# Characteristics of Very Large-Scale Motions in Rough-Bed Open-Channel Flows

^{*}

## Abstract

**:**

## 1. Introduction

_{τ}and persist farther into the outer layers in OCFs than they do in other flows [9]. Similarly, whether different wall types in terms of a smooth or rough bed make a difference is another question. VLSMs, having only been documented rather recently, in the past two decades, still have many open questions, such as regarding their origin, scaling, and dynamics. This study attempts to explore VLSMs in OCFs regarding their scaling and energy contents. Compared with flows over a smooth bed, OCFs over a rough bed are much more complex. In addition to the general impact of Re

_{τ}on turbulent flows, a rough bed introduces other parameters, including relative submergence H/Δ (where Δ refers to the equivalent roughness) and roughness Reynolds number Re

_{Δ}, and their relevance to VLSMs is still debated.

_{τ}. However, they could not identify the dominant contributing factor because their experiments maintained the B/H ratio and Re

_{Δ}constant. To further explore the scaling of VLSMs, Peruzzi et al. [9] conducted experiments under nonuniform flow conditions over a smooth bed and indicated the B/H ratio as the key scale among the B/H ratio, Re

_{τ}, and the nondimensional distance from flume inlet x/H. Because their experiments were performed in smooth-wall conditions, whether relative submergence plays a role in controlling the scaling of VLSMs remains unclear. Providing an answer to this question is one of the main objectives of the present study.

## 2. Experiments

#### 2.1. Experimental Setup

^{3}). The particle images were postprocessed using PIVlab, graphical user interface (GUI)-based PIV software of which the accuracy verification can be found in the literature [18]. The initial interrogation window size was 64 × 64 pixels, and the final window, with 2–4 particles, was 16 × 16 pixels after three passes with a 50% overlap. The accuracy of the resulting velocity vectors was verified using the three standard deviations of the fluctuation velocity. Bad vectors constituted less than 1% for all the flow fields.

_{*}and kinematic viscosity ν, such as U

^{+}= U/u

_{*}and Δx

^{+}= Δxu

_{*}/ν. The prime symbol indicates the root mean square, so the turbulence intensities of the streamwise and wall-normal velocities are u′ and v′, respectively.

_{τ}plays in wall turbulence, the friction Reynolds number differed for different runs. Taking into account that B/H and H/Δ are also possible key parameters in controlling the scaling of VLSMs [19], different values were set for Runs R1–R3. Friction velocity u

_{*}was determined on the basis of the log law. All of the experiments were conducted under fully developed rough and subcritical flow conditions (Fr < 1). Equivalent roughness Δ = 0.67D [20] (where D is the particle diameter) was assumed in this study, and H/Δ > 5 for all runs indicates that all of the flows in this study had large submergence [21].

^{+}was generally less than 1–3 to resolve the small-scale turbulent motions [23] in all runs except for R2, in which ΔT

^{+}was even larger than 5. Therefore, the results for R2 were not used to analyze the fraction of TKE and the Reynolds stress carried by VLSMs owing to its mismatched resolving ability with other runs. The streamwise distance of mean flow movement TU

_{m}was above 1000H, which was sufficiently long to yield convergent spectral results.

#### 2.2. Methods

_{uiuj}(f) in the frequency domain could be obtained by computing the discrete Fourier transform (FT) of the fluctuating velocity signal as follows:

_{ui}(f) is the FT of u

_{i}(u

_{1}= u, u

_{2}= v); * and | | indicate the complex conjugate and modulus, respectively; C is a constant determined by satisfying equation $\overline{{u}_{i}{u}_{j}}={\displaystyle {\int}_{0}^{\infty}}{S}_{{u}_{i}{u}_{j}}\left(f\right)df$.

_{uiuj}(f) can be transformed into wavenumber-based spectral density S

_{uiuj}(k

_{x}) by using Taylor’s frozen turbulence hypothesis [28] through the following relationship:

_{x}is the streamwise wavenumber related to wavelength λ by λ = 2π/k

_{x}, and U (y) is the time-averaged streamwise velocity at y. The relationship between these two spectral densities is as follows:

_{uiuj}(f), S

_{uiuj}(k

_{x}) satisfies equation $\overline{{u}_{i}{u}_{j}}={\displaystyle {\int}_{0}^{\infty}}{S}_{{u}_{i}{u}_{j}}\left({k}_{x}\right)d{k}_{x}$. It can be easily deduced that $\overline{{u}^{2}}={\displaystyle {\int}_{0}^{\infty}{S}_{uu}}\left({k}_{x}\right)d{k}_{x}={\displaystyle {\int}_{0}^{\infty}{k}_{x}{S}_{uu}}\left({k}_{x}\right)d(\mathrm{ln}{k}_{x})$. Given that $\overline{{u}^{2}}$/2 was exactly the mean streamwise TKE, the area enclosed by the curve of k

_{x}S

_{uu}(k

_{x}) plotted in single logarithmic coordinates with the horizontal coordinate was equal to twice the streamwise TKE. Similar conclusions could be drawn regarding the wall-normal TKE and Reynolds stress.

## 3. Results and Discussion

#### 3.1. Turbulence Statistics

^{+}due to the roughness effects (Figure 2a), which can be expressed as follows:

^{+}is the roughness function. The relative zero-plane displacement d/D was in the 0.15–0.3 range [33], and d/D = 0.2 was used in the present study, which is close to the value of 0.217 suggested by Defina [20] and the value of 0.19 suggested by Singh et al. [34]. ΔU

^{+}was related to Re

_{Δ}, and the following relationship suggested by Ligrani and Moffat [35] was used:

_{u}, λ

_{u}, D

_{v}, and λ

_{v}are constants. Kironoto and Graf [37] obtained D

_{u}= 2.04, λ

_{u}= 0.97, D

_{v}= 1.14, and λ

_{v}= 0.76, using least-squares fits to the experimental data for a rough-bed OCF. u′ = $\sqrt{\overline{{u}^{2}}}$, v′ = $\sqrt{\overline{{v}^{2}}}$, u

^{2}/2 and v

^{2}/2 represent the streamwise and wall-normal TKE, respectively, and Equations (7) and (8) show that the roots of twice the time-averaged TKE for rough-bed OCFs collapsed with each other when they were scaled by friction velocities.

#### 3.2. VLSM Scaling in Rough-Bed OCFs

_{LSM}and λ

_{VLSM}, respectively, and their variations along the entire water depth are shown in Figure 4, where the maximal VLSM wavelength for R4 was ~30H, which is much longer than the ~20H in the other runs. Cameron et al. [11] proposed three possible contributions to the VLSM scaling, i.e., B/H, H/Δ, and Re

_{τ}. To unambiguously identify the dominant contributor, the maxima of λ

_{VLSM}are listed in Table 3, and are plotted against B/H, H/D, and Re

_{τ}in Figure 5a–c. For comparison, the experimental data for rough beds (published by Cameron et al. [11] and Zampiron et al. [38]) and for a smooth bed (published by Duan et al. [8]) are also reported. When calculating H/Δ, Δ = D was used in this study for consistency with the results reported by Cameron et al. [11], while Δ ≈ 3.5 k was set for the study by Zampiron et al. [38], who used a hook component with height k ≈ 1.1 mm as the bed roughness. In addition, Δ was assumed to be 0.05 mm for a clear-glass bed [39] in the experiment of Duan et al. [8].

_{τ}varied significantly, shows that the VLSM wavelengths rarely changed. The possibility that Re

_{τ}controlled the VLSM scale could be ruled out, which is consistent with the report of Peruzzi et al. [9]. Comparing R2 in this study with C2 (or C4) in the study by Duan et al. [8], the two runs had almost the same B/H but very different H/Δ, and the VLSM wavelengths were roughly the same. This result suggests that bed roughness probably does not play a role in controlling the VLSM wavelengths, at least for OCFs over bed types with smooth and large relative submergence. After ruling out the roles of H/Δ and Re

_{τ}, B/H appeared to be the controlling scale. Moreover, Ferraro et al. [40] proposed the VLSM size as a function of B/H rather than H/Δ because no significant VLSM variation was observed in their study when Δ varied without varying B/H. However, in their study of flows with low relative submergence, no VLSM-associated peaks were observed.

_{τ}is shown in Figure 5a–c, respectively. The data generally fit the curve relating the VLSM wavelengths to B/H proposed by Cameron et al. [11] despite some experimental scatter at low B/H ratios while deviating significantly from those related to H/Δ and Re

_{τ}. This phenomenon again suggests the importance of B/H in dominating the VLSM scale. The failure to predict λ

_{VLSM}on the basis of H/Δ is evident (Figure 5b), particularly for the smooth OCF case. In this case, the curve fell below 3H, which means that the VLSM wavelengths were even smaller than the LSM ones. Clearly, this is at odds with the facts and impossible.

#### 3.3. VLSM Contributions to TKE and Reynolds Stress

_{uu}and γ

_{uv}with the flow depth are plotted in Figure 6. The data from run R2 were not used for the aforementioned reasons. Figure 6a shows that VLSMs contributed ~60% to the TKE, which is consistent with the contribution of 55–65% reported by Duan et al. [8]. For each run, γ

_{uu}increased from the near-bed region until 0.5H and thereafter maintained a relatively stable value. In addition, an interesting feature is shown in Figure 6a. At the same water depth, γ

_{uu}increased with Re

_{τ}in the 0.1–0.5H region, implying VLSMs are more fully developed in higher Re

_{τ}flows. Given that nearly all hydraulic flows are high-Reynolds-number wall turbulence [19], an increasingly prominent role of VLSMs in the transport of momentum, energy, and mass is to be expected. This trend was also reported by Duan et al. [8] in their smooth-bed OCF study, as shown in Figure 5c. Figure 6b shows that the VLSMs contributed 38–50% to the Reynolds stress fraction, which is less than the reported contribution of 50–60% for smooth-bed OCFs. In contrast to the variation in γ

_{uu}with Re

_{τ}, the variation in γ

_{uv}showed no distinct trend, which is consistent with Yan et al. [7].

_{uu}with Re

_{τ}at y/H = 0.2 and 0.4 are shown in Figure 7. Evidently, γ

_{uu}increased almost linearly with Re

_{τ}. The slopes of the linear fits for y/H = 0.2 and 0.4 were 4.591 × 10

^{−5}and 2.366 × 10

^{−5}, respectively, suggesting that the growth of the VLSM fraction with Re

_{τ}slowed down as the VLSMs developed towards the water surface.

_{uu}corresponding to λ = 20H dropped to ~20%, suggesting the sensitivity of the VLSM-contributed TKE to the separation scale. The difference between these two different separation scales was exactly the amount of the TKE carried by structures with wavelengths in the 3–20H range, that is, ~40%. In addition, the tendency of γ

_{uu}to increase with Re

_{τ}is still present in Figure 8. Such a feature may signal a robust relationship between VLSMs and Re

_{τ}.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Notation

B | flume width |

D | particle diameter |

d | zero-plane displacement from the roughness tops |

Fs | sampling frequency of the velocity fields |

Fu_{i}(f) | FT of u_{i} (u_{1} = u, u_{2} = v) |

f | frequency of FT |

H | water depth |

J | bed slope |

k | bed roughness height |

k_{x} | streamwise wavenumber |

S_{uiuj}(f) | power spectral density of u_{i}u_{j} (u_{1} = u, u_{2} = v) |

T | total image acquisition time |

U | time-averaged velocity in the streamwise direction |

U_{m} | depth-averaged velocity |

u/v | fluctuating velocity in the streamwise or wall-normal direction |

u′/v′ | streamwise or wall-normal turbulence intensities |

u_{*} | friction velocity |

Fr | U_{m}/(gH)^{0.5} = Froude number |

Re | U_{m}H/ν = Reynolds number |

Re_{τ} | u_{*}H/ν = friction Reynolds number |

Re_{Δ} | u_{*}Δ/ν = roughness Reynolds number |

x/y | streamwise or wall-normal direction |

ν | kinematic viscosity |

Δ | equivalent roughness |

Δx/Δy | vector spacing in the streamwise or wall-normal direction |

ΔT | 1/Fs = time interval between successive velocity fields |

ΔU^{+} | roughness function |

γ_{uiuj}(f) | VLSM contributions to the TKE and Reynolds stress (u_{1} = u, u_{2} = v) |

λ | wavelength |

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**Figure 1.**(

**a**) Schematic of the open-channel flume. (

**b**) Definitions of flow depth H and zero-plane displacement d.

**Figure 2.**(

**a**) Wall-normal profile of mean velocity. (

**b**) Roughness function versus the roughness Reynolds number. (

**c**) Wall-normal profiles of turbulence intensities and Reynolds stress.

**Figure 3.**Premultiplied power spectra of streamwise velocity for runs: (

**a**) R1, (

**b**) R2, (

**c**) R3, (

**d**) R4 and (

**e**) R5.

**Figure 5.**Maximal VLSM wavelengths versus (

**a**) flow aspect ratio, (

**b**) relative submergence, and (

**c**) friction Reynolds number. The solid line shows the scaling relationship for the VLSM wavelengths proposed by Cameron et al. [11]. The experimental data for rough beds were from Cameron et al. [11] and Zampiron et al. [38] and the data for a smooth bed was from Duan et al. [8].

**Figure 6.**(

**a**) Streamwise kinetic energy fraction and (

**b**) Reynolds stress fraction contributed by VLSMs, using 3H as a separation scale.

**Figure 7.**Streamwise kinetic energy fraction contributed by VLSMs versus the friction Reynolds number for y/H = 0.2 and 0.4.

Run | J | ν (10^{−6}m^{2}/s) | H (cm) | U_{m} (m/s) | u_{*} (cm/s) | B/H | H/D | H/Δ | Fr | Re | Re_{τ} | Re_{Δ} |
---|---|---|---|---|---|---|---|---|---|---|---|---|

R1 | 0.003 | 0.88 | 1.73 | 0.25 | 2.25 | 14.5 | 4.33 | 6.49 | 0.61 | 4964 | 442 | 68 |

R2 | 0.002 | 0.84 | 2.87 | 0.27 | 2.17 | 8.7 | 7.17 | 10.76 | 0.50 | 9156 | 742 | 69 |

R3 | 0.001 | 0.88 | 4.88 | 0.21 | 1.50 | 5.1 | 12.19 | 18.28 | 0.30 | 11,393 | 831 | 46 |

R4 | 0.003 | 0.85 | 4.86 | 0.45 | 3.30 | 5.1 | 12.14 | 18.21 | 0.65 | 25,601 | 1879 | 104 |

R5 | 0.005 | 0.87 | 4.92 | 0.60 | 4.50 | 5.1 | 12.30 | 18.45 | 0.87 | 34,008 | 2537 | 138 |

^{(a)}J = bed slope, ν = kinematic viscosity, H = water depth above the roughness tops, U

_{m}= depth-averaged velocity, u

_{*}= friction velocity, B/H = flow aspect ratio, H/Δ = relative submergence, Fr = U

_{m}/(gH)

^{0.5}= Froude number, Re = U

_{m}H/ν = Reynolds number, Re

_{τ}= u

_{*}H/ν = friction Reynolds number, and Re

_{Δ}= u

_{*}Δ/ν = roughness Reynolds number.

Run | Image Size (Pixels) | Resolution (Pixels/mm) | F_{s} (Hz) | ΔT^{+} | ΔTU_{m}/H | No. of Image Pairs | TU_{m}/H | Δx^{+}/Δy^{+} |
---|---|---|---|---|---|---|---|---|

R1 | 128 × 640 | 30.77 | 600 | 0.96 | 0.024 | 19,268 × 3 | 1407 | 6.64 |

R2 | 128 × 560 | 17.43 | 100 | 5.61 | 0.093 | 109,784 × 3 | 15,360 | 23.4 |

R3 | 128 × 1600 | 30.77 | 600 | 0.43 | 0.007 | 38,513 × 20 | 5418 | 4.43 |

R4 | 128 × 1600 | 30.77 | 1200 | 1.06 | 0.008 | 38,513 × 12 | 3566 | 10.06 |

R5 | 128 × 1600 | 30.77 | 1400 | 1.66 | 0.009 | 38,513 × 20 | 6744 | 13.41 |

^{(b)}F

_{s}= sampling frequency of the velocity fields, ΔT = 1/Fs = time interval between successive velocity fields, T = total image acquisition time, Δx

^{+}or Δy

^{+}= inner-scaled vector spacing in the streamwise or wall-normal direction.

Author | Run | B/H | H/Δ | Re_{τ} | λ_{VLSM}/H |
---|---|---|---|---|---|

This study | R1 | 14.5 | 4.3 | 442 | 28.5 |

R2 | 8.7 | 7.2 | 742 | 22.0 | |

R3 | 5.1 | 12.2 | 831 | 19.0 | |

R4 | 5.1 | 12.1 | 1879 | 18.0 | |

R5 | 5.1 | 12.3 | 2566 | 20.1 | |

Cameron et al. (2017) [11] | H030 | 39.2 | 1.9 | 1140 | 50.7 |

H050 | 23.5 | 3.1 | 1900 | 39.0 | |

H070 | 16.7 | 4.4 | 2670 | 30.9 | |

H095 | 12.4 | 5.9 | 3590 | 26.0 | |

H120 | 9.8 | 7.5 | 4540 | 22.1 | |

Zampiron et al. (2020) [38] | s000 | 7.9 | 13.2 | 1360 | 25.3 |

Duan et al. (2020) [8] | C1 | 12 | 500 | 614 | 21.3 |

C2 | 8.6 | 700 | 1030 | 23.1 | |

C3 | 9.1 | 660 | 1508 | 29.4 | |

C4 | 8.6 | 1300 | 1903 | 21.5 | |

C5 | 7.2 | 1560 | 2407 | 18.8 |

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**MDPI and ACS Style**

Shen, Y.; Yang, S.; Liu, J.
Characteristics of Very Large-Scale Motions in Rough-Bed Open-Channel Flows. *Water* **2023**, *15*, 1433.
https://doi.org/10.3390/w15071433

**AMA Style**

Shen Y, Yang S, Liu J.
Characteristics of Very Large-Scale Motions in Rough-Bed Open-Channel Flows. *Water*. 2023; 15(7):1433.
https://doi.org/10.3390/w15071433

**Chicago/Turabian Style**

Shen, Ying, Shengfa Yang, and Jie Liu.
2023. "Characteristics of Very Large-Scale Motions in Rough-Bed Open-Channel Flows" *Water* 15, no. 7: 1433.
https://doi.org/10.3390/w15071433