ABSTRACT
Research on the vertical profiles of flow velocity in mountainous river channels is limited, particularly in scenarios where complex bed geometries are absent. Due to the coarse roughness and seepage flow on streambeds composed of gravel, the conventional formulae for flow velocity profiles derived from fluvial river channels do not apply to mountainous river channels. Based on flume experiments with a bed packed with natural gravel and a slope ranging from 0.006 to 0.16, we derived a theoretical formula for flow velocity profiles. This new formula integrates the influence of the subsurface flow and velocity reduction near the water surface, demonstrating a strong alignment with measurements. Our findings indicate that for shallow water flow over rough bed surfaces, the turbulence intensity diminishes along the vertical direction in the near-bed region while remaining relatively constant in the upper water body. Contrary to conventional theories which attribute the increase in flow resistance and the decrease in sediment transport rates in mountainous river channels to form drag, our study emphasizes that the subsurface flow plays a significant role in the overall flow resistance of mountainous river channels and should not be overlooked.
HIGHLIGHTS
The comprehension of flow velocity profiles in mountainous river channels remains limited, especially in scenarios with subsurface flow and secondary flow.
Study challenges conventional theories by highlighting the significant role of subsurface flow in overall flow resistance and sediment transport rates in mountainous river channels.
NOTATIONS
- D50
median particle size
- D84
particle size of which 84% are finer
- ks
roughness height
- S
flume bed slope
- Q
discharge
- W
flume width
- Re
Reynolds number
- Reks
roughness Reynolds number
- Fr
Froude number
- U
average velocity
- u0
seepage velocity at the bed surface
- u*
bed shear velocity
downstream velocity averaged in time
- Lt
mixing length
- ε
Eddy viscosity
- g
gravity acceleration
- h
water depth
- P
thickness of exchange layer
- η
thickness of subsurface layer
- R
specific density
- x, y, z
flume reference frame
- τ
shear stress
- τ*
bed shear stress
- α1, b1
coefficients of Equation (7)
INTRODUCTION
In open-channel flows, the vertical profiles of flow velocity have been extensively studied by numerous researchers (Finley et al. 1966; Nezu & Rodi 1986; Wiberg & Smith 1991; Ferro & Baiamonte 1994; Ferro 2003; Wang et al. 2015; Lamb et al. 2017a, 2017b; Luo et al. 2020, 2022), resulting in various forms of velocity profile formulae, including logarithmic (Nezu & Rodi 1986; Wiberg & Smith 1991; Lamb et al. 2017b; Luo et al. 2022), hyperbolic tangent (Luo et al. 2020), and irregular forms (Finley et al. 1966; Ferro & Baiamonte 1994; Ferro 2003; Wang et al. 2015). Among these, the logarithmic formulae are the most widely applied. However, to date, most studies on velocity profiles have focused on alluvial channels with gentle slopes. Due to the difficulty in integrating the influence of bed roughness and subsurface flow (Yager et al. 2012; Heimann et al. 2015; Schneider et al. 2015), research on velocity profiles in mountainous river channels with steep slopes and rough bed surfaces remains limited. The present situation results in considerable uncertainty in its application, e.g., sediment transport (Ferguson 2007; Rickenmann & Recking 2011; Wu et al. 2023) microorganism transport (Li et al. 2023b) and contaminant transport (Jiang et al. 2022). When applying the conventional formulae of flow velocity profiles derived from fluvial river channels to estimate the sediment transport rates in mountainous river channels, the calculated results may differ by one to two orders of magnitude from actual measurements (Bathurst 2002; Barry et al. 2004; Yager et al. 2012). Owing to the complex geometric structures and shallow depths, velocity profiles in mountainous river channels exhibit non-conventional patterns (Wiberg & Smith 1991; Ferro & Baiamonte 1994; Ferro 2003; Wang et al. 2015). The streambeds of mountainous river channels are often characterized by gravels, pebbles, even boulders, and sometimes sediment aggregates and bedform morphology (Yager et al. 2007; Schneider et al. 2015), which may bring considerable subsurface flow and water surface waves (Finley et al. 1966; Ferro & Baiamonte 1994; Wang et al. 2015), subsequently influences velocity profiles.
In steep channels with gravel beds with relatively shallow flow depths, flow velocities near the bed remain relatively high due to significant subsurface flow, causing deviations from the standard logarithmic velocity profiles (Lamb et al. 2017b; Luo et al. 2020, 2022). Although sometimes velocity profiles on gravel-bed river channels can still be approximated by logarithmic formulas, there is no satisfactory method for determining parameters for different bed surfaces and slopes (Gupta & Paudyal 1985; Nakagawa et al. 1988; Tu et al. 1988). Some scholars have suggested to divide the vertical velocity structure into different regions, but this method is challenging due to the difficulty in determining the thickness of the bottom rough layer (Zagni & Smith 1976; Vedula & Achanta 1985). Lamb et al. (2017a, 2017b) considered variations in bed permeability and roughness across the entire flow depth and developed a one-dimensional velocity model coupling surface and subsurface flows for steep streams with planar rough beds. Luo et al. (2022) considered porosity parameters in his model and achieved numerical solutions for subsurface flow equations.
Field surveys in mountainous river channels indicate that 90% of velocity profile structures deviate from the logarithmic distribution (Byrd et al. 2000), including ‘S’-shaped and Dean-Finley profiles (Ferro & Baiamonte 1994; Ferro & Pecoraro 2000). Through scale analysis of momentum and kinetic energy equations, Byrd et al. (2000) found that the ‘S’-shaped profiles are mainly due to form drag associated with coarse bed roughness. Similar non-conventional velocity profiles can be observed in stream channels with submerged vegetation and large bedform morphologies (Nepf & Ghisalberti 2008; Hu et al. 2013; Wang et al. 2023; Behera et al. 2024). The above studies indicate that the logarithmic velocity distribution is related to small-scale roughness (ks/h < 0.1, in which ks is the roughness height) while non-conventional velocity profiles in mountainous channels are related to large-scale roughness (ks/h > 0.2) (Lamb et al. 2017b). Gravel-bed surfaces, similar to turbulent structures in rigid vegetation (Nepf & Ghisalberti 2008), affect the entire water body, and may result in the shrink of boundary layers (Roy et al. 2004; Lu et al. 2021). Nikora et al. (2007) used Reynolds-averaged Navier–Stokes (RANS) equations to obtain morphological stresses similar to Reynolds stresses, finding that these morphological stresses and turbulence kinetic energy may be responsible for non-conventional velocity profiles. Bathurst (1988) proposed conditions for forming ‘S’-shaped velocity profiles: a slope steepness (S) of approximately 0.01, a ratio of water depth (h) to bed surface particle size (D84, representing particles larger than 84% of the bed) between 1.0 and 4.0, ensuring non-uniform distribution of bed particles. Ferro & Baiamonte (1994) and Ferro (2003) proposed a four-parameter equation (a combination of logarithmic and polynomial equations) to fit ‘S’-shaped profiles. The calculation results of existing formulae for flow resistance in mountainous river channels often very widely (Bathurst 2002). This is partially due to variations in location and significant changes in geometric morphology. Lamb et al. (2017b) conducted experiments in steep-slope, flat-bed channels and derived a vertical velocity profile formula coupled with surface and subsurface flow. The derived relationship of resistance coefficients closely matched field observations, indicating the significant influence of subsurface flow (Emmett 1970; Lamb et al. 2017b; Luo et al. 2022) on flow resistance in mountainous river channels. Similarly, it has been found that surface wave fluctuations also affect the vertical velocity structure in steep-slope shallow water (Emmett 1970; Abrahams & Parsons 1994). Nevertheless, there is limited research examining the impact of this flow on resistance coefficients. Furthermore, it is challenging to accurately measure the relationship between water flow and sediment transport on hillslopes (Qu et al. 2023).
The acoustic Doppler velocity flowmetry (ADV) has been widely used for velocity structure measurements. However, ADV is no longer suitable for velocity measurements for steep bed slopes and rough surfaces (Lamb et al. 2017b). Due to the presence of bubbles, measuring velocities in highly aerated flows is challenging, and the presence of a large amount of air in the water alters energy dissipation and flow transport. This study adopted a conductivity phase-detection technique, which sets no-slip velocity between the water flow and bubbles and be able to describe the flow characteristics of highly aerated flows (Wang & Chanson 2019; Wang et al. 2021; Bai et al. 2022).
This study initially establishes five flow scenarios with a slope of S = 0.01 to examine the velocity distributions on the gravel-bed. Subsequently, experiments were conducted at a slope of S = 0.16, where the ratio of h/D84 still exceeds 1, to investigate the variations in velocity and turbulent kinetic energy distributions compared to those observed on planar smooth bed surfaces. The objectives of this study are:
1. To propose a hybrid model for predicting flow velocity profiles in mountainous channels by coupling surface and subsurface flows.
2. To calculate flow resistance based on the new formula and discuss its implication on sediment transport in mountainous river channels.
METHODS
Experimental setups
Exp. # . | Bed slope, S . | Discharge Q (m3/h) . | Flow Reynolds number, Re (×104) . | Roughness Reynolds Number, (×104) . | Froude number, Fr . | Flow depth, h (m) . | Shields number, . |
---|---|---|---|---|---|---|---|
1 | 6‰ | 196 | 18.9 | 2.05 | 0.37 | 0.28 | 0.02 |
2 | 8‰ | 184 | 17.3 | 2.36 | 0.34 | 0.27 | 0.03 |
3 | 10‰ | 198 | 18.9 | 2.52 | 0.45 | 0.25 | 0.04 |
4 | 12‰ | 195 | 18.6 | 2.84 | 0.37 | 0.28 | 0.04 |
5 | 160‰ | 45 | 4.2 | 3.15 | 1.8 | 0.025 | 0.05 |
6 | 160‰ | 55 | 5.1 | 3.62 | 4.1 | 0.035 | 0.07 |
7 | 160‰ | 65 | 6.0 | 4.25 | 4.9 | 0.045 | 0.1 |
Exp. # . | Bed slope, S . | Discharge Q (m3/h) . | Flow Reynolds number, Re (×104) . | Roughness Reynolds Number, (×104) . | Froude number, Fr . | Flow depth, h (m) . | Shields number, . |
---|---|---|---|---|---|---|---|
1 | 6‰ | 196 | 18.9 | 2.05 | 0.37 | 0.28 | 0.02 |
2 | 8‰ | 184 | 17.3 | 2.36 | 0.34 | 0.27 | 0.03 |
3 | 10‰ | 198 | 18.9 | 2.52 | 0.45 | 0.25 | 0.04 |
4 | 12‰ | 195 | 18.6 | 2.84 | 0.37 | 0.28 | 0.04 |
5 | 160‰ | 45 | 4.2 | 3.15 | 1.8 | 0.025 | 0.05 |
6 | 160‰ | 55 | 5.1 | 3.62 | 4.1 | 0.035 | 0.07 |
7 | 160‰ | 65 | 6.0 | 4.25 | 4.9 | 0.045 | 0.1 |
Five experiments were performed using the same bed, with varying channel-bed slopes (0.006 < S < 0.16). All flows were fully turbulent ( > 103). The Froude numbers () are in a range of 0.34–4.9, i.e., both subcritical and supercritical are considered. Shields numbers (, where R = 1.65 is the submerged specific density of sediment) in our experiments are in a range of 0.02–0.1. The experiment scenarios are summarized in Table 1.
Velocity measurement
A downward-looking ADV was employed to measure the vertical velocity distribution along the centerline of the flume's experimental section (Figure 1(c)). Data from the ADV were collected in four experiment sets, representing conditions with gentle slopes. However, at a bed slope of S = 0.16, the flow became excessively shallow, fast, and aerated, hindering accurate ADV measurements. Consequently, a dual-tip conductivity phase-detection probe (Bai et al. 2022) was utilized for this condition, a tool commonly used in high-energy, shallow, and air mixed streams (Figure 1(d)). The probe, manufactured by The University of Queensland, consisted of two side-by-side needle-shaped sensors of different lengths at the probe's forefront. The phase-detection probe is consistently positioned on the centerline of the flume. Probe displacement in the normal direction is monitored by a mechanical vernier caliper mounted on the probe bracket, with an approximate accuracy of 0.2 mm.
Spatial averaging of velocity profiles was not performed in this study, which will inevitably introduce shape stresses. However, as the ADV-measured velocity profiles were obtained at a uniform position in the flume and the phase-detection probes measured velocity profiles along the flume's centerline (Figure 1(b)), comparisons between these profiles were made to mitigate variations in hydraulic and bed roughness. The ADV was mounted on a vertical screw rod (Figure 1(c)), allowing measurements perpendicular to the flume, with a vertical accuracy of 2 mm. The top of the bed surface (z = 0) was defined as a location 45 mm above the flume bottom of the observation section, with the thickness of the subsurface flow (η) set to 45 mm (Figure 3). Since the cobbles are tightly bound to the bottom of the flume, the exchange layer height P was set to 23 mm. Within this layer, the subsurface flow is strongly influenced by shear from surface layer and turbulence (Lamb et al. 2017b; Rousseau & Ancey 2020). Flow between z = 0 and -P was not measured due to the limitations of the phase-detection probe reaching this area. Rulers were placed along the flume to record differences in height between the water surface and the terrain, used to calculate water depth. The average water depth of the observation section was determined by averaging measurements from the inlet, middle, and outlet sections. Experimental water depths ranged from 0.025 < h < 0.28 m. An electromagnetic flowmetry installed in the inlet pipe of the upstream flume measured flow. Average velocity (U) was calculated using the continuity principle, with the formula , where W = 0.3 m is the flume width and P is the thickness of exchange layer. A honeycomb energy dissipator was placed in the upstream pool to regulate the flow, and a tail valve controlled the outlet opening and the downstream water level, ensuring uniform flow in the observation section.
In Figure 2(a), the gray curves represent ADV-measured data filtered based on the correlation coefficients and signal-to-noise ratio for the three axes, while the black curves depict data with anomalies removed and median-filtered using the Hampel method. All ADV data in our experiment went through this same process to ensure uniform data quality. Additionally, by comparing the velocity calculated via vertical integration of time-averaged velocities from all measurement points above the bed and the average velocity calculated from the electromagnetic flowmeter in the upstream pool pipeline we see consistency. The distribution of depth-averaged velocities obtained from the ADV verticals and those derived from flow rate calculation equations can be seen in Figure 2(b). The comparison shows that the two methods align well.
THEORETICAL DERIVATION
Improved hybrid mixing length model
Model performance evaluation
RESULTS
Surface flow velocity profiles
When the slope is gentle, the coefficient b1 of the divergence function in all the flow velocity profiles in Figure 4(a)–4(d) is not zero. Conversely, b1 equals zero for the downstream flow velocity profiles under conditions where the slope is 0.16. Table 2 records the values of α1, b1 for all fitted curves in Figure 4.
Exp # . | Bed slope, S . | α1 . | β1 . | ks/h . | NSE . |
---|---|---|---|---|---|
1 | 6‰ | 0.25 | 0.3 | 0.56 | 0.82 |
2 | 8‰ | 0.8 | 0.15 | 0.58 | 0.90 |
3 | 10‰ | 0.23 | 1.0 | 0.63 | 0.71 |
4 | 12‰ | 0.7 | 0.4 | 0.56 | 0.34a |
5 | 160‰ | 0.25 | 0 | 6.3 | 0.02 |
6 | 160‰ | 0.18 | 0 | 4.5 | 0.11 |
7 | 160‰ | 0.25 | 0 | 3.5 | 0.03 |
Exp # . | Bed slope, S . | α1 . | β1 . | ks/h . | NSE . |
---|---|---|---|---|---|
1 | 6‰ | 0.25 | 0.3 | 0.56 | 0.82 |
2 | 8‰ | 0.8 | 0.15 | 0.58 | 0.90 |
3 | 10‰ | 0.23 | 1.0 | 0.63 | 0.71 |
4 | 12‰ | 0.7 | 0.4 | 0.56 | 0.34a |
5 | 160‰ | 0.25 | 0 | 6.3 | 0.02 |
6 | 160‰ | 0.18 | 0 | 4.5 | 0.11 |
7 | 160‰ | 0.25 | 0 | 3.5 | 0.03 |
aα1 is larger than 1.
In general, the velocity predicted by the equation that considers near-bed flow closely corresponds with the measured velocity, particularly apparent for experiments where the near-bed flow incurs when the bed slope is larger. Evident in Fi(g)ures 4(a)-(g) under the experimental conditions of changing slope and relative roughness, all velocity profiles essentially comply with the predictions granted by the equation, factoring in the near-bed flow and the near-surface flow dip. The larger the decrease in near-surface velocity, the larger the value of b1.
The mixed length sheds light on the departure of the logarithmic linear region velocity profile near the bed from the logarithmic velocity profile. The incorporation of near-bed flow velocity accounts for the deviation induced by higher subsurface flow velocity, whereas the divergence function elucidates the departure of reduced flow velocity higher from the roughness height ks. In Figure 4(d), at 0.4 to 0.8ks the predicted velocity is higher than the measured velocity. When α1 is greater than 1, we can fit the vertical distribution well, but its influencing factors cannot be explained.
Turbulence intensities
The change in downstream turbulence intensity, as depicted in Figure 5(b), 5(e) and 5(f), reveals a trend of increased intensity from near-bed to 0.2 h, regardless of the considerable variation in slope and flow rate. This may be attributed to the rough particles near the bed impeding water turbulence. For a water depth exceeding 0.2 h, a slight decrease adjacent to the water surface is noticed only in conditions with S = 0.006–0.012, whereas with S = 0.16, no significant decrease is observed near z/h = 1 in Figure 5(e)–5(g), indicating the influence of rough bed elements extends across the entire water depth. The cross-flow turbulence intensity of the flume decreases with increasing water depth, with a maximum value near the bed of approximately 0.6, showing no strong correlation with channel slope. The vertical turbulence intensity in the flume is generally lower than the turbulence intensity of 1 under smooth bed conditions. This suggests that bed rough elements and surface wave action may have a significant impact on the vertical turbulence intensity.
Depth-averaged flow and friction factor
DISCUSSION
Flow velocity profiles
This study proposes a velocity profile equation that considers the interaction between the roughness layer, near-bed flow velocity, and the dip near the surface. This solution better aligns with our observations compared to previous models presented by Lamb et al. (2017b) and Luo et al. (2022). The mixed length model allows deviations from the logarithmic profile within the roughness layer and effectively models the reduction in flow velocity experienced in channels, slopes, and other subsurface flow. While many studies suggest that the decrease in velocity near the flow surface is due to surface wave (Emmett 1970), our experiments show that when the slope S = 0.16, the effect of surface wave was more pronounced than in cases where S = 0.006–0.012, yet the velocity near the flow surface did not decrease significantly.
This discrepancy may be attributed to the contrast in pore and permeability between hillslope and mountain river flow, with the transition from surface water flow to subsurface being smoother in our experimental setup. In our steep-slope experiments, the subsurface velocity was much larger than in the hillslope flow experiments, preventing a reduction in near-surface velocity under shallow flow conditions on steep slopes. As the slope changes, our experiment visually depicts a transition from a relatively steady and uniform water surface to aerated supercritical flow on steep slopes. Despite variations in bed slope or flow conditions, the vertical velocity distribution above the bed can be predicted using a mixed length model (b1 = 0) below the maximum velocity from verticals, considering the mixed length and near-bed flow velocity. While this result suggests that the velocity within the roughness layer is insensitive to relative roughness but responds sensitively to changes in subsurface flow velocity, it supports the findings of Lamb et al. (2017b), but contradicts the assumption presented by Recking (2009).
Although our model, incorporating a divergence function, fits well with the velocity dip near the flow surface, a decrease in velocity was not observed under the steep-slope conditions in our experiment. This implies that the divergence function may not significantly correlate with surface wave.
Implications for sediment transport
The flow resistance of mountainous river channels is influenced by factors such as form drag, grain roughness, and bed structure. There may be significant differences in the sediment transport relationship derived from relative roughness between mountainous river channels and hillslope flow. While most studies focus on the relationship between relative roughness and flow resistance, the presence of bed surface structures in plain rivers may also lead to a reduction in the sediment transport relationship, with little change in relative roughness (Zhang et al. 2021).
Bed surface structures can have a significant impact on sediment transport, especially in cases involving the formation of aggregates or large boulders (Li et al. 2023a). However, there is insufficient evidence to support the claim that bed surface structures are the primary cause of increased flow resistance in mountainous river channels (Lamb et al. 2017b). Emmett (1970) conducted experiments in rivers with gradients exceeding 17‰ and found that the water flow turbulence was significant. Ferro & Pecoraro (2000) found that the flow resistance, derived using the incomplete self-similarity method, is not only related to the relative roughness, but also to the Re.
Additionally, based on our experimental results and referring to Lamb et al. (2008), the formula derived shows that near-bed turbulence intensity varies linearly with mean flow velocity (Figure 5(b)). In our study, the decrease in measured flow resistance is mainly due to the reduction in turbulence intensity compared to that in fluvial rivers, leading to a decrease in mean flow velocity and an increase in the flow resistance, stabilizing sediment transport on the slope.
CONCLUSIONS
The flow velocity profiles on an experimental gravel-bed under different slopes were measured using ADV and a phase-detection probe. Analysis was conducted within a slope range of 0.006–0.16, and flow discharges ranging from 0.01 to 0.06 m3/s. The variations in turbulent intensity and flow resistance parameters were examined. The following conclusions were drawn:
1. The newly developed formulae for flow velocity profiles, considering factors such as near-bed velocity, mixed length under roughness scale, and velocity dip near the water surface, matches well with the measured data.
2. Relative large-scale roughness leads to deviations in the peak positions of turbulent intensity and Reynolds stress from those of a smooth bed surface. Turbulent intensity downstream tends to distribute uniformly across the entire water depth.
3. The flow resistance formula obtained through integration, similar to the ones derived from field measurements in mountainous river channels, aligns well with the measured flow resistance. This suggests that laboratory-scale water flow resistance experiments effectively reflect macro-scale resistance variations. Furthermore, the mixed length formula may overestimate the flow resistance in mountainous river channels due to surface wave fluctuations or secondary flow.
In conclusion, this study highlights the significance of subsurface flow in the velocity profile of mountainous river channels. The limitations of the present study primarily arise from two aspects. Firstly, we did not consider the influences of bed cobble arrangements in our experiment, which brings bed micro-structures and variations in porosity, thereby affecting the parameter α1 in Equation (7) according to Luo et al. (2022). Secondly, our experiment lacks measurements of subsurface flow velocities, preventing the establishment of a direct relationship between velocity profiles and subsurface pore flow. These aspects need to be considered in future research.
ACKNOWLEDGEMENTS
This study was supported by the National Key Research and Development Program of China (2022YFC3203903) and the National Natural Science Foundation of China (No. 52279077). The authors would like to thank Dr Xiaojuan Deng for her help on the figures.
DATA AVAILABILITY STATEMENT
Data cannot be made publicly available; readers should contact the corresponding author for details.
CONFLICT OF INTEREST
The authors declare there is no conflict.