the standardized time average velocity distribution. The typical statistics of the irregularly arranged rough-bed OCF are consistent with previous classical research. The friction Reynolds number and roughness coefficient have a certain modulation effect on the radial and vertical velocity pulsations.
- (2)
- The double peak phenomenon of LSMs and VLSMs exists in the present irregularly arranged rough-bed OCF, which is similar to the results of the smooth-bed OCF and the regular rough-bed OCF. Under the same aspect ratio ranges, the wavelengths of VLSMs on the irregularly arranged rough-bed OCF appear minimal among different bed scenarios (i.e., smooth-bed OCF, regular rough-bed OCF, and irregularly arranged rough-bed OCF), and the result of the smooth-bed OCF is the opposite.
- (3)
- The contributions of VLSMs to the turbulent kinetic energy and the Reynolds shear stress in the irregularly arranged rough-bed OCF are greater than those in the smooth-bed OCF. More than 60% of the turbulent kinetic energy and 40% of the Reynolds shear stress originate from VLSMs in the present irregularly arranged rough-bed OCF. Similar to those in the smooth-bed OCF, VLSMs in the present, irregularly arranged rough-bed OCF also play major roles in contributing to the turbulent energy and Reynolds shear stress.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**a**) Schematic of an open-channel flume (not to scale) and (

**b**) PIV system arrangement (a small light guided glass plate (10 cm long, 5 cm wide and 0.2 cm thick) is arranged right above the water surface to eliminate the surface fluctuations).

**Figure 2.**Characteristics of the rough-bed: (

**a**) realistic view of the rough-bed surface (18 × 15 cm

^{2}); (

**b**) rough-bed terrain reconstructed by structure from motion (SFM) technology; (

**c**) schematic of the height of the rough-bed surface; (

**d**) probability distribution of bed surface height.

**Figure 3.**Time-averaged velocity profiles at different flow cases (Tip indicates that the starting point of the measurement is at the top of the pebble particles): (

**a**) the time-averaged absolute velocity profiles; (

**b**) the normalized velocity profiles processed by the friction velocity.

**Figure 7.**Wall-normal distributions of streamwise wavelengths corresponding to the peaks of LSMs and VLSMs.

**Figure 9.**Cumulative contribution fractions of different scale structures to the streamwise kinetic energy (g

_{uu}) and Reynolds shear stress (g

_{uv}). (

**a**,

**d**) H055Re

_{t}1780; (

**b**,

**e**) H075Re

_{t}2760; (

**c**,

**f**) H100Re

_{t}4110.

**Figure 10.**Wall-normal profiles of the contributions of VLSMs to (

**a**) turbulent kinetic energy and (

**b**) Reynolds shear stress.

Case | H (m) | B/H | u (m^{2}/s) | U_{m} (m/s) | u* (m/s) | F_{r} | Re | Re_{t} | k_{s}^{+} |
---|---|---|---|---|---|---|---|---|---|

H055Re_{t}1780 | 0.055 | 10.2 | 1.03 × 10^{−6} | 0.428 | 0.034 | 0.583 | 22,757 | 1785 | 442 |

H075Re_{t}2760 | 0.075 | 7.5 | 1.03 × 10^{−6} | 0.506 | 0.038 | 0.590 | 36,714 | 2762 | 494 |

H100Re_{t}4110 | 0.100 | 5.6 | 1.03 × 10^{−6} | 0.586 | 0.042 | 0.592 | 56,671 | 4110 | 546 |

^{(a)}H: water depth; B/H: width-depth ratio; υ: kinematic viscosity; U

_{m}: depth mean velocity; u*: friction velocity, determined based on the log law with the von Kármán constant k = 0.412 and additive constant A = 5.26; F

_{r}, Froude number; Re = U

_{m}H/υ, Reynolds number; Re

_{τ}= u*H/υ, friction Reynolds number; k

_{s}

^{+}= k

_{s}u*/υ, dimensionless roughness height.

Case | Image Size (Pixels) | Resolution (Pixel/mm) | F_{s} | T (s) | Number of Images | Δx^{+} | TU_{m}/h | ΔTU_{m}/h | ΔT^{+} | y* (mm) |
---|---|---|---|---|---|---|---|---|---|---|

H055Re_{t}1780 | 1024 × 128 | 0.062 | 800 | 1879.25 | 1503,400 | 4.22 | 14,616 | 0.0097 | 1.24 | 0.0323 |

H075Re_{t}2760 | 1496 × 128 | 0.062 | 800 | 1544.51 | 1235,610 | 7.52 | 10,422 | 0.0084 | 1.81 | 0.0267 |

H100Re_{t}4110 | 1600 × 128 | 0.062 | 800 | 1684.99 | 1347,990 | 11.99 | 9872 | 0.0073 | 2.46 | 0.0229 |

^{(a)}F

_{s}, sampling frequency of the velocity fields; T, total image acquisition time; Δx

^{+}, inner-scaled vector spacing in the streamwise direction; ΔT

^{+}, time interval between successive velocity fields; y* = υ/u*, the viscous length scales.

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**MDPI and ACS Style**

Wang, Y.; Zhang, P.; Yang, S.; Hu, C.; Jin, J.; Zhang, R.
Characteristics of Large-Scale Coherent Structures on Irregularly Arranged Rough-Bed Open-Channel Flows. *Water* **2023**, *15*, 1105.
https://doi.org/10.3390/w15061105

**AMA Style**

Wang Y, Zhang P, Yang S, Hu C, Jin J, Zhang R.
Characteristics of Large-Scale Coherent Structures on Irregularly Arranged Rough-Bed Open-Channel Flows. *Water*. 2023; 15(6):1105.
https://doi.org/10.3390/w15061105

**Chicago/Turabian Style**

Wang, Yongqiang, Peng Zhang, Shengfa Yang, Chunhong Hu, Jianling Jin, and Rangang Zhang.
2023. "Characteristics of Large-Scale Coherent Structures on Irregularly Arranged Rough-Bed Open-Channel Flows" *Water* 15, no. 6: 1105.
https://doi.org/10.3390/w15061105