Listen

Experimental investigations were conducted to analyze the effect of Reynolds numbers on turbulent flow properties in a nonuniform sand bed channel. Steady flow simulations were performed over the nonuniform sand bed channel, considering five Reynolds numbers within the range of 36500–53886. This article endeavors to delineate the influence of Reynolds number on turbulent flow properties through meticulous laboratory studies. Observations revealed that higher Reynolds numbers corresponded to increased longitudinal velocity. As the Reynolds number increases by 10 to 47%, various turbulent flow properties exhibit distinct trends. Specifically, the longitudinal velocity, longitudinal turbulent intensity, vertical turbulent intensity, turbulent kinetic energy, Reynolds shear stress, and Taylor scale show increases ranging from 5 to 30%, 15 to 25%, 15 to 20%, 25 to 60%, 20 to 40%, and 35 to 45%, respectively. Taylor scale analysis indicated higher magnitudes associated with higher Reynolds numbers. In-depth examinations of turbulent anisotropy, third-order moments of velocity fluctuations, kurtosis, turbulent kinetic energy production, and dissipation provided additional insights into flow behavior across different Reynolds numbers. This study contributes to a more comprehensive understanding of flow dynamics in nonuniform sand bed channels under varying Reynolds number conditions, bridging the gap between laboratory studies and real-world scenarios.

Listen

  • Reynolds number variations impact turbulent flow characteristics.

  • Longitudinal velocity and turbulent intensity increase with higher Reynolds numbers.

  • Reynolds shear stress and turbulent kinetic energy also rise with increasing Reynolds numbers.

  • Taylor scale reveals larger magnitudes associated with higher Reynolds numbers.

  • Insights into turbulent anisotropy and kurtosis enhance flow dynamics understanding.

kinetic energy budget, Reynolds number, Taylor scale, turbulent flow, turbulent intensity, velocity
Listen

Numerous researchers have delved into experimental studies of turbulent channel flow (Johansson & Alfredsson 1982; Kumar & Sharma 2022; Singh et al. 2023; Sahoo et al. 2023). A focal point for many investigations has been the Reynolds number's influence on skin friction and mean flow, as comprehensively reviewed by Dean (1978). Zanoun et al. (2009) conducted a review of experimental studies on turbulent channel flow, emphasizing the geometric challenges inherent in achieving well-resolved measurements at high Reynolds numbers. Notably, factors such as high aspect ratio and development length contribute to the increased costliness of attaining extremely high Reynolds numbers in channel flow facilities compared to pipe or boundary layer flow (Schultz & Flack 2013; Omar et al. 2022).

Experimental investigations into turbulent channel flow include Laufer's seminal study (1951), which explored streamwise turbulence statistics up to a Reynolds number based on channel height and bulk mean velocity Reynolds number (Re) of 62,000. Comte-Bellot & Craya (1965) extended these measurements to Re = 230,000, noting that even higher-order moments of streamwise and vertical velocities scaled on inner variables out to y+ = 100. Despite their contributions, both studies lacked spatial resolution. Wei & Willmarth (1989) addressed this limitation by conducting an extensive study using laser Doppler velocimetry, extending their analysis to Re = 40,000. They observed an increase in streamwise Reynolds normal stress (RNS) within the buffer layer with increasing Reynolds number, contrary to previous investigations. Pearson & Antonia (2001) delved into the Reynolds number dependence of turbulent velocity and pressure increments. They observed that the Kolmogorov-normalized moments of longitudinal and transverse velocity increments increased with Reynolds number within each range studied. Direct numerical simulations (Hoyas & Jiménez 2006; Omar et al. 2021a, 2021b) and experiments (Ng et al. 2011; Omar & Kumar 2021; Gaur et al. 2023) support Wei and Willmarth's findings. Deardorff (1970) described a three-dimensional numerical model for investigating turbulent shear flow within a channel at large Reynolds numbers. These studies collectively contribute to our understanding of turbulent channel flow dynamics. Hoyas & Jiménez (2008) numerically investigated the Reynolds stress budgets in turbulent channels, focusing on the effects of Reynolds number.

Furthermore, many experimental studies were performed on different Reynolds numbers to investigate the influence of Reynolds number in alluvial channels. Afzal et al. (2009) conducted three sets of experiments for Reynolds number in the range of 23,000 < Re < 72,000, and turbulent intensity and Reynolds shear stress distribution show that the effect of Reynolds number can be significant in the open-channel flow. Rapp & Manhart (2011) performed two-dimensional Particle image velocimetry (PIV) measurements in a water channel across Reynolds numbers ranging from 5,600 to 37,000. They validated their findings using point-by-point one-dimensional laser Doppler anemometry measurements. Schultz & Flack (2013) found that the skin-friction coefficient follows a power law for Re < 60 000, while at higher Reynolds numbers, it is best characterized by a logarithmic law with κ = 0.40 and A = 5.0. Essel et al. (2014) conducted experiments to investigate the low Reynolds number effect on the open-channel flow over a transverse square rib by using PIV. They observed that mean velocities were independent of Reynolds number in the recirculation region but reduced at the reattachment point, and turbulent kinetic energy (TKE) increased beyond the center of the recirculation region with increasing Reynolds number. Kähler et al. (2016) conducted high-resolution PIV and particle tracking velocimetry measurements in a water tunnel at Reynolds numbers of 8,000 and 33,000, offering a precise database for near-wall flow feature analysis. Török et al. (2019) introduced a novel method, which can separate sand or gravel-dominated bed load transport in rivers with mixed-size bed material, and the method was verified with field and laboratory data, both performed at nonuniform bed material. They found that the shear Reynolds number-based method operates more reliably than the Shield-Parker diagram. Wall-resolved large eddy simulations of turbulent flows over periodic hills conducted by Zhou et al. (2021) shed light on the Reynolds number's impact on flow statistics. They observed a decrease in the friction coefficient's magnitude with increasing Reynolds number, while the pressure coefficient exhibited an opposite trend. Moreover, higher Reynolds numbers gave rise to smaller turbulence structures, and mean velocities and Reynolds stresses demonstrated asymptotic behaviors with Reynolds number escalation. Bed roughness exerts a significant influence on turbulent flow within open channels, particularly as it interacts near the bed (Penna et al. 2020; Omar et al. 2021a, 2021b; Omar & Kumar 2021). Investigating Reynolds stress anisotropy in the open-channel flow amidst rigid emergent vegetation, Kumar et al. (2023) unveiled insights suggesting that flow anisotropy can be elucidated through anisotropic invariant maps. These maps indicate axis-symmetric expansion in nonvegetated zones, contrasting with axis-symmetric contraction observed within vegetation zones. Previous research has delved into comprehending the dynamic behavior of sinuous or meandering river systems through laboratory experiments (Devi et al. 2022; Kumar et al. 2022) and numerical studies (Omar 2015; Liu et al. 2021; Kadia et al. 2022).

Understanding how Reynolds number influences turbulent flow over a sand bed channel holds significant importance for realistic applications. Gao et al. (2020) explored Reynolds number effects on dynamics within the recirculation zone using wall-modeled large eddy simulations, considering Reynolds numbers up to Re = 105. Their study revealed that the length of the separation bubble behind the hill decreases as Reynolds number increases. In a different vein, Song & Eaton (2004) conducted experiments on a separating, reattaching, and recovering boundary layer in a closed-loop wind tunnel mounted inside a pressure vessel. They proposed empirical Reynolds number scaling for mean velocity and Reynolds stresses across various flow regions. Their findings indicate that the mean flow exhibits weak dependency on Reynolds number, whereas turbulence quantities strongly correlate with Reynolds number.

In summary, there exists a necessity for improved experimental data validation under controlled conditions, specifically focusing on understanding the impact of Reynolds number variations on flow turbulence. With this goal in mind, it is suggested to conduct experimental investigations to examine the influence of Reynolds number on turbulent flow over rough surfaces in open channels. From the previous studies, it is observed that the impact of Reynolds number on turbulent flow statics in open channels is yet to be explored. Therefore, the objective of this study is to systematically investigate the influence of Reynolds number variations on various turbulent flow properties within an open-channel flow context. Through measurements encompassing streamwise velocity, vertical velocity, streamwise turbulent intensity, vertical turbulent intensity, turbulent anisotropy, turbulent kinetic energy, Reynolds shear stresses, third-order moment of velocity fluctuations, kurtosis, Taylor scale, turbulent kinetic energy production, and turbulent kinetic energy dissipation, the research aims to discern how these parameters evolve across a spectrum of Reynolds numbers. The study encompasses five distinct Reynolds numbers spanning from 36,500 to 53,886, providing a comprehensive exploration of the Reynolds number's impact on turbulent flow characteristics within the open-channel flow regime. Reynolds number is very large in natural rivers (typically Re ⩾ 106) where flows are almost always turbulent (Malverti et al. 2008). Although smaller than in natural rivers, Reynolds number Re in present experiments is kept sufficiently high to ensure fully turbulent flow. In accordance with the ranges suggested by the previous literature, five different Reynolds number values were selected for the present experiments. In experimental flumes, maintaining a consistent Reynolds number range ensures that the flow conditions remain within a certain regime, facilitating more controlled and predictable experiments. The current study sheds light on how turbulent flow events are intricately linked with Reynolds number in open channels. Understanding turbulence is crucial as it plays a pivotal role in shaping morphodynamic changes by entraining and depositing sediments. By bridging the gap between laboratory studies and real-world scenarios, the research contributes to practical insights for managing nonuniform sand bed channels under varying flow conditions. This has implications for engineering designs, environmental management, and hydraulic infrastructure planning. Overall, these findings provide a comprehensive picture of how Reynolds numbers influence turbulent flow properties, offering valuable insights for both theoretical understanding and practical applications.

Listen
The experiments were conducted within a tilting straight rectangular flume measuring 20 m in length, 1 m in width, and 0.72 m in depth. Upstream of the flume, a tank measuring 2.8 m long, 1.5 m wide, and 1.5 m deep was utilized to straighten the flow before its entry into the flume. The channel bed was covered with sand, and wooden baffles were strategically placed at the upstream collection tank to mitigate highly turbulent flow, ensuring smoother flow entry into the channel. The upstream section was smoothed for 2 m to further stabilize the flow. The bed slope was uniformly set at 0.001 across all experiments and D50 of nonuniform sand bed 0.0005 m, which was used in the bed material of a open channel. Flow within the flume comes from the inlet tank, gradually released by a valve connected to the overhead storage tank. Since it is recirculating flume, the water is drained into an underground tank and pumped by three 10 HP centrifugal pumps to the upstream overhead storage tank. Discharge measurements were facilitated by a rectangular notch located downstream of the flume, where the coefficient of discharge (Cd) was found to be 0.82. The percentage deviation of flow discharge measured by rectangular notch was around ± 1.25%, which is low (Barman & Kumar 2022). In addition, water depth within the flume was monitored using a digital point gauge positioned at the center of the test section. Flow depth regulation was achieved through a tailgate mechanism positioned downstream of the channel. Further details about the experimental setup can be obtained from the study by Sharma & Kumar (2018). In the present work, the experimental setup considered five different discharges corresponding to five Reynolds numbers: 36,500 (R1), 40,215 (R2), 44,337 (R3), 48,878 (R4), and 53,886 (R5), respectively, arranged in the ascending order. Here, Reynolds number is calculated as Re = uR/υ, where u is the average velocity of the flow, R is the hydraulic radius, and υ is the kinematics viscosity of water. While the study investigated a range of Reynolds numbers, other influential parameters such as sediment size distribution and bed slope were not varied. This limitation could restrict the generalizability of the findings, as turbulence characteristics might vary differently under different combinations of parameters. Throughout the experiments, subcritical flow conditions were meticulously maintained. Further details regarding the experimental setup and parameters are presented in Table 1. A schematic diagram of the experimental setup is illustrated in Figure 1.
Table 1

Flow parameters for experimentations

Experiment No.Discharge (m3/s)Flow depth (m)Reynolds number (Re)Froude number
1. 0.037 0.107 36,500 0.333 
2. 0.040 0.112 40,215 0.343 
3. 0.044 0.115 44,337 0.363 
4. 0.049 0.117 48,878 0.390 
5. 0.054 0.119 53,886 0.419 
Experiment No.Discharge (m3/s)Flow depth (m)Reynolds number (Re)Froude number
1. 0.037 0.107 36,500 0.333 
2. 0.040 0.112 40,215 0.343 
3. 0.044 0.115 44,337 0.363 
4. 0.049 0.117 48,878 0.390 
5. 0.054 0.119 53,886 0.419 
View Large
Figure 1
Schematic diagram of tilting flume and laboratory setup.
View largeDownload slide

Schematic diagram of tilting flume and laboratory setup.

Figure 1
Schematic diagram of tilting flume and laboratory setup.
View largeDownload slide

Schematic diagram of tilting flume and laboratory setup.

Close modal

Instantaneous velocity data were acquired using a Nortek 3D Acoustic Doppler Velocimeter (ADV) at the downstream cross-section located at 5.5 m within the channel. A sampling rate of 200 Hz was employed for data collection, capturing readings at the channel center across a 5.5-m cross-section. However, it is important to note that the ADV could not capture data within 5 cm below the water surface due to limitations. During the data collection, the channel was in the mobile condition. As the sediment movement is so small, it is unlikely to distort velocity estimates significantly, except when the instrument is being used too close to a boundary and one or more of the measurements are made where the sampling volume includes part of the boundary. A four-beam downlooking ADV probe, named Nortek® Vectrino, was used to measure the instantaneous velocity components at a point. A sampling rate of 200 Hz was used for the data acquisition. It worked with an acoustic frequency of 10 MHz having an adjustable cylindrical sampling volume of 6 mm diameter and 1–4 mm height. The sampling length in the near-boundary flow was set with a lowest height of 1 mm. A sampling length of 1 mm was found to be adequate to capture the correct descriptions in the shear layer and near-boundary zones (Sharma et al. 2020). The measuring location was 5 cm below the probe. Hence, the flow field 5 cm below the free surface could not be measured. To validate the accuracy of the ADV, the standard deviation (Sd) of various flow characteristics near the channel bed was assessed. Around 15 pulses of velocity data were collected at z/h = 0.1, as outlined in Table 2, indicating low standard deviation across different flow characteristics. In Table 2, ⁠, , and represent the time-averaged velocities in the streamwise, spanwise, and vertical directions, respectively, while ⁠, , and denote the velocity fluctuations in the streamwise, spanwise, and vertical directions. ⁠, ⁠, and are the root mean square (rms) of ⁠, , and , respectively. Here, the term standard uncertainty of the mean refers to the standard deviation of the mean for a set of several repeated pulses of instantaneous velocities. Standard uncertainty is calculated as ⁠, where n is the number of measurements in the set. The data presented in Table 2 are within ±0.5% error for the time-averaged velocities and rms quantities, affirming the capability of the 200 Hz frequency of measurements by Vectrino. This confirms the accuracy of the ADV and validates its suitability for measurements. Data acquisition duration was set at 3 min for all experiments. This experiment was continued for 12–24 h with sediment particles remaining in the state of incipient motion throughout the channel test reach. These experiments were continued for several hours until the channel geometry and physical characteristics of bed features reached the equilibrium condition. To ensure reliable data, the signal-to-noise ratio was maintained at 15 or above, and a correlation of 70% between transmitted and received signals was adhered to as a cutoff value. The velocity data obtained from the ADV exhibit spikes due to interface issues between transmitted and received signals. To address this, the acceleration thresholding method proposed by Goring & Nikora (2002) was employed to filter out these spikes from the velocity data. In the given coordinate system, the z-axis (where z = 0) serves as the reference for the bed surface, with positive values indicating an upward direction. The x-axis aligns with the centerline of the flume, so x = 0 denotes the measurement location and positive values extend in the streamwise direction. The y-axis represents the transverse direction, with positive values to the right. Hence, the coordinates of the measuring location are (0, 0, z).

Table 2

Uncertainty associated with ADV data

Standard deviation 4.31 × 10−3 9.62 × 10−4 4.33 × 10−4 1.05 × 10−3 9.32 × 10−4 3.44 × 10−4 
Uncertainty % 0.33 0.061 0.85 0.091 0.072 0.041 
Standard deviation 4.31 × 10−3 9.62 × 10−4 4.33 × 10−4 1.05 × 10−3 9.32 × 10−4 3.44 × 10−4 
Uncertainty % 0.33 0.061 0.85 0.091 0.072 0.041 
View Large

Listen

Velocity distribution

Listen
Figure 2(a) and 2(b) illustrate the longitudinal and vertical velocity distributions, respectively, of the uniform flow setup in the downstream section (5.5 m) across different Reynolds numbers. The velocity distributions are plotted against the normalized depth of flow (z/h). In Figure 2(a), it is evident that longitudinal velocities within the channel increase toward the free surface, exhibiting a consistent distribution profile across all cases. Lower values of longitudinal velocity are observed near the bed surface, while higher values are attained toward the free surface. However, the magnitude of velocity varies depending on the Reynolds number in the channel. Specifically, the smallest Reynolds number (R1) exhibits the lowest magnitude of velocity distribution throughout the flow depth, with the magnitude increasing by 5–30% as the Reynolds number increases. This observation underscores the significant influence of Reynolds number on the velocity profile. As Reynolds number increases, the mean velocity profile becomes fuller, and the shape factor decreases correspondingly. This phenomenon contrasts with the laminar flat plate flow, where viscosity dominates across the entire layer, and the shape factor remains independent of Reynolds number (Gad-el-Hak & Bandyopadhyay 1994). In Figure 2(b), the distribution of vertical velocities shares a similar nature, with minimal differences in strength observed except for the smallest Reynolds number (R1 = 36,500) considered in this study. For R1, the profile of vertical velocities forms almost a straight vertical line, with its strength nearly zero (or very small negative values) near the bed surface. The strength slightly increases in the downward direction toward the free surface. In contrast, for Reynolds numbers R2–R5, the strength of vertical velocities is predominantly positive (indicating upward flow) and increases toward the free surface. At lower Reynolds numbers, the boundary layer thickness tends to be smaller. In the context of the experiment, this may result in a more pronounced influence of viscous effects, causing the flow to adhere more closely to the solid boundaries. As a result, the vertical velocity component near the boundaries could be suppressed, leading to an overall lower vertical velocity. In the case of Figure 2, higher Reynolds numbers may lead to the development of turbulent structures that induce more significant vertical motion within the flow domain, contributing to higher vertical velocities. The increase in longitudinal velocity with increasing Reynolds number in open-channel flows is attributed to enhanced turbulent mixing, boundary layer development, turbulent shear stress, and the presence of secondary currents (Pearson & Antonia 2001). These factors collectively influence the velocity distribution across the flow depth and result in higher velocities at higher Reynolds numbers.
Figure 2
Depth profiles of mean longitudinal velocities (u) and vertical velocities (w).
View largeDownload slide

Depth profiles of mean longitudinal velocities (u) and vertical velocities (w).

Figure 2
Depth profiles of mean longitudinal velocities (u) and vertical velocities (w).
View largeDownload slide

Depth profiles of mean longitudinal velocities (u) and vertical velocities (w).

Close modal

Turbulent intensity

Listen
Figure 3 presents the root mean square of velocity fluctuations for the uniform flow setup across Reynolds numbers ranging from 36,500 to 53,886. In Figure 3(a), the distribution of longitudinal turbulent intensity against Reynolds number is depicted for normalized depths of flow at z/h = 0.025, 0.2, and 0.52. The results indicate that the magnitude of longitudinal turbulent intensities generally increases by 15–25% with higher Reynolds numbers in the ranges of 36,500–53,886, except for Reynolds number R2. Initially, there is a decrease from R1 to R2, followed by an increase toward higher Reynolds numbers. At Reynolds number R2, the flow might be undergoing a transition from one type of turbulent regime to another. Such transitional flows can exhibit complex behavior where the relationship between turbulent intensities and Reynolds number may not follow a straightforward pattern. In certain flow regimes, specific instabilities might arise at particular Reynolds numbers, leading to fluctuations or suppression of turbulent intensities. These instabilities could be related to flow separation, vortex shedding, or other flow phenomena characteristic of the specific Reynolds number regime. Longitudinal turbulent intensity is notably higher in the inner layer near the bed surface (z/h = 0.025) and decreases toward z/h = 0.52 for each specific Reynolds number. Comparable investigations by Moser et al. (1999) and Hussain & Reynolds (1975) support these findings, indicating that longitudinal turbulent intensity tends to increase with rising Reynolds numbers. In Figure 3(b), the distribution of vertical turbulent intensity against Reynolds number is presented for normalized depths of flow at z/h = 0.025, 0.2, and 0.52. Vertical turbulent intensity increases by 15–20% with increasing Reynolds numbers in the ranges of 36,500–53,886, with lower values observed for the smallest Reynolds number and higher values for the greatest Reynolds number at each specific depth from the bed surface. When comparing depths from the bed surface, the vertical turbulent intensity is lower near the bed surface (z/h = 0.025) and higher value is achieved at the junction of the inner layer and the outer layer (z/h = 0.2). This trend signifies that longitudinal turbulent intensity is stronger near the bed surface, decreases toward the free surface, and increases with higher Reynolds numbers. The present results are agreed with those of Wei & Willmarth (1989). The Reynolds number increases and turbulence intensity generally increases, leading to enhanced vertical mixing and more vigorous exchange of fluid particles between different flow layers (Zhou et al. 2021). The increase in turbulent intensity with increasing Reynolds number in a sand bed channel has significant implications for sediment transport and channel morphology.
Figure 3
Variation of longitudinal (σu) and vertical (σw) turbulent intensity with Reynolds number.
View largeDownload slide

Variation of longitudinal (σu) and vertical (σw) turbulent intensity with Reynolds number.

Figure 3
Variation of longitudinal (σu) and vertical (σw) turbulent intensity with Reynolds number.
View largeDownload slide

Variation of longitudinal (σu) and vertical (σw) turbulent intensity with Reynolds number.

Close modal

Turbulent anisotropy

Listen
Figure 4 illustrates the anisotropy in the center of the channel for Reynolds numbers ranging from 36,500 to 53,886 at normalized depths from the bed surface (z/h = 0.025, 0.2, 0.52). Anisotropy is defined as the ratio of vertical turbulent intensity to longitudinal turbulent intensity (Kumar et al. 2023). As longitudinal intensity typically exceeds vertical intensity in a flow, the anisotropic ratio inversely reflects the anisotropic behavior exhibited by individual cases. It is observed that the anisotropy ratio varies across all three different depths. Deshpande & Kumar (2016) and Sharma & Kumar (2021) corroborated this variation. Near the bed surface (z/h = 0.025), the magnitude of flow anisotropy for Reynolds numbers (36,500–53,886) ranges from 0.25 to 0.3. At depth (z/h = 0.2), the magnitude increases with increasing Reynolds numbers, ranging between 0.325 and 0.35. For the outer layer (z/h = 0.52), the magnitude is higher than in the inner layer, ranging from 0.35 to 0.45. The turbulent anisotropy is higher toward the free surface by 30–45 % than near the bed surface. The ratio experiences slight changes and does not follow any specific trend for different Reynolds numbers at z/h = 0.025. However, at z/h = 0.2 and 0.52, the ratio tends to increase with increasing Reynolds numbers except for R3. This indicates that the ratio increases toward the free surface for all Reynolds numbers, suggesting that the region away from the bed surface is more anisotropic than the region near the bed surface. While flow anisotropy provides insight into the extent of anisotropy at different depths for different Reynolds number cases, it does not account for lateral intensity.
Figure 4
Variation of turbulent anisotropy with Reynolds number.
View largeDownload slide

Variation of turbulent anisotropy with Reynolds number.

Figure 4
Variation of turbulent anisotropy with Reynolds number.
View largeDownload slide

Variation of turbulent anisotropy with Reynolds number.

Close modal

Turbulent kinetic energy

Listen
Figure 5 depicts the variation of turbulent kinetic energy against Reynolds number at normalized depths of z/h = 0.025, 0.2, and 0.52. Turbulent kinetic energy (k) is defined as the cumulative effect of turbulent intensity, represented by the equation: k = 0.5⁠, where k = turbulent kinetic energy and are turbulent intensities in the longitudinal, lateral, and vertical directions, respectively (Kumar & Sharma 2022). The results indicate that the magnitude of turbulent kinetic energy increases with the rising Reynolds number. Turbulent kinetic energy experiences an increase ranging from 25 to 60% across Reynolds numbers within the range of 36,500 to 53,886. Comparing normalized depths, higher magnitudes are observed near the bed surface (z/h = 0.025), which gradually decreases toward the free surface. In the inner layer (z/h = 0.025 and 0.2), the magnitude increases in the ascending order of Reynolds numbers (36,500–53,886), ranging between 15 and 25 cm2/s2. The magnitude is slightly higher at z/h = 0.025 than at z/h = 0.2 for Reynolds numbers (40,215–53,886). In the outer layer (z/h = 0.52), the magnitude slightly increases with the increasing Reynolds numbers, ranging within 10–15 cm2/s2, which is smaller than the turbulent kinetic energy in the inner layer. The results indicate that the inner layer exhibits a higher turbulent kinetic energy compared to the outer layer or the region near the free surface, with an increase ranging from 25 to 50%, and it also increases with higher Reynolds numbers. This is because the bed surface acts as a source of turbulent energy due to the interaction between the flow and the roughness elements, leading to enhanced turbulence production in this region. As Reynolds number increases, turbulence intensity typically increases, leading to higher levels of turbulent kinetic energy. This is because higher Reynolds numbers are associated with greater flow inertia relative to viscous forces, allowing for more vigorous turbulent motion and energy transfer within the flow. Turbulent kinetic energy provides insight into the extent of turbulence for different Reynolds numbers at different normalized depths.
Figure 5
Variation of turbulent kinetic energy with Reynolds number.
View largeDownload slide

Variation of turbulent kinetic energy with Reynolds number.

Figure 5
Variation of turbulent kinetic energy with Reynolds number.
View largeDownload slide

Variation of turbulent kinetic energy with Reynolds number.

Close modal

Reynolds shear stress

Listen

Reynolds shear stress provides insight into momentum exchange resulting from interactions among fluctuating velocities in different flow layers (Tang et al. 2021). The fluctuations in longitudinal and vertical velocities denoted as and , respectively, represent the random quantities in the open-channel flow. Their mean product yields the Reynolds shear stress, which is expressed as ⁠, where ρ is the density of water.

Figure 6 illustrates the variation of Reynolds shear stress across Reynolds numbers ranging from 36,500 to 53,886 at normalized depths of z/h = 0.025, 0.2, and 0.52. The results indicate that Reynolds shear stress increases with escalating Reynolds numbers. The Reynolds shear stress increases by 20 to 40% as the Reynolds number increases from 36,500 to 53,886. Near the bed surface (z/h = 0.025), its magnitude is higher and decreases toward the outer layer (z/h = 0.52). Reynolds shear stress is notably stronger in the inner layer, particularly close to the bed surface by 30–45%, compared to the outer layer. The influence of Reynolds number on Reynolds shear stress appears more pronounced for higher Reynolds numbers and closer to the bed surface or inner layer, while it is relatively weaker for smaller Reynolds numbers and toward the free surface or outer layer. This observation may be attributed to the minimum velocity within the inner layer, which creates substantial velocity differences, particularly for higher Reynolds numbers, resulting in heightened momentum exchange compared to the outer layer at lower Reynolds numbers. Consequently, Reynolds shear stress increases with higher Reynolds numbers, and its strength is more prominent in the inner layer. The current findings align well with those of Zhou et al. (2021). This agreement suggests intensified turbulent motions and enhanced momentum exchange between fluid layers, consequently resulting in elevated levels of Reynolds shear stress. The interaction between the flow and the channel bed results in intensified turbulent motions and increased momentum transfer, leading to higher levels of shear stress. The strong shear stress in the inner layer results in increased flow resistance and energy dissipation rates near the bed surface (Song & Eaton 2004; Kumar & Sharma 2022). Reynolds shear stress also plays a crucial role in mixing processes, facilitating the exchange of momentum and mass between different flow layers.
Figure 6
Variation of Reynolds shear stress (RSS) with Reynolds number.
View largeDownload slide

Variation of Reynolds shear stress (RSS) with Reynolds number.

Figure 6
Variation of Reynolds shear stress (RSS) with Reynolds number.
View largeDownload slide

Variation of Reynolds shear stress (RSS) with Reynolds number.

Close modal

Third-order moments of velocity fluctuation

Listen
According to Kumar et al. (2023), turbulence flow skewness can be expressed as the third-order moments of velocity fluctuation. The third-order moment reflects the symmetric Gaussian distribution about the mean or zero skewness. A positive skewness indicates a distribution shift to the right, while a negative skewness suggests a distribution shift to the left, relative to the mean (Kumar et al. 2023). According to Raupach (1981), the third-order moments of velocity fluctuation can be determined using Equation (1).
(1)
where and ⁠.

Here denotes the streamwise flux of RNS in the streamwise direction and defines the vertical flux of RNS in the vertical direction.

Figure 7(a) and 7(b) illustrate the moments of velocity fluctuations in the streamwise flux and the vertical flux of RNS, respectively, for Reynolds numbers ranging from 36,500 to 53,886 at normalized depths (z/h = 0.025, 0.2, 0.52). In Figure 7(a), M3,0 is more pronounced in the inner layer, especially near the bed surface (z/h = 0.025), with its strength slightly decreasing with increasing Reynolds numbers. Near the bed surface, M3,0 changes from positive to negative values with increasing Reynolds numbers, indicating that the streamwise flux of RNS moves in the streamwise direction near the bed surface. At z/h = 0.2 and 0.52, M3,0 becomes negative, suggesting that the streamwise flux of RNS moves in the upstream direction, with its magnitude (in the negative direction) increasing with higher Reynolds numbers. M3,0 changes from positive to negative from the bed surface to the free surface. It is decreasing by 100 to 200% with higher Reynolds numbers within the range of 36,500–53,886. Figure 7(b) reveals that M0,3 is consistently positive throughout the flow depth (i.e., z/h = 0.025, 0.2, and 0.52) for all Reynolds numbers in the range of 36,500–53,886, indicating that the vertical flux of RNS moves in the upward direction. The strength of M0,3 is higher in the outer layer than the inner layer and increases with increasing Reynolds numbers. In comparison, M3,0 is higher near the bed surface and decreases with increasing Reynolds numbers, while M0,3 is smaller near the bed surface and increases with increasing Reynolds numbers. Higher Reynolds numbers endorse stronger and organized turbulent structures, resulting in a reduction in velocity asymmetry and, consequently, a decrease in skewness (Pearson & Antonia 2001). The inner layer of turbulent flow, particularly near the bed, experiences strong velocity gradients and turbulent mixing. Near the bed surface, the presence of boundary effects and complex flow interactions contribute to asymmetric velocity distributions, leading to higher skewness values in this region compared to the outer layers. Near the bed surface, where velocity gradients are most pronounced, boundary effects play a significant role in generating asymmetric velocity distributions and higher skewness values (Kumar et al. 2023). As Reynolds number increases, these boundary effects become less pronounced, leading to a decrease in skewness.
Figure 7
Variation of third-order moment with Reynolds number.
View largeDownload slide

Variation of third-order moment with Reynolds number.

Figure 7
Variation of third-order moment with Reynolds number.
View largeDownload slide

Variation of third-order moment with Reynolds number.

Close modal

Kurtosis

Listen

The fourth-order correlation, also known as kurtosis, serves to characterize the intermittency of turbulence. The kurtosis in the longitudinal direction, denoted as ‘K(u)’, and the vertical direction, denoted as ‘K(w)’, can be calculated as follows:

(2)

The magnitude of K(u) and K(w) should ideally reflect the behavior expected in the case of isotropic turbulence with the Gaussian distribution (Sharma et al. 2021).

Figure 8(a) and 8(b) depict the distribution of kurtosis in the longitudinal and transverse directions, respectively, against Reynolds number ranging from 36,500 to 53,886 for normalized depths (z/h = 0.025, 0.2, and 0.52). In Figure 8(a), the longitudinal kurtosis is higher for z/h = 0.2 and smaller for z/h = 0.52. The distribution of longitudinal kurtosis initially increases from R1 to R2 and then decreases with increasing Reynolds number. In Figure 8(b), the vertical kurtosis is higher for R1 and smaller for R2–R5 in the outer layer (z/h = 0.52) compared to the inner layer. The distribution of vertical kurtosis does not follow any specific pattern; the magnitude randomly varies with increasing Reynolds number for all three normalized depths. The vertical kurtosis is influenced by the intricate interplay of various flow parameters and turbulent structures within the channel. These dynamics can be highly complex and variable, leading to a lack of clear patterns in the distribution of kurtosis. Near-wall effects and interactions with the channel bed or boundaries can influence the distribution of vertical kurtosis. These effects may introduce complexities that obscure any discernible patterns in the kurtosis distribution. In nonuniform sand bed channels, flow stratification due to sediment transport or variations in flow velocity profiles can further complicate the distribution of kurtosis. Different flow layers or regions may exhibit distinct kurtosis patterns, contributing to the overall lack of a specific pattern. The obtained values suggest a deviation from the normal distribution and indicate a higher level of turbulence intermittency in the flow region. This confirms that sweep and ejection are responsible for the bursting events that strongly contribute to turbulence production. The high levels of skewness and kurtosis reflect high levels of intermittency in the particle trajectories. Notably, when K(u) > 3, it implies a profile with a peaky signal pattern of intermittent turbulence events, while K(u) < 3 suggests a flatter pattern (Sharma et al. 2021). The experimental value of K(u) indicates a minimal value, suggesting that the signal tends to yield a flatter pattern for the flow interacting with the bed surface. The value of kurtosis provides a quantitative measure of departures, reflecting the skewness and peakedness of the velocity distribution. The distribution of a non-Gaussian velocity with high kurtosis may indicate the existence of coherent structures or localized turbulence phenomena within the flow. Higher values of kurtosis may suggest the increased flow variability and sediment transport capacity, whereas smaller values of kurtosis may indicate more stable flow conditions with reduced potential of sediment transport. Kurtosis can also provide insights into the spatial and temporal variability of turbulent structures and mixing processes within the flow. Variations in kurtosis along the length of channel or across different flow conditions may indicate changes in turbulent energy production and dissipation, which can influence flow dynamics and sediment transport patterns.
Figure 8
Variation of kurtosis with Reynolds number.
View largeDownload slide

Variation of kurtosis with Reynolds number.

Figure 8
Variation of kurtosis with Reynolds number.
View largeDownload slide

Variation of kurtosis with Reynolds number.

Close modal

Taylor scale

Listen
Turbulent flow is characterized by the formation of eddies, which vary widely in size and can be comparable to the depth of the channel through which the flow occurs. This section examines the Taylor micro-scale for Reynolds numbers ranging from 36,500 to 53,886, specifically at normalized depths of z/h = 0.02, 0.2, and 0.52. The Taylor micro-scale represents the length scale of the inertial subrange, where inertial effects predominantly control eddy motions. It provides an average estimation of the size of eddies at a specific point within the flow.
(3)
where v is the kinematic viscosity of water and is the longitudinal velocity fluctuations. The value of TKE dissipation rate is found as follows (Krogstad & Antonia 1999):
(4)
Figure 9 illustrates the Taylor micro-scale within a sand bed channel under uniform flow conditions across various Reynolds numbers. In the Reynolds number range of 36,500–53,886 and at normalized depths of z/h = 0.02, 0.2, and 0.52, the Taylor micro-scale exhibits distinct behaviors. Initially, its magnitude decreases as the Reynolds number increases from 36,500 to 40,215. However, beyond this threshold (40,215–53,886), the magnitude of the Taylor scale increases by 35–45% with the rising Reynolds number for all normalized depths. It is noteworthy that concerning the normalized depth (z/h), the Taylor scale's magnitude is smaller near the bed surface (z/h = 0.02), and higher at z/h = 0.2 for every Reynolds number. This implies that the magnitude of the Taylor scale increases from the vicinity of the bed surface toward z/h = 0.2 by 40–90%, after which it decreases in the outer layer toward the free surface by 10–40%. The mean velocity of the section significantly influences the Taylor scale, as evident in the dissipation rate equation (Equation (4)). The flow characteristics within the sand bed channel, across different Reynolds numbers and depths, ultimately impact the average size of eddies within the system.
Figure 9
Variation of Taylor scale with Reynolds number.
View largeDownload slide

Variation of Taylor scale with Reynolds number.

Figure 9
Variation of Taylor scale with Reynolds number.
View largeDownload slide

Variation of Taylor scale with Reynolds number.

Close modal

Turbulent kinetic energy budget

Listen
In a two-dimensional flow field, the turbulent energy budget is characterized by turbulent production denoted as and turbulent dissipation ε as expressed by Equation (4). Here, tp represents the TKE production, while tD signifies the TKE dissipation. Figure 10 displays the distribution of tp and tD for various Reynolds numbers ranging from 36,500 to 53,886, observed at normalized depths z/h = 0.02, 0.20, and 0.520. In Figure 10(a), turbulent production tp exhibits an increase near the bed surface (at z/h = 0.02), followed by a rapid decrease. It eventually stabilizes at a nearly constant, albeit small, magnitude for z/h = 0.52, consistent across different Reynolds numbers within the range of 36,500–53,886. The distribution of tp does not display a specific trend concerning Reynolds number across all normalized depths. This heightened turbulence near the bed surface correlates with increased turbulence intensities and greater momentum transfer across the nonuniform sand bed. A positive value of tp indicates the conversion of energy from the time-averaged flow to turbulence (Sharma & Kumar 2021). Figure 10(b) illustrates the distribution of turbulent kinetic energy dissipation (tD), which is higher near the bed surface and diminishes towards the free surface. In the outer layer (z/h = 0.52), tD is smaller compared to the inner layer flow for each Reynolds number. Notably, the distributions of tD exhibit a distinct lag compared to those of tp. The influence of sediment entrainment is evident in the near-bed distributions of tp and tD, where the lag is reversed, indicating tD>tp. To elaborate, the effect of sediment entrainment reduces tp while increasing tD. The reduction of tp near the bed surface impacts velocity distributions significantly. The crucial nature of these dynamics near the bed surface suggests an increased rate of sediment transport due to gains in turbulence production.
Figure 10
Variation of turbulent energy production and dissipation with Reynolds number.
View largeDownload slide

Variation of turbulent energy production and dissipation with Reynolds number.

Figure 10
Variation of turbulent energy production and dissipation with Reynolds number.
View largeDownload slide

Variation of turbulent energy production and dissipation with Reynolds number.

Close modal

The measured and calculated values of the flow characteristics for Reynolds number in the range of 36,500 ≤ Re ≤ 53,886 in each index provides the important information, and characteristics are summed up in Table 3 by comparing the trend of each index for Reynolds number (36,500 ≤ Re ≤ 53,886) and normalized flow depth (z/h = 0.025, 0.2, and 0.52) in a sand bed straight rectangular channel. The relevance of the effect of Reynolds number research is appropriate for identifying the sensitivity of the turbulent flow characteristics for the various Reynolds numbers (Afzal et al. 2009; Rapp & Manhart 2011; Zhou et al. 2021), turbulence in a water tunnel for near-wall feature analysis (Kähler et al. 2016), which is important to examine the turbulence model. The development of a series of models with experimental datasets in the range of Reynolds number is a main aim in the improvement of turbulence modeling. Therefore, the current work provides the crucial information on the modification turbulence structure for different Reynolds number. The information needed to know the effect of Reynolds number in the open-channel flow is therefore provided in this study.

Table 3

Flow characteristics and their trend

Sl. No.Flow characteristicsResults (36,500 ≤ Re ≤ 53,886)
1. u Increases 
w Minimal differences in strength except R1 
2.  Increases and higher toward free surface 
 Increases and higher in inner layer 
3.  Increases except R3 and smaller near bed 
4. K Increases and smaller toward free surface 
5. RSS Increases and higher near bed 
6. M30 Decreases and smaller toward free surface with negative magnitude 
M03 Increases and higher toward free surface 
7. Ku Increases then decrease and smaller toward free surface 
Kw Randomly varies 
8.  Increases and smaller near bed 
9.  Randomly varies and higher near bed 
tD Decreases and higher near bed 
Sl. No.Flow characteristicsResults (36,500 ≤ Re ≤ 53,886)
1. u Increases 
w Minimal differences in strength except R1 
2.  Increases and higher toward free surface 
 Increases and higher in inner layer 
3.  Increases except R3 and smaller near bed 
4. K Increases and smaller toward free surface 
5. RSS Increases and higher near bed 
6. M30 Decreases and smaller toward free surface with negative magnitude 
M03 Increases and higher toward free surface 
7. Ku Increases then decrease and smaller toward free surface 
Kw Randomly varies 
8.  Increases and smaller near bed 
9.  Randomly varies and higher near bed 
tD Decreases and higher near bed 
View Large

Listen

Laboratory experiments were conducted to explore the flow characteristics within a nonuniform sand bed channel across various Reynolds numbers. The study encompassed diverse turbulent properties including velocity profiles, turbulent intensity, turbulent anisotropy, Reynolds shear stress, Taylor scale, turbulent kinetic energy production, and turbulent kinetic energy dissipation. Reynolds numbers ranging from 36,500 to 53,886 were investigated across different normalized depths. The following conclusion is achieved from the study:

  • (i)

    The analysis revealed an increase in longitudinal velocity with escalating Reynolds numbers, whereas smaller vertical velocities were observed for lower Reynolds numbers.

  • (ii)

    Turbulent intensity, Reynolds shear stress, turbulent kinetic energy, and Taylor scale exhibited increments with rising Reynolds numbers.

  • (iii)

    Longitudinal turbulent intensity and Reynolds shear stress were notably pronounced near the bed surface, while vertical turbulent intensity, turbulent anisotropy, and Taylor scale were diminished in proximity to the bed surface across all Reynolds numbers.

  • (iv)

    Furthermore, turbulent kinetic energy production, dissipation, and third-order moments of velocity fluctuations in the longitudinal direction were higher near the bed surface.

  • (v)

    Taylor scale demonstrated an increasing trend, while third-order moments of velocity fluctuations in the longitudinal direction decreased with increasing Reynolds numbers.

Kurtosis in both longitudinal and vertical directions was also analyzed in the study. The examination of turbulent kinetic energy budgets across different Reynolds number setups facilitated the clarification of production and dissipation regions within the channel. Consequently, the findings underscored the influence of Reynolds number on turbulent flow properties in nonuniform sand bed channels.

This article offers insights into how flow behaviors vary across different Reynolds numbers. It is crucial to note that real-world river conditions often deviate from uniform Reynolds numbers, exhibiting diverse distributions across cross-sections.

Listen

This research received no specific grant from any funding agency

Listen

This article does not contain any studies with human participants or animals performed by any of the authors.

Listen

Informed consent was obtained from all individual participants included in the study.

Listen

All relevant data are included in the paper or its Supplementary Information.

Listen

The authors declare there is no conflict.

Afzal
B.
,
Faruque
M. A.
 &
Balachandar
R.
2009
Effect of Reynolds number, near-wall perturbation and turbulence on smooth open-channel flows
.
Journal of Hydraulic Research
47
(
1
),
66
–
81
.
Google Scholar
Crossref
Search ADS
 
Comte-Bellot
G.
 &
Craya
A.
1965
Ecoulement Turbulent Entre Deux Parois Paralleles
.
Service de documentation scientifique et technique de l'armement
,
New York
, p.
159
.
Google Scholar
 
Dean
R. B.
1978
Reynolds number dependence of skin friction and other bulk flow variables in two-dimensional rectangular duct flow
.
Journal of Fluids Engineering
100
(
2
),
215
–
223
.
Google Scholar
Crossref
Search ADS
 
Essel
E. E.
,
Atamanchuk
K.
,
d'Auteuil
S.
 &
Tachie
M. F.
2014
Low Reynolds Number Effect on Open Channel Flow Over a Rib
. In:
Fluids Engineering Division Summer Meeting
, Vol.
46230
.
American Society of Mechanical Engineers
,
Chicago, IL
, p.
V01CT16A010
.
Google Scholar
Crossref
Search ADS
 
Gad-el-Hak
M.
 &
Bandyopadhyay
P. R.
1994
ReynoldsNumber Effects in Wall-Bounded Turbulent flows
47
,
307
–
365
.
Google Scholar
 
Gaur
S.
,
Omar
P. J.
 &
Eslamian
S.
2023
Advantage of grid-free analytic element method for identification of locations and pumping rates of wells
. In:
Handbook of Hydroinformatics
(Eslamian, S. & Eslamian, F., eds.)
.
Elsevier
,
Amsterdam
, pp.
1
–
10
.
Google Scholar
Crossref
Search ADS
 
Goring
D. G.
 &
Nikora
V. I.
2002
Despiking acoustic Doppler velocimeter data
.
Journal of hydraulic engineering
128
(
1
),
117
–
126
.
Google Scholar
Crossref
Search ADS
 
Hussain
A. K. M. F.
 &
Reynolds
W. C.
1975
Measurements in fully developed turbulent channel flow
.
ASME J. Fluids Eng.
97
,
568
–
580
.
Google Scholar
Crossref
Search ADS
 
Johansson
A. V.
 &
Alfredsson
P. H.
1982
On the structure of turbulent channel flow
.
Journal of Fluid Mechanics
122
,
295
–
314
.
Google Scholar
Crossref
Search ADS
 
Kähler
C. J.
,
Scharnowski
S.
 &
Cierpka
C.
2016
Highly resolved experimental results of the separated flow in a channel with streamwise periodic constrictions
.
Journal of Fluid Mechanics
796
,
257
–
284
.
Google Scholar
Crossref
Search ADS
 
Krogstad
P. Å.
 &
Antonia
R. A.
1999
Surface roughness effects in turbulent boundary layers
.
Experiments in Fluids
27
(
5
),
450
–
460
.
https://doi.org/10.1007/s003480050370
.
Google Scholar
Crossref
Search ADS
 
Kumar
A.
,
Kumar
P.
 &
Sharma
A.
2022
Structure of Turbulent Flow in a Meander Gravel Bed Channel
. In
International Conference on River Corridor Research and Management
.
Springer Nature Singapore
,
Singapore
, pp.
107
–
116
.
Google Scholar
Crossref
Search ADS
 
Laufer
J.
1951
Investigation of turbulent flow in a two-dimensional channel (No. NACA-TR-1053)
.
Malverti
L.
,
Lajeunesse
E.
 &
Métivier
F.
2008
Small is beautiful: Upscaling from microscale laminar to natural turbulent rivers
.
Journal of Geophysical Research: Earth Surface
113
(F4
),
F04004
.
Google Scholar
 
Moser
R. D.
,
Kim
J.
 &
Mansour
N. N.
1999
Direct numerical simulation of turbulent channel flow up to ret5590
.
Phys. Fluids
11
,
943
–
945
.
Google Scholar
Crossref
Search ADS
 
Ng
H. C. H.
,
Monty
J. P.
,
Hutchins
N.
,
Chong
M. S.
 &
Marusic
I.
2011
Comparison of turbulent channel and pipe flows with varying Reynolds number
.
Experiments in Fluids
51
,
1261
–
1281
.
Google Scholar
Crossref
Search ADS
 
Omar
P. J.
2015
Geomatics techniques based significance of morphometric analysis in prioritization of watershed
.
International Journal of Enhanced Research in Science Technology and Engineering
4
(
3
),
15
–
16
.
Google Scholar
 
Omar
P. J.
,
Gaur
S.
,
Dwivedi
S. B.
 &
Dikshit
P. K. S.
2021a
Development and application of the integrated GIS-MODFLOW model
.
Fate and Transport of Subsurface Pollutants
24
,
305
–
314
.
Google Scholar
Crossref
Search ADS
 
Omar
P. J.
,
Shivhare
N.
,
Dwivedi
S. B.
,
Gaur
S.
 &
Dikshit
P. K. S.
2021b
Study of methods available for groundwater and surfacewater interaction: A Case Study on Varanasi, India
. In
The Ganga River Basin: A Hydrometeorological Approach
(Chauhan, M. S. & Prasad Ojha, C. S., eds.). Springer, Cham, pp. 67–83
.
Google Scholar
 
Omar
P. J.
,
Rai
S. P.
 &
Tiwari
H.
2022
Study of morphological changes and socio-economic impact assessment: A case study of Koshi River
.
Arabian Journal of Geosciences
15
(
17
),
1426
.
Google Scholar
Crossref
Search ADS
 
Pearson
B. R.
 &
Antonia
R. A.
2001
Reynolds-number dependence of turbulent velocity and pressure increments
.
Journal of Fluid Mechanics
444
,
343
–
382
.
Google Scholar
Crossref
Search ADS
 
Rapp
C.
 &
Manhart
M.
2011
Flow over periodic hills: An experimental study
.
Experiments in Fluids
51
(
1
),
247
–
269
.
Google Scholar
Crossref
Search ADS
 
Sahoo
S.
,
Khuntia
J. R.
,
Devi
K.
,
Sai Prasad
B. S.
 &
Kumar Khatua
K.
2023
Turbulence modelling for depth-averaged velocity and boundary shear stress of a dense rigid grass bed open channel
.
AQUA – Water Infrastructure, Ecosystems and Society
72
(
9
),
1748
–
1769
.
Google Scholar
Crossref
Search ADS
 
Schultz
M. P.
 &
Flack
K. A.
2013
Reynolds-number scaling of turbulent channel flow
.
Physics of Fluids
25
,
2
.
Google Scholar
Crossref
Search ADS
 
Singh
A. N.
,
Mudgal
A.
,
Tripathi
R. P.
 &
Omar
P. J.
2023
Assessment of wastewater treatment potential of sand beds of River Ganga at Varanasi, India
.
AQUA – Water Infrastructure, Ecosystems and Society
72
(
5
),
690
–
700
.
Google Scholar
Crossref
Search ADS
 
Tang
X.
,
Rahimi
H.
,
Guan
Y.
 &
Wang
Y.
2021
Hydraulic characteristics of open-channel flow with partially-placed double layer rigid vegetation
.
Environmental Fluid Mechanics
21
,
317
–
342
.
Google Scholar
Crossref
Search ADS
 
Wei
T.
 &
Willmarth
W. W.
1989
Reynolds-number effects on the structure of a turbulent channel flow
.
Journal of Fluid Mechanics
204
,
57
–
95
.
Google Scholar
Crossref
Search ADS
 
Zanoun
E. S.
,
Nagib
H.
 &
Durst
F.
2009
Refined cf relation for turbulent channels and consequences for high-Re experiments
.
Fluid Dynamics Research
41
(
2
),
021405
.
Google Scholar
Crossref
Search ADS
 
Zhou
Z.
,
Wu
T.
 &
Yang
X.
2021
Reynolds number effect on statistics of turbulent flows over periodic hills
.
Physics of Fluids
33
(
10
),
105124
.
Google Scholar
Crossref
Search ADS
 
© 2024 The Authors
This is an Open Access article distributed under the terms of the Creative Commons Attribution Licence (CC BY-NC-ND 4.0), which permits copying and redistribution for non-commercial purposes with no derivatives, provided the original work is properly cited (http://creativecommons.org/licenses/by-nc-nd/4.0/).